MATH0060
Lecture Notes
Carlo Marinelli
This version: 9.III.2021

Contents
Preface 5
Chapter 1. Preliminaries 7
Chapter 2. A market model on a finite probability space 9
2.1. Introduction 9
2.2. Price processes and trading strategies 9
2.3. Contingent claims, completeness and arbitrage 11
2.4. Pricing by no-arbitrage I 15
2.5. Pricing by no-arbitrage II 17
2.6. Single period models 19
2.7. Notes 20
2.8. Exercises 20
Chapter 3. The binomial model 23
3.1. The one-period model 23
3.2. The multiperiod model 24
3.3. Continuous-time limit 26
3.4. The formula of Black and Scholes 28
3.5. Complements 30
3.6. Notes 31
3.7. Exercises 31
Chapter 4. Finance in continuous time 33
4.1. Notions of no-arbitrage and the fundamental theorem of asset pricing 33
4.2. Superreplication 35
4.3. Attainable claims 37
4.4. Brownian market model 37
4.5. Exercises 39
Chapter 5. Portfolio optimization 43
5.1. Utility functions 43
5.2. Maximization of the expected utility of final wealth 43
5.3. Optimization and no-arbitrage 44
5.4. A closer look 45
5.5. The case of incomplete markets 47
5.6. Notes 48
5.7. Exercises 48
Bibliography 49
3

Preface
These lecture notes are preliminary and will be corrected and/or expanded when needed. They are
meant to serve as a summary of the topics covered in class, and should not be redistributed. Please
report any error or misprint to the author.
5

CHAPTER 1
Preliminaries
Let (Ω,F ,P) be a probability space endowed with a filtration F = (Fn)n∈N, where Fn is a sub-σ-algebra
of F for every n ∈ N. Given a measurable space (E, E), an E-valued stochastic process X is a family of
E-valued random variables (Xn)n∈N. In other words, Xn : Ω→ E is an (F , E)-measurable function for
every n ∈ N. The following notions of measurability of discrete-time stochastic processes are standard.
1.0.1. Definition. A stochastic process X with values on a measurable space E is called F-adapted if
Xn is Fn-measurable for every n ∈ N. The process X is called F-predictable if Xn is Fn−1-measurable
for all n > 1.
Let us introduce some notation. For any stochastic process Y with values in a vector space, let us
set
∆Yt := Yt − Yt−1, t > 1,
Let E be a vector space endowed with a scalar product 〈·, ·〉. If X and Y are E-valued processes, the
real stochastic process X · Y defined by
(X · Y )t :=
t∑
u=1
〈
Xu,∆Yu
〉
=
t∑
u=1
〈
Xu, (Yu − Yu−1)
〉
, t > 1,
is called the (discrete-time) stochastic integral of X with respect to Y .
In order to simplify notation, note that any element f of E can be identified with the linear form
〈f, ·〉, so that we can write, with a minor abuse of notation, 〈f, g〉 = fg for any g ∈ E. We shall use
this notational convention whenever confusion cannot arise.
The simple identity
∆(XY )t = Xt∆Yt +∆XtYt−1, t > 1,
is quite useful. With the convention that X− denotes the process (Xn−1)n>1, this can be written as
∆(XY ) = X∆Y + Y−∆X.
This identity implies
(XY )t = X0Y0 +
t∑
u=1
∆(XY )u
= X0Y0 +
t∑
u=1
Xu∆Yu +
t∑
u=1
Yu−1∆Xu
= X0Y0 + (X · Y )t + (Y− ·X)t,
which can also be written in the equivalent more symmetric form
XY = X0Y0 +X− · Y + Y− ·X + [X,Y ],
where the process [X,Y ] is defined by
[X,Y ]t :=
t∑
u=1
∆Xu∆Yu, t > 1.
7

CHAPTER 2
A market model on a finite probability space
»Kann man aus Nichts Etwas machen?« fragte er
lauernd, worauf ich mißtrauisch den Kopf schüttelte.
Er nickte traurig.
Friedrich Dürrenmatt, Das Bild des Sisyphos.
2.1. Introduction
In this chapter we deal with a discrete-time market model with finite horizon T ∈ N. In particular, the
market can be described as follows: at each time t = 1, . . . , T agents (market participants) can trade
(buy and sell) 1 + d financial assets. One may think, for instance, to a market where is a risk-free cash
account and d risky assets, such as stocks, are traded. More generality, however, is allowed, as will be
soon clear.
The uncertainty underlying the market will be modeled by a finite probability space (Ω,F ,P),
where Ω = {ω1, ω2, . . . , ωN}, N ∈ N, F is the power set of Ω,, and P has full support, i.e. P({ωn}) > 0
for all n ∈ {1, . . . , N}. The flow of information available to all agents will be modeled by a filtration
F = (Ft)t=0,1,...,T such that FT ⊆ F . The informal economic interpretation of this setup is that
Ft contains all information available to market participants up to and including time t, and that no
information gets lost with time, i.e. all information contained in Ft−1 is also contained in Ft.
2.2. Price processes and trading strategies
We consider a financial market where agents can trade 1 + d assets at times t ∈ {1, . . . , T − 1}. The
prices of the tradeable assets are assumed to be exogenous, i.e. not influenced by trading. We shall
denote the price at time t ∈ {0, . . . , T} of asset k ∈ {0, 1, . . . , d} by Sˆkt .
2.2.1. Assumption. The stochastic process Sˆ : Ω× {0, 1, . . . , T} → R1+d defined by
Sˆ = (Sˆt)t∈{0,1,...,T} = (Sˆ0t , Sˆ
1
t , . . . , Sˆ
d
t )t∈{0,1,...,T},
is adapted, with Sˆ0t > 0 for all t and Sˆ
0
0 = 1.
The 0-th asset is assumed to have a strictly positive price at all times because it will be used as
a numéraire. Although it is common to choose as numéraire the risk-free cash account, there is no
compelling reason to do so: for the developments to come, the strict positivity of its price process is
enough.
Concerning trading by market participants, it is natural to assume that at time t their investment
decisions can only be based on the information available prior to time t: between time t− 1 and time t
an agent holds Hˆkt units of the k-th asset, and the decision is taken at time t− 1. Let us formalize this
observation.
2.2.2. Definition. A trading (or portfolio) strategy Hˆ is an R1+d-valued predictable process
Hˆ = (Hˆt)t∈{1,...,T} = (Hˆ0t , Hˆ
1
t , . . . , Hˆ
d
t )t∈{1,...,T}.
The portfolio value Vˆ associated to the trading strategy Hˆ is the real-valued process
Vˆt := 〈Hˆt, Sˆt〉 :=
d∑
k=0
Hˆkt Sˆ
k
t , t = 1, . . . , T.
9
10 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
We shall use only trading strategies of the type defined next.
2.2.3. Definition. A strategy Hˆ is said to be self-financing if it satisfies
〈Hˆt, Sˆt〉 = 〈Hˆt+1, Sˆt〉 ∀t = 1, . . . , T − 1. (2.2.1)
Note that (2.2.1) is equivalent to
d∑
k=0
Hˆkt Sˆ
k
t =
d∑
k=0
Hˆkt+1Sˆ
k
t .
The meaning of (2.2.1) is that by changing the porfolio from Hˆt to Hˆt+1 there is neither input not
outflow of funds, i.e. the investor only “rearranges” his holdings without adding or subtracting funds
to the investment. Note also that the initial investment required for a self-financing strategy Hˆ can be
expressed, thanks to (2.2.1) and to the assumption Sˆ00 = 1, as
Vˆ0 = 〈Hˆ1, Sˆ0〉 = Hˆ01 +
d∑
k=1
Hˆk1 Sˆ
k
0 .
2.2.4. Exercise. Show that the set of self-financing trading strategies is a vector space.
2.2.5. Proposition. Let Hˆ be a trading strategy. The following assertions are equivalent:
(a) Hˆ is self-financing;
(b) 〈∆Hˆt+1, Sˆt〉 = 0 for all t ∈ {1, . . . , T − 1};
(c) Vˆt = Vˆ0 + (Hˆ · Sˆ)t for all t ∈ {1, . . . , T}.
Proof. The equivalence of (a) and (b) is just a reformulation of the definition of self-financing strategy.
Let us show that (b) and (c) are equivalent: the integration by parts formula yields Vˆ = Vˆ0+Hˆ ·Sˆ+Sˆ−·Hˆ,
hence (c) is equivalent to (Sˆ− · Hˆ)t = 0 for every t > 1, where
(Sˆ− · Hˆ)t =
t∑
u=1
〈Sˆu−1,∆Hˆu〉, t > 1,
from which it immediately follows that (b) implies (c). Conversely, (Sˆ− · Hˆ)t = 0 for every t > 1 implies
that
〈∆Hˆt+1, Sˆt〉 = (Sˆ− · Hˆ)t+1 − (Sˆ− · Hˆ)t = 0.
We are now going to discount the price processes Sˆ1, . . . , Sˆd in terms of the numéraire, i.e. we
introduce the adapted Rd-valued stochastic process S defined by
St = (S
1
t , . . . , S
d
t ) :=
( Sˆ1t
Sˆ0t
, · · · , Sˆ
d
t
Sˆ0t
)
, t = 0, . . . , T,
to which we shall refer as the process of discounted prices. Analogously, the discounted portfolio value
process V is defined as
Vt :=
Vˆt
Sˆ0t
, t = 0, . . . , T.
Note that V0 = Vˆ0 by the convention that Sˆ
0
0 = 1. Moreover, the integration by parts formula yields
V = V0 + Hˆ · Sˆ
Sˆ0
+
Sˆ−
Sˆ0−
· Hˆ,
where ( Sˆ−
Sˆ0−
· Hˆ
)
t
=
t∑
u=1
1
Sˆ0u−1
〈∆Hˆu, Sˆu−1〉 = 0
and, by the obvious identity ∆(Sˆ0/Sˆ0) = 0,
Hˆ · Sˆ
Sˆ0
= H · S,
2.3. CONTINGENT CLAIMS, COMPLETENESS AND ARBITRAGE 11
where H = (Hˆ1, . . . , Hˆd) is the Rd-valued predictable process of portfolio holding in the assets 1 to d.
We thus have the representation of the discounted portfolio value process V as
V = V0 +H · S.
Unsurprisingly, the discounted portfolio value does not depend on Hˆ0, simply because the price of the
0-th asset remains constant in discounted terms (that is, in terms of itself!). The random variable
(H · S)t can be interpreted as the discounted cumulative gain (or loss) up to time t corresponding to
the trading strategy Hˆ.
Let H and SF denote the vector space of Rd-valued predictable processes and the vector space of
self-financing trading strategies, respectively.
2.2.6. Lemma. The vector space of self-financing trading strategies SF is isomorphic to the vector space
H× L0(Ω,F0.P).
Proof. Recall that the initial value of a portfolio strategy Hˆ is given by
V0 = Vˆ0 = 〈Hˆ1, Sˆ0〉 = Hˆ01 +
d∑
k=1
Hˆk1 Sˆ
k
0 = Hˆ
0
1 +
d∑
k=1
Hk1S
k
0
and note that the map
SF −→ L0(Ω,F0,P)
Hˆ 7−→ Hˆ01 +
d∑
k=1
Hˆk1 Sˆ
k
0 = Hˆ
0
1 + 〈H1, S0〉
is a homomorphism. Since the map Hˆ 7→ H is clearly a homomorphism from SF to H, it follows that
the map
Φ: SF −→ H× L0(Ω,F0,P)
Hˆ 7−→ (H, Hˆ01 + 〈H1, S0〉)
is a homomorphism of vector spaces, which is easily seen to be injective. Let us show that it is also
surjective: let H ∈ H and a ∈ L0(Ω,F0,P), and set Hˆk := Hk for all k = 1, . . . , d, Hˆ01 := a− 〈H1, S0〉.
Moreover, let Hˆ0t , t > 2, be defined through the self-financing condition
Hˆ0t Sˆ
0
t +
d∑
k=1
Hˆkt Sˆ
k
t = Hˆ
0
t+1Sˆ
0
t +
d∑
k=1
Hˆkt+1Sˆ
k
t .
We have thus shown that Φ is an isomorphism of vector spaces.
The lemma implies that for any predictable process H modeling the portfolio holdings in the
securities 1 to d and an initial endowment a, there exists a unique self-financing trading strategy with
initial value equal to a. In other words, given H and a as said, H can be complemented in a unique
way by a real predictable process Hˆ0 modeling the holding in the 0-th security such that (Hˆ0, H) is a
self-financing trading strategy with initial value equal to a.
Thanks to the isomorphism established by the lemma, from now on we shall work, whenever it is
convenient, in discounted terms, i.e. representing discounted portfolio value processes of self-financing
strategies with initial value equal to a by processes of the type a+H · S.
2.3. Contingent claims, completeness and arbitrage
One of our main interests is the pricing of so-called financial derivatives (contingent claims). These are
contracts that guarantee a flow of payments (Xˆt)
T
t=1 which depend on the price evolution of the basic
underlying assets, e.g. Xˆt = φ(Sˆ0, . . . , Sˆt) for a given function φ. For the time being, we are going to
content ourselves with contingent claims entitling to a single payoff at time T , so that Xˆt = 0 for all
t < T , Xˆ := XˆT .
2.3.1. Definition. A contingent claim is a random variable on the probability space (Ω,F ,P).
12 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
A central question is to determine what a fair price for a contingent claim would be. Let us start
with the following observation: suppose that we can find an initial investment (or endowment) V0 and
a self-financing strategy such that X(ω) = V0 + (H · S)T (ω) for all ω ∈ Ω, where X := Xˆ/Sˆ0T . It is
then clear that there is no difference between holding the contingent claim or holding the self-financing
portfolio. In fact, independently of the evolution of asset prices, at time T the value of the contingent
claim and the value of the portfolio associated to H will be identical. Thus V0 is a very good candidate
to be considered as fair price of X (or equivalently of Xˆ). Let us formalize this reasoning.
2.3.2. Definition. A contingent claim X is called attainable if there exist V0 ∈ R and an Rd-valued
predictable process H = (Ht)
T
t=1 such that
V0 + (H · S)T = X.
In this case we shall say that V0 is a fair price of X. Moreover, a market is called complete if every
contingent claim is attainable.
The above reasoning does not guarantee that the fair price of an attainable claim is unique. In
order to make sure that this is actually the case (a quite natural requirement indeed), we have first to
introduce another very important concept, which will keep us busy for a while.
2.3.3. Definition. We say that arbitrage possibilities exist if there exists a predictable process H with
values in Rd such that (H · S)T (ω) > 0 for all ω ∈ Ω, and (H · S)T (ω) > 0 for at least one ω ∈ Ω. A
market is free of arbitrage if it does not admit any arbitrage possibility.
In other words, we want to rule out the following situation: starting from zero initial capital, there
exists a trading strategy such that the resulting portfolio value at time T is non-negative and not
identically equal to zero. A market where such possibilities exist would clearly make no (economic)
sense. If a market does not admit any arbitrage possibility, then we shall say that it satisfies the (NA)
condition.
The first important consequence of the absence of arbitrage is a uniqueness principle for the price
of attainable contingent claims.
2.3.4. Proposition (Law of one price). If the market satisfies the (NA) condition, then every attainable
contingent claim admits a unique fair price.
Proof. We proceed by contradiction. Suppose that
X = V0 + (H · S)T = V¯0 + (H¯ · S)T ,
and assume that V0 < V¯0. Then (
(H − H¯) · S)
T
= V¯0 − V0 > 0,
i.e. investing according toH−H¯ gives rise to an arbitrage opportunity, which contradicts the hypotheses.
The case V0 > V¯0 is completely similar.
Let us also prove that H1 ·S = H2 ·S (identity of processes!): we have (H1 ·S)T = H2 ·S)T , define
A =
{
(H1 · S)t < (H2 · S)t
} ∈ Ft
and H := (H1 −H2)1A1]t,T ], which is a predictable process. Here t is the first t for which (H1 · S)t 6=
(H2 · S)t. Since ((H1 −H2) · S)T = 0 implies
0 = 1A((H
1 −H2) · S)T = 1A((H1 −H2) · S)t + (H · S)T ,
where, by definition of A, 1A((H
1 −H2) · S)t < 0, hence (H · S)T > 0, which contradicts the absence
of arbitrage.
We shall denote by K the subset of L0(Ω,FT ,P) defined as
K :=
{
(H · S)T : H ∈ H
}
,
i.e. K is the set of contingent claims that are attainable with zero initial capital.
2.3. CONTINGENT CLAIMS, COMPLETENESS AND ARBITRAGE 13
We shall also use the convex cones C0 ⊆ L0(Ω,FT ,P) and C ⊆ L∞(Ω,FT ,P) of super-replicalble
claims at price zero, defined as
C0 :=
{
g ∈ L0(Ω,FT ,P) : ∃f ∈ K, f > g
}
= K − L0+
and
C :=
{
g ∈ L∞(Ω,FT ,P) : ∃f ∈ K, f > g
}
= C0 ∩ L∞(Ω,FT ,P).
Note that C0 ⊇ C ⊇ L∞− .
We can now give a geometric definition of the no-arbitrage property.
2.3.5. Definition. A financial market S is free of arbitrage if
K ∩ L0+ = {0}. (2.3.1)
The following characterization of the no-arbitrage property is often useful.
2.3.6. Lemma. Condition (2.3.1) is equivalent to C ∩ L∞+ = {0}.
Proof. Assume that C ∩ L∞+ = {0}. If f ∈ K ∩ L0+, then f ∧ n converges to f pointwise as n → +∞
and f ∧ n ∈ C ∩ L∞+ , hence f ∧ n = 0 for every n ∈ N, thus also f = 0. The converse implication is
even more immediate.
2.3.7. Proposition. Assume that (NA) holds. Then C0 ∩ (−C0) = K. In particular, C ∩ (−C) =
K ∩ L∞.
Proof. The inclusion K ⊆ C0∩(−C0) is obvious: if g ∈ K then g ∈ C0, and, recalling that K is a vector
space, also −g ∈ K, hence g ∈ −C0. We have to prove the inclusion K ⊇ C0∩(−C0): if g ∈ C0∩(−C0),
then there exist f1 ∈ K and h1 ∈ L0+ such that g = f1 − h1, as well as f2 ∈ K and h2 ∈ L0+ such that
−g = f2 − h2. This implies K ∋ f1 + f2 = h1 + h2 ∈ L0+, hence, by the (NA) condition, h1 = h2 = 0,
which yields g = f1 ∈ K.
2.3.8. Proposition. Assume that (NA) holds. The set C of bounded contingent claims superreplicable
at price zero is closed in L∞(Ω,FT ,P).
Proof. We shall first prove that C0 = K + L
0
− is closed in L
0. Let (kn + hn)n∈N be a sequence in C0,
with kn ∈ K and hn ∈ L∞− for every n ∈ N, converging in L0. Identifying L0 with RN , let us assume,
by contradiction, that kn is unbounded, i.e. that ‖kn‖ → ∞ as n→∞. Setting
kˆn :=
kn
‖kn‖ ∈ K, hˆn :=
hn
‖kn‖ ∈ L
0
−, zˆn :=
kn + hn
‖kn‖ ,
one has zˆn → 0 as n→∞ and ‖kˆn‖ = 1 for every n ∈ N, which implies that (kˆn) admits a subsequence,
denoted by the same symbol, converging to kˆ of norm one. Therefore (hˆn) also admits a subsequence
converging to hˆ with kˆ + hˆ = 0. Since both K and L∞− are closed, one has kˆ ∈ K and hˆ ∈ L0−, hence
kˆ = −hˆ ∈ K ∩L0+ = {0}. This is a contradiction because ‖kˆ‖ = 1. Let us now show that C = C0 ∩L∞
is closed in L∞. If (gn) is a sequence in C converging to g ∈ L∞, then, a fortiori gn converges to g in
L0, and g ∈ C0 by the previous reasoning, hence g ∈ C.
2.3.9. Definition. A probability measure Q on (Ω,F) is called a martingale measure for S if S is a
martingale with respect to Q.
We shall denote the sets of martingale measures for S that are absolutely continuous with respect
to P by Ma, and the subset of Ma of measures that are equivalent to P by Me. Both sets are convex.
Since we have assumed that Ω is finite and P has full support (i.e. P(ω) > 0 for all ω ∈ Ω), then Q is
equivalent to P if and only if it has full support. In particular, in this setting one simply has
dQ
dP
(ω) =
Q(ω)
P(ω)
∀ω ∈ Ω.
14 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
It will sometimes be convenient to identify the probability measure Q with its Radon-Nikodym derivative
dQ/dP ∈ L1(Ω,F ,P). Recall that S is a martingale with respect to Q if one has St ∈ L1(Ω,Ft,Q) for
all t ∈ {0, . . . , T} and
EQ
[
St+1|Ft
]
= St ∀t = 0, 1, . . . , T − 1.
The following result is known as the Fundamental Theorem of Asset Pricing (FTAP).
2.3.10. Theorem. A model S satisfies the no-arbitrage condition if and only if there exists an equivalent
martingale measure for S.
For the proof of the FTAP we need results on discrete-time stochastic integrals and on separation
of convex sets.
2.3.11. Proposition. A stochastic process M on a filtered probability space (Ω,F ,F,P), adapted to F ,
is a P-martingale if and only if EP(H ·M)T = 0 for any predictable process H.
Proof. (a) If M is a martingale and H is predictable, then H ·M is a martingale by an application of
conditional expectation.
(b) Conversely, assume that EP(H ·M)T = 0 for any predictable processH. In particular, let k ∈ 1, . . . , T
be arbitrary, and take A ∈ Fk. Then the process H defined by
Ht(ω) =
{
0, t 6= k + 1,
1A(ω), t = k + 1
is predictable, because Hk+1 is Fk-measurable. In particular, by (a), one has
0 = EP(H ·M)T = EP
(
(Mk+1 −Mk)1A
)
.
Since A was arbitrary, this is equivalent to EP[Mk+1|Fk] =Mk. The process M is hence a martingale,
because k was also arbitrary.
The following corollary is not directly needed in the proof of Theorem 2.3.10, but it will be very
useful soon.
2.3.12. Corollary. Assume that S is free of arbitrage. Let Q be a probability measure on FT . The
following assertions are equivalent:
(i) Q is a martingale measure for S;
(ii) EQ f = 0 for all f ∈ K;
(iii) EQ g 6 0 for all g ∈ C.
Proof. The equivalence of (i) and (ii) is just the statement of the previous proposition. If g ∈ C, then
there exist f ∈ K such that f > g, hence EQ g 6 0 for every martingale measure Q, that is (ii) implies
(iii). The converse implication, the only one that needs the NA hypothesis, follows immediately by the
identity K = C ∩ (−C).
2.3.13. Theorem (Hahn-Banach, geometric version). Let E be a Banach space, A ⊂ E and B ⊂ E
non-empty convex disjoint subsets, with A compact and B closed. Then there exists a closed hyperplane
that strictly separates A and B.
For a proof see e.g. [2].
Proof of Theorem 2.3.10. Let us assume that Q is an equivalent martingale measure for S. We are
going to show that K ∩ L0+ = {0}. Let f ∈ K ∩ L0+. Then f = (H · S)T , with H predictable, hence
EQ f = 0 by Proposition 2.3.11. Moreover, since f > 0 and Q(ω) > 0 for all ω > 0, EQ f = 0 implies
f = 0.
Let us now prove that the converse implication is true. Let us introduce the space
P :=
{ N∑
k=1
µk1{ωk}, µk > 0,
∑
µk = 1
}
,
2.4. PRICING BY NO-ARBITRAGE I 15
that is, P is the convex hull of the so-called Arrow-Debreu securities. Note that P ⊂ L∞+ \ {0} and is
compact. Invoking the Hahn-Banach theorem, we can strictly separate the closed convex set K and the
compact convex set P . In particular, there exists a linear functional Q˜ ∈ L1 and real numbers α < β
such that
〈Q˜, f〉 6 α ∀f ∈ K and 〈Q˜, f〉 > β ∀f ∈ P.
Since K is a vector space, it can be easily seen that one can take α = 0, and that the first inequality
in the previous display is actually an equality. Moreover, since β > 0, one has η := 〈Q˜, e〉 > 0, where
e := (1, 1, . . . , 1), hence Q := Q˜/η is a probability measure. Since f ∈ K is arbitrary, Proposition 2.3.11
implies that Q is an equivalent martingale measure for S.
From now on we shall always assume that (NA) holds.
We are now going to prove the equivalence of single-period and multi-period no-arbitrage.
2.3.14. Theorem. Let S be a discounted price process. The following are equivalent:
(a) S satisfies the no-arbitrage property;
(b) for any t ∈ {1, . . . , T}, there does not exist η ∈ L0(Ω,Ft−1,P;Rd) such that 〈η,∆St〉 ∈ L0+ \ {0}.
Proof. (b) ⇒ (a): assume, by contradiction, that S admits arbitrage. In particular, there exists a
predictable process H such that, setting
t := min
{
s ∈ {1, . . . , T} : Vs := (H · S)s ∈ L+0 \ {0}
}
,
one has t 6 T . For such t, it must hold Vt−1 = 0 or P
(
Vt−1 < 0
)
> 0. In the former case one has
〈Ht,∆St〉 = Vt − Vt−1 = Vt ∈ L+0 \ {0},
therefore η := Ht contradicts (b). In the latter case, setting η := Ht1{Vt−1<0}, one has
〈η, St − St−1〉 = 〈Ht, St − St−1〉1{Vt−1<0}
= (Vt − Vt−1)1{Vt−1<0} > −Vt−11{Vt−1<0} > 0,
again contradicting (b).
(a) ⇒ (b): let us assume, by contradiction, that there exist such t and η. One can then define the
predictable process H by Hs := η if s = t, and Hs = 0 for all s 6= t, thus obtaining
(H · S)T = 〈η,∆St〉 ∈ L+0 \ {0},
which contradicts (a).
2.4. Pricing by no-arbitrage I
Given a random variable f and a number a, we are going to use the notation
Kf,a = span
(
K, f − a).
2.4.1. Definition. Let X ∈ L∞(Ω,F ,P) be a bounded contingent claim. A number pi is called an
arbitrage-free price of X if KX,pi satisfies the NA condition, i.e. if
KX,pi ∩ L∞+ = {0}.
The set of all arbitrage-free prices of X will be denoted by Π(X). The infimum and the supremum
of this set will be denoted by pi(X) and pi(X), respectively.
2.4.2. Remark. Arbitrage-free prices can also be introduced as follows: a contingent claimX ∈ L∞(Ω,F ,P)
has arbitrage-free price pi if there exists an adapted process Sd+1 such that Sd+10 = pi, S
d+1
T = X, and
(S, Sd+1) satisfies the NA property. In fact, simply setting Sd+1t := EQ[X|Ft], it is easy to show that the
extended market (S, Sd+1) satisfies the above conditions. Conversely, if (S, Sd+1) has the NA property,
the FTAP yields an equivalent measure Q such that Sk, k = 1, . . . , d, d + 1 are Q-martingale, hence
Q ∈ Q and EQX = EQ Sd+1T = pi.
We now establish a description of the set Π(X) in terms of the set Q of equivalent martingale
measures.
16 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
2.4.3. Theorem. Let X ∈ L∞(Ω,F ,P). One has
Π(X) =
{
EQX : Q ∈ Q
}
.
In particular, Π(X) is a bounded interval.
Proof. Let a = EQX, with Q ∈ Q. Then EQ(X − a) = 0, hence KX,a ∩ L∞+ = {0}. In fact, f ∈ KX,a
means that there exist α, β ∈ R and a predictable process H such that f = α(H ·S)T +β(X−a), which
in turn implies EQ f = 0. If, moreover, f ∈ L∞+ , then it must be f = 0. We have thus proved that{
EQX : Q ∈ Q
} ⊆ Π(X). Let us now show that the reverse inclusion also holds: let pi be such that
KX,pi∩L∞+ = {0}. By the Hahn-Banach argument used in the proof of FTAP, one obtains a probability
measure Q ≈ P such that EQ g = 0 for all g ∈ KX,pi ⊃ K. In particular, EQ g = 0 for all g ∈ K, hence
Q is a martingale measure, and EQ(X − pi) = 0, i.e. pi = EQX.
Recall that, if X is attainable, then Π(X) reduces to a singleton, i.e. EQX is independent of Q ∈ Q.
The next theorem implies, among other things, that this property characterizes attainable claims, i.e.
X ∈ K if and only if EQX = 0 for all Q ∈ Q.
2.4.4. Theorem. Let X ∈ L∞(Ω,F ,P). The following “dichotomy” holds:
(a) if X is attainable, then Π(X) is a singleton;
(b) if X is not attainable, then Π(X) is an open interval.
Proof. It is enough to prove (b). We are going to show that, for any pi ∈ Π(X), there exist pi0 and
pi1 ∈ Π(X) such that pi0 < pi < pi1. Let pi = EQX and set Xt := EQ[X|Ft], so that
X = X0 +
T∑
t=1
∆Xt = X0 +
T∑
t=1
(Xt −Xt−1).
By hypothesis X is not reachable, hence there exists t 6 T such that Xt −Xt−1 6∈ Kt, where
Kt :=
{〈η,∆St〉 : η Ft-measurable}.
Note that Kt is a convex closed linear subspace of L
∞(Ω,Ft,P), hence also of L∞(Ω,Ft,Q). By the
Hahn-Banach theorem, we can strictly separate Kt and the compact convex set {∆Xt}, thus obtaining
that there exists ζ ∈ L1(Ω,Ft,Q) such that
〈ζ, f〉 < 〈ζ,∆Xt〉 ∀f ∈ Kt,
or, equivalently,
EQ ζf < EQ ζ∆Xt ∀f ∈ Kt.
Since Kt is linear, we deduce that EQ ζf = 0 for all f ∈ Kt, hence EQ ζ∆Xt > 0. Let us assume,
Without loss of generality, that |ζ| < 1/3, and define the measure Q˜ by
dQ˜
dQ
= ζ˜ , ζ˜ := 1 + ζ − EQ[ζ|Ft−1].
It is immediate to check that ζ˜ > 0 and EQ ζ˜ = 1, so that Q˜ is a probability measure equivalent to Q.
Let us show that pi1 := EQ˜X > EQX = pi: one has
EQ˜X = EQ ζ˜X = EQX + EQ ζX − EQX EQ[ζ|Ft−1],
where, recalling that ζ is Ft-measurable, EQ ζX = EQ ζ EQ[X|Ft] = EQ ζXt, and
EQX EQ[ζ|Ft−1] = EQ EQ[X|Ft−1]EQ[ζ|Ft−1]
= EQXt−1 EQ[ζ|Ft−1] = EQ EQ[ζXt−1|Ft−1] = EQ ζXt−1,
hence
EQ˜X = EQX + EQ ζ(Xt −Xt−1) > EQX,
because EQ ζ(Xt − Xt−1) by the Hahn-Banach separation argument above. In order to prove that
pi1 ∈ Π(X), we need to show that Q˜ is a martingale measure, i.e. that EQ˜[Sk−Sk−1|Fk−1] = 0 for all k.
2.5. PRICING BY NO-ARBITRAGE II 17
To this purpose, we distinguish three cases: (i) if k < t, then Q˜ = Q on Fk, hence the claim is verified.
In fact, since k 6 t− 1 implies Fk ⊆ Ft−1, one has, for any A ∈ Fk,
Q˜(A) = EQ ζ˜1A = EQ 1A + EQ ζ1A − EQ 1A EQ[ζ|Ft−1]
= Q(A) + EQ ζ1A − EQ EQ[ζ1A|Ft−1] = Q(A).
(ii) If k = t, we recall that, by the Hahn-Banach argument above, one has EQ ζf = 0 for all f ∈ Kt.
Therefore, for any A ∈ Ft−1, so that 1A is Ft−1-measurable, one has
EQ 1A(St − St−1)ζ = 0,
which is equivalent to EQ[(St − St−1)ζ|Ft−1] = 0. Since EQ[ζ˜|Ft−1] = 1, it holds
EQ˜[(St − St−1)|Ft−1] = EQ[(St − St−1)ζ˜|Ft−1]
= EQ[(St − St−1)|Ft−1] + EQ[(St − St−1)ζ|Ft−1]
− EQ[(St − St−1)|Ft−1] EQ[ζ|Ft−1]
= 0.
(iii) Let k > t. One has
EQ˜[Sk − Sk−1|Fk−1] =
1
EQ[ζ˜|Fk−1]
EQ[(Sk − Sk−1)ζ˜|Fk−1],
where EQ[ζ˜|Fk−1] = ζ˜ because ζ˜ is Ft-measurable and k > t implies Ft ⊆ Fk−1. For the same reason
one also has EQ[(Sk − Sk−1)ζ˜|Fk−1] = ζ˜ EQ[(Sk − Sk−1)|Fk−1], hence
EQ˜[Sk − Sk−1|Fk−1] = EQ[(Sk − Sk−1)|Fk−1] = 0.
This concludes the proof that there exists pi1 ∈ Π(X) such that pi1 > pi. The proof that there also exists
pi0 ∈ Π(X) such that pi0 < pi, one simply sets pi0 := EQ¯X, with
dQ¯
dQ
:= 2− dQ˜
dQ
.
In particular, Q¯ is equivalent to Q because 1/3 < |dQ¯/dQ| < 5/3. All other properties can be easily
verified.
2.4.5. Corollary. An arbitrage-free market is complete if and only if the set of equivalent martingale
measures is a singleton.
Proof. Assume that Q = {Q}. Then X ∈ L∞(Ω,F ,P) must be attainable because, if it were not, the
set {EQX}Q∈Q would be an open interval, which is clearly impossible.
If the market is complete, then, for any A ∈ F , 1A is attainable. Therefore, for any Q1, Q2 ∈ Q,
one has
Q1(A) = EQ1 1A = EQ2 1A = Q2(A),
i.e. Q1 = Q2.
2.5. Pricing by no-arbitrage II
Let us consider the duality between L∞(Ω,FT ,P) and L1(Ω,FT ,P) induced by the bilinear form
L∞ × L1 −→ R
(f, g) 7−→ 〈f, g〉 := E fg.
Recall that all Lp(Ω,FT ,P) spaces are isomorphic with RN if |Ω| = N , as we assume in this section.
The set C ⊂ L∞ of contingent claims that are superreplicable at price zero is a closed convex cone, as
it follows by the identity C = K +L∞− . In particular, the bipolar theorem implies that the bipolar C
◦◦
of C coincides with C.
The following result gives a fundamental duality relation between superreplicable claims at price
zero and equivalent martingale measures.
18 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
2.5.1. Proposition. Let S satisfy NA. Then C◦ is the cone generated by the set of martingale measures
absolutely continuous with respect to P. Therefore, for any g ∈ L∞, the following facts are equivalent:
(i) g ∈ C;
(ii) EQ g 6 0 for every martingale measure Q absolutely continuous with respect to P.
(iii) EQ g 6 0 for every martingale measure Q equivalent to P.
Proof. That (i) implies (ii) has already been proved in Corollary 2.3.12, and obviously (ii) implies (iii).
Since equivalent martingale measures are dense in the set of martingale measures, (iii) implies (ii). It
thus remains only to show that (ii) implies (i). To this purpose, note that, since C is a cone, one has
C◦ =
{
Q ∈ L1 : 〈Q, g〉 6 0 ∀g ∈ C}.
Although C◦ is not a set of probability measures, it is certainly a subset of L1+ because C ⊃ L∞− . From
this it easily follows that C◦ is the cone generated by Ma. Since
C◦◦ =
{
g ∈ L∞ : 〈Q, g〉 6 0 ∀Q ∈ C◦},
the bipolar theorem yields
C = C◦◦ =
{
g ∈ L∞ : EQ g 6 0 ∀Q ∈Ma
}
.
Recalling that K = C ∩ (−C), one immediately obtains the
2.5.2. Corollary. Let S satisfy NA and f ∈ L∞. The following facts are equivalent:
(i) f ∈ K;
(ii) EQ f = 0 for all Q ∈Ma;
(iii) EQ f = 0 for all Q ∈Me.
It immediately follows that if f ∈ L∞ is such that EQ f = a for all Q ∈ Me, then there exists a
predictable process H such that f = a+ (H · S)T .
Another consequence of the above developments is the following characterization of complete mar-
kets, under the assumptions of NA.
2.5.3. Theorem. Assume that S satisfies NA. The market is complete if and only if there exists a
unique equivalent martingale measure.
Proof. SinceMe contains only one element Q, it follows by the above observation that f = a+(H ·S)T .
The reverse implication is trivial.
With the results just obtained we can prove that the set of no-arbitrage prices of a non-attainable
claim is an open interval.
Proof of Theorem 2.4.4. Let X be a bounded non-attainable contingent claim and Π := {EQX; Q ∈
Me} its set of no-arbitrage prices, whose infimum and supremum we shall denote by pi0 and pi1, re-
spectively. Recall that Π is an interval, bounded because X is bounded. Let us show that pi1 6∈ Π: one
obviouly has
EQ(X − pi1) = EQX − sup
Q∈Me
EQX 6 0 ∀Q ∈Me,
hence X − pi1 ∈ C by Proposition 2.5.1. Then, by definition of C, there exists f ∈ K such that
f > X − pi1. Assuming, by contradiction, that pi1 ∈ Π, i.e. that there exists Q1 ∈ Me such that
pi1 = EQ1 X, one has EQ1 [f − (X−pi1)] = 0, hence f = X−pi1 because f > X−pi1 and Q1 is equivalent
to P. But this implies that X is attainable at price pi1, which is absurd, hence Π is open on the right.
The same reasoning applied to −X shows that Π is also open on the left.
2.6. SINGLE PERIOD MODELS 19
2.6. Single period models
In this subsection we specialize some of the results obtained above to answer basic questions of no-
arbitrage and completeness in a one-period model (i.e. with T = 1). We still assume that Ω =
{ω1, . . . , ωN}, with N <∞.
Let us begin with a motivativating example: assume N = 3 and d = 1, with
S10 = 5, S
1
1(ω) =


3, ω = ω1,
4, ω = ω2,
6, ω = ω3,
and let us look for an arbitrage strategy H ∈ R. By definition, H is an arbitrage strategy if it satisfies
(H · S)1 =


−2H > 0,
−H > 0,
H > 0,
with at least one strict inequality. This system of inequalities admits only the trivial solution H = 0,
which is clearly not an arbitrage strategy, thus the market model is free of arbitrage. This implies,
by the fundamental theorem of asset pricing, that an equivalent risk neutral measure exists, i.e. that
EQ S1 = S0, or equivalently
0 = EQ∆S = −2q1 − q2 + q3,
where we have set, as usual, qi = Q({ωi}), i = 1, 2, 3. We are thus led to the system{
−2q1 − q2 + q3 = 0
q1 + q2 + q3 = 1, q1 > 0, q2 > 0, q3 > 0,
which admits the family of solutions
(q1, q2, q3) = (2a− 1, 2− 3a, a), 1
2
< a <
2
3
,
i.e. there exist (infinitely) many equivalent risk neutral measures.
Let us now check whether the model is complete: again by definition, we need to find V0 ∈ R and
H ∈ R that satisfy
V0 + (H · S)1 = X ⇐⇒


V0 − 2H = x1,
V0 −H = x2,
V0 +H = x3,
where X is identified with the vector in R3 with components x1, x2 and x3. Simple computations reveal
immediately that the system of three equations with two unknowns does not admit a solution, i.e. the
market model is not complete. Suppose now that we add a second risky asset to the market with
S20 = 10, S
2
1(ω) =


12, ω = ω1,
13, ω = ω2,
7, ω = ω3.
One can again verify, in complete analogy to the case with one risky asset, that the model is free of
arbitrage. Let us check completeness: we need to solve the system
V0 + (H · S)1 = X ⇐⇒


V0 − 2H1 + 2H2 = x1,
V0 −H1 + 3H2 = x2,
V0 +H
1 − 3H2 = x3,
which admits the unique solution
V0 =
x2 + x3
2
, H1 = −3x1 + 5
8
x2 − 41
8
x3, H
2 = −x1 + 3
8
x2 − 15
8
x3.
Extending the reasoning to models with general N and d we arrive at the following principle.
20 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
2.6.1. Proposition. An arbitrage-free model is complete if and only if N is equal to the number of
independent vectors in 

1
1
...
1

 ,


∆S11(ω1)
∆S11(ω2)
...
∆S11(ωN )

 , . . . ,


∆Sd1 (ω1)
∆Sd1 (ω2)
...
∆Sd1 (ωN )

 .
Proof. In fact completeness is equivalent to the solvability of the system (written in vector notation)
V0


1
1
...
1

+H1


∆S11(ω1)
∆S11(ω2)
...
∆S11(ωN )

+ · · · +Hd


∆Sd1 (ω1)
∆Sd1 (ω2)
...
∆Sd1 (ωN )

 =


x1
x2
...
xN

 ,
which admits a (unique) solution if and only if the rank of the span of the above vectors is equal to
N .
In other words, we have completeness in a model with N possible states if we have d = N − 1
non-redundant risky assets plus the risk-free money-market account.
2.7. Notes
The material in this chapter is standard and has been adapted in part from [3, 4, 9].
2.8. Exercises
2.8.1. Exercise. In the setting of a financial market in discrete time on a finite probability space,
consider the set C of super-replicable claims at price zero, i.e.
C =
{
g ∈ L0 : ∃f ∈ K, g > f},
where K = {(H · S)T : H predictable}. Show that:
(a) the no-arbitrage property (NA) holds if and only if C ∩ L0+ = {0};
(b) K ⊂ C ∩ (−C).
Assuming that NA holds, show that
(c) C ∩ (−C) ⊂ K.
2.8.2. Exercise. In the setting of a financial market in discrete time on a finite probability space,
consider the set C of bounded super-replicable claims at price zero, i.e.
C =
{
g ∈ L∞(Ω,F ,P) : ∃f ∈ K, f > g},
where K = {(H · S)T : H predictable}. Show that:
(a) S is a martingale with respect to the measure Q if and only if EQ g 6 0 for all g ∈ C.
Assuming that the market is free of arbitrage, show that
(b) a bounded contingent claim g is super-replicable if and only if EQ g 6 0 for all equivalent martingale
measures Q.
Hint. For (b) it might be useful to show first that, denoting by Q the set of all equivalent martingale
measures, and setting pi := supQ∈Q EQ g, the set of contingent claims K
g,pi does not satisfy the no-
arbitrage property.
2.8.3. Exercise. In the setting of an arbitrage-free financial market model S in discounted terms and
discrete time on a finite probability space (Ω,F ,P), consider the set C of bounded super-replicable
claims at price zero, i.e.
C =
{
g ∈ L∞(Ω,F ,P) : ∃f ∈ K, f > g},
where K = {(H · S)T : H predictable}.
Let us denote the set of absolutely continuous martingale measures for S by Qa.
2.8. EXERCISES 21
(a) Show that, for any probability measure Q on (Ω,F), one has Q ∈ Qa if and only if EQ g 6 0 for
all g ∈ C.
Let V be a finite-dimensional vector space with dual V ′, and denote their duality form by 〈·, ·〉. If
D ⊆ V is a convex cone, its polar D◦ is defined as
D◦ :=
{
g ∈ V ′ : 〈g, f〉 6 0 ∀f ∈ D}.
(b) Show that the polar of C coincide with the convex cone generated by Qa.
(c) Let g ∈ L∞. Assuming that C◦◦ := (C◦)◦ = C, show that g ∈ C if and only if EQ g 6 0 for all
Q ∈ Qa.
Hint. For (a), recall a characterization of Qa in terms of K. For (b), prove first that C◦ ⊆ L1+. Recall
that a subset C of a (real) vector space V is a convex cone if αv + βw ∈ C for all v, w ∈ V and α,
β > 0.
2.8.4. Exercise. Consider a one-period model with Ω = {ω1, ω2, ω3}, on which a risk-free asset and a
risky asset S1 are given, with
S10 = 1, S
1
1(ω) =


0.5, ω = ω1,
0.75, ω = ω2,
1.25, ω = ω3.
(a) Prove that the market is free of arbitrage and find an equivalent risk-neutral measure.
Suppose that a further risky asset S2 is introduced (so that two risky assets are traded, in addition to
the risk-free asset) with
S20 = 2, S
2
1(ω) =


3, ω = ω1,
1.75, ω = ω2,
2, ω = ω3.
(b) Determine whether the market is still free of arbitrage and, if it is, compute an equivalent risk-
neutral measure.
(c) Determine whether the contingent claim
X =
(
1
2S
2
1 − S11
)+
.
is attainable and, if possible, compute its fair price.
2.8.5. Exercise. Consider a one-period model of a financial market in discounted terms with Ω =
{ω1, ω2, ω3}, on which two risky assets S1, S2 are traded, with
S10 = 1, S
1
1(ω) =


3/4, ω = ω1,
5/4, ω = ω2,
1/2, ω = ω3,
S20 = 1, S
2
1(ω) =


5/4, ω = ω1,
1/2, ω = ω2,
1 + a, ω = ω3,
where a ∈ R.
(a) Find all values of a such that the market is free of arbitrage.
(b) Let a = 0. Is the market free of arbitrage? Is it complete? Are your conclusions in contradiction
with the so-called second fundamental theorem? Find all values of a such that the market is
complete.
(c) Let a = 0. Is it possible to introduce a further asset S3 such that the resulting enlarged market is
free of arbitrage and complete?
2.8.6. Exercise. Consider a single-period model of a financial market in discounted terms with two
risky assets on a probability space Ω = {ω1, ω2, ω3}, with
S10 = 4, S
1
1(ω) =


8, ω = ω1,
6, ω = ω2,
3, ω = ω3,
22 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE
and
S20 = 7, S
2
1(ω) =


10, ω = ω1,
8, ω = ω2,
4, ω = ω3.
Show that
(a) the law of one price holds;
(b) the market is not free of arbitrage.
Hint. Recall that the law of one price holds if the following is not possible: there exist real numbers
V0, V¯0, V0 6= V¯0 and self-financing strategies H, H¯ such that V0 + (H · S)1 = V¯0 + (H¯ · S)1.
CHAPTER 3
The binomial model
3.1. The one-period model
The results of this paragraph are obviously a special (very simple!) case of those treated in the last
section. However, they are written in a self-contained way for the convenience of the reader.
Let us assume that the following assumptions are in force throughout this subsection, unless oth-
erwise stated:
(1) only one risky asset is traded;
(2) T = 1, i.e. t = 0, 1;
(3) Ω = {ωd, ωu}, P({ωd}) > 0, P({ωu}) > 0;
(4) Sˆ00 = 1 and Sˆ
0
1 = 1 + r, r > 0.
(5) Sˆ11(ω) =
{
Sˆ10 dˆ, ω = ωd,
Sˆ10 uˆ, ω = ωu,
with uˆ > dˆ > 0.
Let us first determine under what condition is the market free of arbitrage. To that purpose, we observe
that we have
∆S11(ω) =
Sˆ11(ω)
Sˆ01
− S0 =


S10
( dˆ
1 + r
− 1
)
= S10d, ω = ωd,
S10
( uˆ
1 + r
− 1
)
= S10u, ω = ωu,
where
d :=
dˆ
1 + r
− 1, u := uˆ
1 + r
− 1.
Therefore we can also write
(H · S)1 =


HS10d, ω = ωd,
HS10u, ω = ωu,
(3.1.1)
from which one can immediately deduce that the market is free of arbitrage if and only if u > 0 > d,
or equivalently if and only if uˆ > 1 + r > dˆ. In view of this observation, we shall assume from now on
that u > 0 > d, i.e. we shall assume that the model is free of arbitrage.
We are now going to prove that the model is complete, i.e. that every contingent claim is attainable.
In particular, let X be the discounted payoff of a contingent claim, i.e. X : Ω→ R, and set
xd := X(ωd), xu := X(ωu).
3.1.1. Proposition. The contingent claim X is attainable.
Proof. We have to prove that there exist V0 and a self-financing portfolio H such that X = V0+(H ·S)1.
Taking (3.1.1) into consideration, we have to solve the linear system{
V0 +HS
1
0d = xd
V0 +HS
1
0u = xu
for the two unknowns V0 and H. A simple computation reveals that
H =
xu − xd
S10(u− d)
, V0 =
u
u− dxd −
d
u− dxu,
thus showing that the contingent claim X is attainable.
23
24 3. THE BINOMIAL MODEL
By the law of one price, V0 is the unique fair price of the contingent claim X. The following
observation is of fundamental importance: setting
qd :=
u
u− d =
uˆ− (1 + r)
uˆ− dˆ , qu := −
d
u− d =
1 + r − dˆ
uˆ− dˆ ,
since qd, qu > 0 with qd + qu = 1, we can define a new probability measure Q on Ω as
Q({ωd}) = qd, Q({ωu}) = qu,
thus obtaining the representation
V0 = EQ[X].
3.2. The multiperiod model
Let T ∈ N, Ω := {−1, 1}T . Denoting the generic element of Ω by ω = (ω1, ·, ωT ), let Yt : ω 7→ ωt be the
projection on the t-th coordinate, and define the dynamics of S = S1 as
St := S0
t∏
k=1
(1 +Rk), Rt(ω) :=
{
u, if ωt = 1,
d, if ωt = −1,
Note that Rt = ∆St/St−1, which is compatible, also from the notational point of view, with the
treatment of the single-period case. We endow Ω with the σ-algebra F equal to its power set, and with
the filtration (Ft) where F0 = {∅,Ω} and Ft is the σ-algebra generated by Y1, . . . , Yt for all t > 1.
Clearly this is the same σ-algebra generated by S1, ·, St. Finally, let P be a probability measure on F
whose only null set is the empty set. The model (in discounted terms) just constructed is called the
(multiperiod) binomial model.
3.2.1. Proposition. The binomial model is free of arbitrage if and only if u > 0 > d. If this is the
case, then
(a) the model is complete;
(b) the (discounted) returns R1, . . . , RT are independent and identically distributed with respect to Q,
with
Q(Rt = u) =
−d
u− d , Q(Rt = d) =
u
u− d .
Proof. A measure Q on (Ω,FT ) is a martingale measure if and only if, by definition,
St = EQ[St+1|Ft] = EQ[St(1 +Rt+1)|Ft] = St + St EQ[Rt+1|Ft],
that is, if and only if
0 = EQ[Rt+1|Ft] = uQ(Rt+1 = u|Ft) + dQ(Rt+1 = d|Ft).
This is possible if and only if
Q(Rt+1 = u|Ft) = qu := −d
u− d , Q(Rt+1 = d|Ft) = qd :=
u
u− d
(in fact, 0 ∈ [d, u] can be written in a unique way as a linear combination of d and u). Then the random
variables R1, . . . , RT are independent and identically distributed with respect to Q. Moreover, since Q
is explicit, it is unique. Existence and uniqueness of a martingale measure is thus proved. It is evident
that Q is equivalent to P if and only if qu ∈ ]0, 1[, i.e. if and only if u > 0 > d.
Let us now prove the converse implication: assume u > 0 > d and constrcut Q equivalent to P on
(Ω,FT ) by
Q({ω}) = qku(1− qu)T−k,
where k is the number of components of ω = (ω1, ·, ωT ) that are equal to one. Then R1, . . . , RT are
independent and identically distributed on Ω under Q. The proof that Q is a martingale measure follows
immediately by the independence of returns.
3.2. THE MULTIPERIOD MODEL 25
Let us now discuss pricing and hedging. Consider a discounted contingent claim
X = g(S0, S1, . . . , ST ),
and define the process V as Vt := V0 + EQ[X|Ft].
3.2.2. Proposition. There exists a function vt such that Vt = vt(S0, . . . , St) for all t ∈ J .
Proof. One has
X = g(S0, S1, . . . , ST ) = g
(
S0, S1, . . . , St, StSt+1/St, . . . , StST /St
)
,
which implies
Vt − V0 = EQ
[
g
(
S0, . . . , St, StSt+1/St, . . . , StST /St
)∣∣Ft].
Note that St+k/St is independent of Ft (under Q) and has the same law as Sk/S0 =
∏k
j=1(1 + Rj).
Hence, by the usual properties of conditional expectation, setting
vt(x0, . . . , xt) := EQ g
(
x0, xt, xtS1/S0, . . . , xtST−t/S0
)
,
we have Vt − V0 = vt(S0, . . . , St).
Note that VT = X and Vt = EQ[Vt+1|Ft], so we obtain a recursive formula for vt: it is immediate
that
vT (x0, . . . , xT ) = g(x0, . . . , xT ),
and
vt(x0, . . . , xt) = quvt+1(x0, . . . , xt, xtu) + qdvt+1(x0, . . . , xt, xtd).
Everything simplifies if X = g(ST ):
Vt = vt(St), vt(xt) =
T−t∑
k=0
(
T − t
k
)
qkuq
T−t−k
d g
(
xtu
kdT−t−k
)
.
Through a graphical approach based on the so-called binomial tree, one can give a (heuristic) proof
of the following fact: let X be a contigent claim of the form X = g(S1T ). Then X is attainable, i.e.
there exist V0 ∈ R and a predictable process H such that
X = V0 + (H · S)T . (3.2.1)
Let us briefly recall that we have obtained V0 and H by a constructive method based on backwards
induction: in fact, let us write
V0 + (H · S)T−1 +HT∆ST = VT−1 +HT∆ST = X.
Then we can apply the results on the one-period model to compute HT and VT−1. Then we can treat
VT−1 as a contingent claim to replicate (it is in fact a random variable!), and we try to obtain, again
using the techniques developed in the one-period case, VT−2 and HT−1 such that
VT−2 +HT−1∆ST−1 = VT−1.
Iterating this procedure, we will finally obtain V0 and a strategy H such that (3.2.1) holds.
In fact a more general result holds true.
3.2.3. Proposition. An arbitrage-free multiperiod binomial model is complete, i.e. every contingent
claim X admits the representation (3.2.1).
Absence of arbitrage implies, see Theorem 2.3.10, that EQ(H · S)T = 0, hence that V0 = EQX,
which gives a pricing principle for the contingent claim X. In order to make this relation operational,
we would need to be able to compute EQX. To this purpose, note that
Sˆ1T = S
1
0 uˆ
Y dˆT−Y ,
26 3. THE BINOMIAL MODEL
where Y is the random variable counting how many times the asset price has increased. It is then
clear that, under the risk neutral measure Q, Y has a binomial distribution with parameters T and
qu := Q(Z1 = uˆ), i.e.
Q(Y = y) =
(
T
y
)
qyu(1− qu)T−y, y = 0, 1, . . . , T.
Therefore we can write
V0 = EQX = EQ g(S
1
T ) = EQ g
(
(1 + r)−T Sˆ1T
)
=
T∑
y=0
(
T
y
)
qyu(1− qu)T−yg
(
(1 + r)−TS10 uˆ
ydˆT−y
)
.
This formula can be easily implemented numerically and is therefore much more practical to use than
the binomial tree. Nonetheless the two approaches are equivalent.
3.3. Continuous-time limit
Suppose we subdivide a fixed period of time [0, T ] (so T is physical time, not number of periods as in
the previous subsection) into N subintervals of length T/N ,
[0, T ] =
N⋃
k=1
[
(k − 1) T
N
, k
T
N
]
.
Given a contingent claim X, we can compute a fair price piN (X) for it based on the N -period binomial
model. The following natural question can be asked: using finer and finer binomial trees, i.e. letting
N tend to infinity (but keeping T fixed!), do we get a limiting price? The answer is positive, and we
give only the final answer. Let us only briefly mention that the so-called continuous-time limit can be
rigorously justified applying a suitable version of the central limit theorem.
3.3.1. Lemma. Let X
(n)
1 , . . . , X
(n)
n on (Ωn,Fn,Pn) be such that
(a)
∣∣X(n)k ∣∣ 6 cn Pn-a.s., with cn → 0 as n→∞;
(b)
∑
k EPn X
(n)
k → m ∈ R as n→∞;
(c)
∑
k varPn X
(n)
k → σ2 ∈ R as n→∞,
Then
n∑
k=1
X
(n)
k
d−→ m+ σZ,
where Z is a standard Gaussian random variable.
3.3.2. Theorem. Let R
(n)
k , k = 1, . . . , n be a sequence of independent random variable with respect to
Qn such that
−1 < an 6 R(n)k 6 bn,
with an → 0 and bn → 0 as n→∞, and
σ2n :=
1
T
n∑
k=1
varR
(n)
k → σ2 > 0.
Then the distribution with respect to Qn of the discounted prices S
(n)
n converges weakly (i.e., in the
sense of measures) to
S0 exp
(
σWT − 1
2
σ2T
)
,
where WT is a centered Gaussian random variable with variance T .
3.3. CONTINUOUS-TIME LIMIT 27
Proof. Let S0 = 1 (without loss of generality). Recall that
log(1 + x) = x− 1
2
x2 + ρ(x)x2,
and one easily shows that −1 < a 6 x 6 b implies |ρ(x)| 6 δ(a, b), with δ(a, b) → 0 as a, b → 0. We
apply this to
S(n)n =
n∏
k=1
(
1 +R
(n)
k
)
,
obtaining (omitting the superscripts for simplicity)
logSn =
n∑
k=1
log
(
1 +Rk
)
=
n∑
k=1
(
Rk − 1
2
R2k + ρ(Rk)R
2
k
)
,
where
∑
ρ(Rk)R
2
k 6 δ(an, bn)
∑
R2k by the assumptions on (Rk), hence, taking into account that
EQn Rk = 0,
EQn
n∑
k=1
ρ(Rk)R
2
k 6 δ(an, bn)
n∑
k=1
varQn Rk.
The right-hand side converges to zero as n → ∞ because so does δ(an, bn) and
∑
varQn Rk → σ2T .
Recalling Slutsky’s theorem, it is thus enough to show that
n∑
k=1
(
Rk − 1
2
R2k
) d−→ −1
2
σ2T + σ
√
TZ,
where Z is a standard Gaussian random variable. Setting Xk := Rk −R2k/2, one has |Xk| 6 cn + c2n/2,
with cn := an ∨ bn, and
EQn
∑
k
Xk → −1
2
σ2T.
Moreover, since ∑
k
EQn |Rk|p 6 cp−2n
∑
k
EQn R
2
k → 0 ∀p > 2,
we immediately infer that
varQn
∑
k
Xk → σ2T.
The proof is concluded appealing to the previous lemma.
3.3.3. Proposition. Let rN = rT/N , and σ > 0 a fixed number. Set
uˆN = e
σ
√
T/N , dˆN = e
−σ
√
T/N ,
and let Sˆ1T (N) be the distribution of Sˆ
1
T in a N -period binomial model under the risk neutral measure.
Then
Sˆ1T (N)
d−→ S10 exp
(
σWT + (r − σ2/2)T
)
(3.3.1)
as N →∞, where d→ denotes convergence in distribution and WT is random variable with distribution
N(0, T ).
Note that one has, as N →∞,
√
NrN → 0,
√
N(dˆN − 1)→ −σ
√
T ,
√
N(uˆN − 1)→ σ
√
T ,
hence (NA) is satisfied for N large enough.
Let us assume that Xˆ = g(Sˆ1T ), so that
piN (X) = EQ
[ g(Sˆ1T (N))
(1 + rN )N
]
.
28 3. THE BINOMIAL MODEL
Then we obtain, taking into account the fundamental limit
lim
N→∞
(1 + rT/N)N = erT ,
that the sequence of prices piN (X) converges, namely
EQ
[ g(Sˆ1T (N))
(1 + rN )N
]
→ e−rT EQ[g(Sˆ1T )] =
e−rT√
2pi
∫ +∞
−∞
g
(
S10e
σ
√
T y+rT−σ2T/2)e−y2/2 dy.
3.3.4. Remark. Strictly speaking, the above convergence follows as a consequence (or by definition itself)
of convergence in distribution only if g is continuous and bounded. However, it is not difficult to prove
that it is enough to assume that g is continuous and |g(y)| 6 C(1 + |y|n) for some n ∈ [0, 2[.
3.4. The formula of Black and Scholes
Let us consider a call option, so that Xˆ = g(Sˆ1T ) with g(x) = (x − Kˆ)+. Then the price of X in the
continuous-time limit can be written as
e−rT√
2pi
∫ +∞
−∞
(
S10e
σ
√
T y+rT−σ2T/2 − Kˆ)+e−y2/2 dy.
Note that the integrand is zero for all y such that
y 6 − logS
1
0/Kˆ + (r − σ2/2)T
σ
√
T
=: −ζ−.
Then we can write, setting K = e−rT Kˆ,
e−rT√
2pi
∫ +∞
−∞
(
S10e
σ
√
T y+rT−σ2T/2 − Kˆ)+e−y2/2 dy
=
e−rT√
2pi
∫ +∞
−ζ−
(
S10e
σ
√
T y+rT−σ2T/2 − Kˆ)e−y2/2 dy
= S10
∫ +∞
−ζ−
1√
2pi
eσ
√
T y−σ2T/2−y2/2 dy −K
∫ +∞
−ζ−
1√
2pi
e−y
2/2 dy
= S10
∫ +∞
−ζ−
1√
2pi
e−(y−σ
√
T )2/2 dy −K
∫ ζ−
−∞
1√
2pi
e−y
2/2 dy.
Setting ζ+ := ζ−+ σ
√
T we finally obtain, by a change of variable in the first integral, that the price of
a call option is given by
S10
∫ ζ+
−∞
1√
2pi
e−y
2/2 dy − e−rTK
∫ ζ−
−∞
1√
2pi
e−y
2/2 dy = S10Φ(ζ+)− e−rT KˆΦ(ζ−),
where, as usual, Φ(x) := (2pi)−1/2
∫ x
−∞ e
−y2/2 dy denotes the cumulative distribution function of a
standard normal random variable. The above formula is the celebrated Black-Scholes formula.
It is often important to be able to compute (or at least estimate accurately) the sensitivity of the
price of a contingent claim with respect to several parameters of interest. In particular, we would like
to estimate the change in value of a contingent claim given a change in the underlying asset price, for
instance to have an idea of how risky is our exposure to derivative products. Similarly, to bound the
risk of computing derivatives’ prices with respect to a misspecified model, we would like to estimate the
sensitivity of the price of a contingent claim with respect to changes in the parameters of the model.
While these notions make perfect sense for any contingent claim, we shall focus on European call options
in a Black-Scholes market with one risky asset. Setting
pic(x, t, σ, r, Kˆ) = xΦ(ζ+(x, t))− e−rtKˆΦ(ζ−(x, t)),
3.4. THE FORMULA OF BLACK AND SCHOLES 29
the following “greeks” can be defined:
∆ =
∂pic
∂x
, Γ =
∂2pic
∂x2
,
Θ =
∂pic
∂t
, ρ =
∂pic
∂r
,
V = ∂pic
∂σ
,
where V, which is not a Greek letter, goes under the name of “vega”. While computing the above partial
derivatives involves only elementary computations, they are quite long and involved. However, ∆ can be
obtained in a concise and elegant way by Euler’s theorem on homogeneous functions. In fact, note that
pic is an homogeneous function with respect to the variables x and Kˆ, and therefore we have (omitting
the dependence of pic on other parameters for convenience of notation)
pic(x, Kˆ) = x
∂pic
∂x
+ Kˆ
∂pic
∂Kˆ
,
from which it immediately follows that
∆ = Φ(ζ+),
hence also
Γ =
ϕ(ζ+)
xσ
√
t
.
Note that ∆ ∈ [0, 1] and Γ > 0, so that, in particular, x 7→ pic(x) is increasing (call option prices are
increasing with respect to the price at time zero of the underlying asset) and
|pic(x)− pic(y)| 6 |x− y|,
i.e. the value of a call option changes less than the value of the underlying in absolute terms. However,
Γ > 0 implies that x 7→ pic(x) is strictly convex, hence
pic(y)− pic(x)
y − x >
pic(x)− pic(0)
x
=
pic(x)
x
, y > x,
thus also
pic(y)− pic(x)
pic(x)
>
y − x
x
, y > x,
and similarly
pic(y)− pic(x)
pic(x)
<
y − x
x
, y < x.
In other words, the value of a call option changes more than the value of the underlying in relative
terms. The latter phenomenon is called leverage effect.
Rather lenghty computations show that
Θ =
xσ
2
√
t
ϕ(ζ+) + e
−rtKˆrΦ(ζ−),
ρ = e−rtKˆtΦ(ζ−),
V = x
√
tϕ(ζ+).
Note that Θ > 0, i.e. call option prices are increasing in the time to maturity (this corresponds to the
intuition that waiting longer, there is more probability that the option will end up in-the-money), and
V > 0, i.e. σ 7→ pic is increasing, corresponding to the intuition that higher asset price variability will
increase the probability of the event S1T > K.
A simple computation (do it!) shows that ∆, Γ and Θ satisfy the equation
Θ(t, x) =
1
2
σ2x2Γ(t, x) + rx∆(t, x)− rpic(t, x),
30 3. THE BINOMIAL MODEL
hence R2+ ∋ (t, x) 7→ pic(t, x) satisfies the partial differential equation (PDE)

∂pic
∂t
− 1
2
σ2x2
∂2pic
∂x2
− rx∂pic
∂x
+ rpic = 0
pic(0, x) = (x− Kˆ)+.
This is only a special case of a more general result.
3.4.1. Proposition. The price v(t, x) of a European contingent claim with payoff function g ∈ C0(R+)
satisfying |g(x)| 6 N(1 + |x|)p for some N , p > 0, satisfies the PDE

∂v
∂t
− 1
2
σ2x2
∂2v
∂x2
− rx∂v
∂x
+ rv = 0
v(0, x) = g(x).
(3.4.1)
Proof. We have
v(t, x) = e−rt EQ g(Sˆ1t ) =
∫ +∞
−∞
g(y)ψt(y) dy,
where ψt is the density of Sˆ
1
t = x exp
(
σ
√
tZ + (r − σ2/2)t). In particular we get
v(t, x) =
1
σ
√
t
∫ +∞
−∞
ϕ
( log y − (r − σ2/2)t− log x
σ
√
t
)
g(y) dy,
from which (3.4.1) follows by differentiation under the integral sign.
3.4.2. Remark. An equivalent formulation of equation (3.4.1) is often found in the literature, where one
writes an equation for u(t, x) := v(T − t, x), t ∈ [0, T ]. Note that u(t, x) is the price at time t of a
contingent claim with expiration T , conditional on Sˆ1t = x. A simple computation shows that u satisfies
the following PDE endowed with a terminal condition:

∂u
∂t
+
1
2
σ2x2
∂2u
∂x2
+ rx
∂u
∂x
− ru = 0
u(T, x) = g(x).
This equation is the celebrated Black-Scholes PDE.
3.5. Complements
The following observation is almost trivial, but it can be very helpful in several situations. Suppose
that we want to price and replicate a contingent claim X which can be decomposed into two “simpler”
contingent claims X1 and X2, in the sense that X = X1+X2. Then, if both X1 and X2 are attainable,
writing
X1 = V 10 + (H
1 · S)T , X2 = V 20 + (H2 · S)T ,
we also get X = V0 + (H · S)T , with
V0 := V
1
0 + V
2
0 , H := H
1 +H2.
This reasoning extends of course to contingent claims that can be represented as a (finite) sum of simpler
contingent claims.
3.5.1. Remark. The above argument is also completely independent of the market model used, and is
clearly not limited to the binomial model. The only hypotheses we have made is, apart of absence of
arbitrage, the attainability of the claims X, X1 and X2.
Let us give a few examples. Observing that
(x−K)+ − (K − x)+ = (x−K)+ − (x−K)− = x−K,
we see that
EQ(K − S1T )+ = EQ(S1T −K)+ − EQ(S1T −K) = EQ(S1T −K)+ − S10 +K,
3.7. EXERCISES 31
i.e. we can express the price of a put option in terms of the price of a call option with the same strike
price and of the price of the risky asset. This relation is often called put-call parity.
Similarly, a straddle option is a contigent claim with a payoff function g(x) = |x−K|, i.e. a straddle
is a combination of a call and a put option with the same strike price.
Slightly more complicated is the butterfly spread option, whose payoff function is given by g(x) =
(L − |x −K|)+. One can show that buying a butterfly spread is equivalent to buying one call option
with strike price K −L, one call option with strike price K +L, and selling two call options with strike
price K.
3.6. Notes
A worked-out example on pricing and hedging using binomial trees can be found in [1, Ch. 2]. A
rigourous derivation of the continuous-time limit is given, among others, in [4]. An intuitively appealing
discussion about delta hedging and delta-gamma hedging is given in [1, Ch. 9]. Many examples of
options that are combination of simpler options can be found in [5]1.
3.7. Exercises
3.7.1. Exercise. A straddle is a European contingent claim with payoff function φ(x) = |x− Kˆ|, where
Kˆ > 0 is called strike price. Consider a Black-Scholes model with parameters Sˆ0 = 4, σ = 30%, T = 1,
r = 2%.
(a) Determine the price of a straddle with strike price Kˆ = 5;
(b) construct a three-period binomial approximation for the dynamics of S;
(c) in the setting of (b), determine the price of the straddle.
1Any edition is ok, no matter how old it is.

CHAPTER 4
Finance in continuous time
Throughout this section we shall assume that a filtered probability space (Ω,F , (Ft),P) satisfying the
so-called usual conditions is given, on which a d-dimensional adapted stochastic process S = (S1, . . . , Sd)
is defined, modeling the discounted prices of d securities.
We shall make the following additional assumption.
4.0.1. Assumption. The Rd-valued process of discounted prices is a semimartingale.
This assumption, which seems to be made just to be able to use the powerful machinery of stochastic
calculus, turns out to be implied by no-arbitrage considerations. In other words, it is natural to assume
that discounted prices are semimartingales.
As in the discrete-time setting, if H is an Rd-valued predictable process such that Hkt denotes the
number of units of the k-th asset held at time t, the corresponding portfolio value is Vt =
∑
kH
k
t S
k
t .
4.0.2. Definition. An Rd-valued predictable process H, integrable with respect to S, is a self-financing
strategy if V = V0 + H · S. A self-financing trading strategy H is called admissible if there exists a
constant α > 0 such that P
(
(H · S)t > −α
)
= 1 for all t > 0.
The set of admissible strategies will be denoted by H.
Note that we have used as definition of self-financing strategies what was in fact a property of such
strategies in the discrete-time case. The admissibility requirement is needed to rule out paradoxical
situations such as the possibility of using doubling strategies (a problem that does not arise in the
discrete-time setting).
We are now going to discuss a few general results in the semimartingale setting, mostly without
proof.
4.1. Notions of no-arbitrage and the fundamental theorem of asset pricing
In order to define no-arbitrage and a slightly strengthened notion thereof, we first introduce the sets of
attainable and super-replicable claims at price zero. Let T ∈ R+ denote a final time horizon.
The set of replicable claims (with zero initial investment) is
K :=
{
(H · S)T : H ∈ H
}
,
and the set of bounded superreplicable claims (with zero initial investment) is
C :=
{
g ∈ L∞ : ∃f ∈ K : f > g}.
Note that both K ⊂ L0 and C ⊂ L∞ are convex cones. Moreover, setting C0 := K − L0+, one has
C = C0 ∩ L∞. The closure of C in L∞ will be denoted by C. We can now formulate the notions of
no-arbitrage, which is formally identical to the discrete-time case, and a stronger condition which is, in
a sense to be made clear soon, a more appropriate one in the present context. In the following we shall
identify a market model in discounted terms with the semimartingale modeling the price processes of
its traded assets.
4.1.1. Definition. The semimartingale S satisfies the no-arbitrage (NA) condition if K ∩ L0+ = {0}.
4.1.2. Definition. The semimartingale S satisfies the no-free lunch with vanishing risk (NFLVR)
condition if C ∩ L∞+ = {0}.
4.1.3. Exercise. Show that NA is equivalent to C ∩ L∞+ = {0} and that NFLVR implies NA.
33
34 4. FINANCE IN CONTINUOUS TIME
A free lunch with vanishing risk is a bounded positive claim X, different from zero, for which there
exists a sequence (gn) of super-replicable claims at price zero such that
lim
n→∞
ess sup
ω∈Ω
∣∣X(ω)− gn(ω)∣∣ = 0.
If such an X ∈ L∞+ \ {0} and (gn) ⊂ C exist, then g−n (ω) converges to zero uniformly with respect to
ω ∈ Ω as n→∞. This explains the meaning of the expression “free lunch with vanishing risk”.
One can hence say that S satisfies the NFLVR property if a free lunch with vanishing risk does not
exist.
Equivalent martingale measures, or, better said, suitably extended versions thereof, as one can
expect, play a fundamental role also in continuous time. Unfortunately it is no longer true that the
existence of an equivalent martingale measure is equivalent to the NA condition. However, an analogous
characterization holds.
4.1.4. Definition. A probability measure Q on FT that is equivalent to P is called
(a) an equivalent martingale measure if S is a Q-martingale;
(b) an equivalent local martingale measure if S is Q-local martingale;
(c) an equivalent σ-martingale measure if S if a Q-σ-martingale.
We can now state a general version of the fundamental theorem of asset pricing.
4.1.5. Theorem. A semimartingale S satisfies the NFLVR property if and only if there exists an equiv-
alent σ-martingale measure.
Proof of sufficiency. Let S be a Q-σ-martingale, that is, by definition, there exist an Rd-valued Q-
martingale M and an R+-valued predictable process φ, integrable with respect to M in the sense of
semimartingales, such that S = φ ·M . Moreover, let H ∈ H. Then H · S = (Hφ) ·M is bounded
from below because H is admissible, hence the Ansel-Stricker lemma implies that H · S is a local
martingale, thus also a supermartingale by an application of the Fatou lemma. This in turn implies
that EQ(H · S)T 6 0, hence also, since H is arbitrary, that EQ g 6 0 for every g ∈ C. Let (gn) be a
sequence in C converging to X in L∞. Then EQ gn 6 0 implies EQX 6 0. In particular, if X ∈ L∞+ , it
must be equal to zero.
The proof of necessity is much harder to prove and we refer to the specialized literature (see [3, 6]).
We just mention the main steps: first one proves that NFLVR implies that C ⊂ L∞ is weak*-closed in
L∞. Therefore, by the Kreps-Yan separation theorem (a consequence of the Hahn-Banach theorem),
there exists a probability measure Q equivalent to P such that EQ g 6 0 for all g ∈ C. If S is bounded,
for any s < t, A ∈ Fs, and a ∈ R, we have a(St − Ss)1A ∈ C, which implies EQ(St − Ss)1A = 0, i.e. S
is a Q-martingale. The case of locally bounded S can be dealt with by localization techniques, while
the general case of a general semimartingale S requires ad hoc techniques using the theory of random
measures.
If the semimartingale S is locally bounded (in particular, if S has continuous paths), the FTAP
can be formulated without σ-martingales.
4.1.6. Corollary. A locally bounded semimartingale S satisfies the NFLVR condition if and only if
there exists an equivalent local martingale measure.
As mentioned in the introduction, the assumption that discounted asset prices are described by a
semimartingale is completely natural. The following result, the proof of which is omitted, is a rigorous
formulation of this fact.
4.1.7. Theorem. Let S be an adapted càdlàg process. If S is locally bounded and satisfies NFLVR for
simple integrands, then S is a semimartingale.
4.2. SUPERREPLICATION 35
4.2. Superreplication
Throughout this section we assume that S is a locally bounded semimartingale satisfying the NFLVR
condition.
Our goal is to characterize the minimal superreplication price of a contingent claim bounded from
below and, as a consequence, a characterization of attainable claims.
We denote the set of absolutely continuous and equivalent local martingale (probability) measures
by M a and M e, respectively. By the identification of a probability measure absolutely continuous
with respect to P with its Radon-Nykodym derivative, both sets can be viewed as subsets of L1. More
precisely, M a is contained in the intersection of the unit ball of L1 with L1+, and M
e is a dense subset
of M a (in the topology of L1).
The first result is the general version of (a part of) Proposition 2.5.1.
4.2.1. Proposition. The polar C◦ of C is the convex cone generated by M a or, in other words,
M
a = C◦ ∩ {f ∈ L1+, ‖f‖L1 = 1}.
Proof. Let Q ∈ M a and H be an admissible strategy, so that H · S is bounded from below, hence a
Q-local martingale by the Ansel-Stricker criterion, as well as a Q-supermartingale, which implies that
EQ(H ·S)T 6 EQ(H ·S)0 = 0. For any g ∈ C there exists an admissible strategyH such that (H ·S)T > g,
hence EQ g 6 0, i.e. Q ∈ C◦. We have thus proved that M a ⊆ C◦ ∩
{
f ∈ L1+, ‖f‖L1 = 1
}
. In order to
prove the converse inclusion, let (Tn) be a localizing sequence of stopping time such that S
Tn is bounded
for every n. For any s < t, A ∈ Fs, and a ∈ {1,−1}, the random variable a1A(STnt − STns ) belongs to
C. Then for any probability measure Q ∈ C◦ one has EQ 1ASTnt = EQ 1ASTns , hence STns = E[STnt |Fs],
i.e. S is a Q-local martingale.
4.2.2. Corollary. M a is closed in L1.
The proof is an exercise, solved next for the lazy reader.
Proof. Let (fn) ∈ M a converge to f in L1. We need to show that f ∈ M a. One has E fng 6 0 for
all g ∈ C. Then E fg 6 0 for all g ∈ C. Since C ⊇ L∞− , then f > 0. Finally, fn → f in L1 implies
‖fn‖ → ‖f‖ by |‖fn‖ − ‖f‖| 6 ‖fn − f‖ , so f ∈ M a.
The definition of super-replication price for a claim X is the same as in Chapter 2, i.e. a ∈ R
such that there exists an admissible strategy H such that a + (H · S)T > X. We are going to show
that for every claim bounded from below there exists a minimum super-replication price that can be
characterized in terms of the set of equivalent local martingale measures. As a first step, we prove the
result for bounded claims. To this purpose, we shall use the fact that C is weakly* closed in L∞, hence,
a fortiori, C is closed in L∞ (recall that its weak* topology is coarser than the topology induced by its
norm). Moreover, we recall that the dual of L∞ with the weak* topology is L1.
4.2.3. Lemma. Let X ∈ L∞(FT ). Then
sup
Q∈Me
EQX = sup
Q∈Ma
EQX = inf{x : ∃k ∈ K, x+ k > X}.
and the infimum is attained, i.e. it is a minimum.
Proof. The supremum of EQX over M
a coincides with the supremum over M e because the latter set
is dense in the former. Note that
inf{x : ∃k ∈ K, x+ k > X} = inf{x : ∃g ∈ C, x+ g > X}
= inf{x : ∃g ∈ C, x+ g = X}
= inf{x : X − x ∈ C},
as all sets involved are identical and non-empty, since they include all x > ‖X‖L∞ . Let x ∈ R and
g ∈ C be such that x + g > X, hence EQX 6 x + EQ g 6 x for every Q ∈ M a by Proposition 4.2.1,
thus also
sup
Q∈Ma
EQX 6 x
36 4. FINANCE IN CONTINUOUS TIME
and
sup
Q∈Ma
EQX 6 inf{x : ∃g ∈ C, x+ g > X}.
We shall prove the converse inequality by a separation argument. Let b ∈ R with b < inf{x : ∃g ∈
C, x + g > X}, so that X − b 6∈ C. Since C is convex and weakly* closed and the singleton {X − b}
is convex and weakly* compact in L∞, by the Hahn-Banach theorem there exists an element q of the
dual of L∞ with the weak* topology, i.e. of L1, such that
〈q,X − b〉 > 〈q, g〉 ∀g ∈ C.
Since g is a cone, it must be 〈q, g〉 6 0 for all g ∈ C, and the inclusion L∞− ⊂ C implies q ∈ L1+.
Moreover, since 0 ∈ C, we have 〈q,X − b〉 > 0 and q 6= 0. Then the probability measure
Q :=
q
‖q‖L1
· P
is absolutely continuous with respect to P and satisfies EQ g 6 0 for every g ∈ C, i.e. Q ∈ C◦, hence
Q ∈ M a by Proposition 4.2.1. Moreover, EQX > b, hence, by arbitrariness of b, we have shown that
for every y < inf{x : ∃g ∈ C, x+ g > X} there exists Q ∈ M a such that EQX > y. This implies that
sup
Q∈Ma
EQX > inf{x : ∃g ∈ C, x+ g > X}.
In fact, assume by contradiction that z := supQ∈Ma EQX is strictly less than the infimum on the
right-hand side. Then there exists Q ∈ M a such that EQX > z = supQ∈Ma EQX, which is impossible.
Finally, the infimum in inf{x : X − x ∈ C} is attained because C is closed in L∞, hence
{x : X − x ∈ C} = ((−C) +X) ∩ R
is also closed.1
To extend the proof to a claim bounded from below, we shall use the following facts, that we recall
without proof: if S is a locally bounded semimartingale satisfying NFLVR, then (a) the set Ka of
discounted portfolio values (H · S)T bounded from below by a ∈ R is bounded in L0 for every a ∈ R;
(b) if (kn) is a sequence in Ka, then there exists a sequence (gn) in the convex hull of (km)m>n, g ∈ L0,
and h ∈ Ka such that gn → g in L0, and h dominates g and is maximal in Ka. Maximality in Ka is
meant in the sense of the order of L0, i.e. if there exists h′ ∈ Ka, h′ > h, then h′ = h.
4.2.4. Theorem. Let X ∈ L0(FT ) be a claim bounded from below, i.e. there exists a constant α ∈ R
such that X > α. Then
sup
Q∈Me
EQX = sup
Q∈Ma
EQX = inf{x : ∃k ∈ K, x+ k > X}.
and the infimum is attained, i.e. it is a minimum.
Proof. Let us set, for compactness of notation, A(X) := {x ∈ R : ∃k ∈ K, x + k > X}. If A(X) is
empty and supQ∈Me EQX = +∞, there is nothing to prove. If A(X) is not empty, then, by a reasoning
already used in the proof of the previous lemma, EQX 6 x for every Q ∈ M e and every x ∈ A(X). Let
us hence assume from now on that A(X) is not empty. The inequality supQ∈Me EQX 6 inf A(X) can
be proved as in the previous lemma. In order to prove the converse inequality, let b be a real number
strictly larger than supQ∈Me EQX. For every n ∈ N, b is also strictly larger than supQ∈Me EQ(X ∧ n),
hence the previous lemma yields
b > sup
Q∈Me
EQ(X ∧ n) = inf{x : ∃k ∈ K, x+ k > X},
hence there exist xn ∈ R and kn ∈ K such that xn + kn > X ∧ n and xn < b. The sequence (xn) is
then bounded (from above by b and from below by α), hence it admits a subsequence converging to
x ∈ [α, b]. Moreover,
kn > X ∧ n− xn > α− b ∀n ∈ N,
1This simple statement can of course be proved using sequences, but it is true in any topological vector space that
the translate of a closed set is closed.
4.4. BROWNIAN MARKET MODEL 37
i.e. the sequence (kn) belongs to Kα−b, which is a convex bounded subset of L0. Then there exists a
sequence (gn), with gn in the convex hull of (km)m>n, such that gn → g in L0, with g ∈ L0 dominated
by a maximal element h of Kα−b. We have thus shown that there exist x ∈ R and h ∈ K such that
x+ h > X. This in turn implies that
b > inf A(X) ∀b > sup
Q∈Me
EQX,
therefore, recalling that supQ∈Me EQX 6 inf A(X), we conclude that supQ∈Me EQX = inf A(X). To
finish the proof we need to show that the infimum of A(X) is attained: let (xn) be a minimizing
sequence, so that xn + hn > X, hn ∈ Kα−b, and xn → x := inf A(X). Using a reasoning based on
sequences of convex combinations of (hn), completely similar to the one used above, we obtain that
there exists h ∈ K such that x+ h > X, hence x ∈ A(X).
4.3. Attainable claims
The assumption that S is a locally bounded semimartingale satisfying the NFLVR property is still in
place.
4.3.1. Definition. A contingent claim X ∈ L0(FT ) bounded from below is attainable if there exists
x ∈ R and a maximal element k of K such that x+ k = X.
We have the following characterization of attainability.
4.3.2. Theorem. A contingent claim X ∈ L0(FT ) bounded from below is attainable if and only if there
exists an equivalent local martingale measure Q∗ such that
EQ∗ X = sup
Q∈Me
EQX.
The proof relies on a characterization of extremal elements of K that we report without proof.
4.3.3. Lemma. Let k = (H · S)T be an element of K. The following statements are equivalent:
(a) k is maximal in K;
(b) there exists an equivalent local martingale measure Q such that EQ k = 0;
(c) there exists an equivalent local martingale measure Q such that H · S is a uniformly integrable
martingale with respect to Q.
Proof of Theorem 4.3.2. Let X be an attainable claim, i.e. X = x + (H · S)T , with x ∈ R and
(H · S)T maximal in K. By the previous lemma there exists Q∗ ∈ M e such that H · S is a uniformly
integrable Q∗-martingale, hence EQ∗ X = x. Since EQX 6 x for every Q ∈ M e, it follows that
EQ∗ X = supQ∈Me EQX.
Let us now assume that there exists Q∗ ∈ M e such that EQ∗ X = supQ∈Me EQX. Theorem 4.2.4
implies that
x := EQ∗ X = min{z ∈ R : ∃k ∈ K, z + k > X},
i.e. there exists and admissible H such that x + (H · S)T > X. Recalling that H · S is a Q∗-
supermartingale,
x = EQ∗ X 6 x+ EQ∗(H · S)T 6 x,
hence EQ∗(H ·S)T = 0. By Lemma 4.3.3 (H ·S)T is thus maximal in K. Moreover, x+(H ·S)T −X > 0
and EQ∗ [x+ (H · S)T −X] = 0 implies x+ (H · S)T = X, that is X is attainable.
4.4. Brownian market model
LetW be an Rm-valued standard Wiener process, [and let us assume that F is the filtration associated to
W ]. Moreover, let b be an Rd-valued adapted [bounded] process and σ an L(Rm,Rd)-valued predictable
[bounded] process. If the undiscounted price process satisfies
Sˆit = Sˆ
i
0 +
∫ t
0
Sˆisb
i
s ds+
∫ t
0
Sˆisσ
i
s dWs, i = 1, . . . , d,
38 4. FINANCE IN CONTINUOUS TIME
then we say that Sˆ is a Brownian market model (in undiscounted terms). Identifying a vector v ∈ Rd
with the d× d matrix diag(v), the equation can also be written in the more compact form
Sˆt = Sˆ
i
0 +
∫ t
0
Sˆsbs ds+
∫ t
0
Sˆsσs dWs.
Alternatively (and equivalently, but perhaps more “canonically”), setting A = diag(b1, . . . , bd) and
B : Ω× R+ × Rd −→ L(Rm;Rd)
(ω, t, v) 7−→ diag(v)σt(ω),
the equation for Sˆ can be written as
Sˆ = Sˆ0 +
∫ t
0
AsSˆs ds+
∫ t
0
BsSs dWs.
An application of Itô’s formula (or the definition of stochastic exponential) yields
Sˆit = Sˆ
i
0 exp
(∫ t
0
(
bis − ‖σis‖2/2
)
ds+
∫ t
0
σis dWs
)
.
Assuming that the numéraire 0-th asset is a strictly positive continuous finite-variation process with
price process
Sˆ0t = exp
(∫ t
0
rs ds
)
,
it follows by the integration by parts formula that
Sit = S
i
0 +
∫ t
0
Sis(b
i
s − rs) ds+
∫ t
0
Sisσ
i
s dWs, i = 1, . . . , d,
hence
Sit = S
i
0 exp
(∫ t
0
(
bis − rs − ‖σis‖2/2
)
ds+
∫ t
0
σis dWs
)
.
By the Girsanov theorem, for any h ∈ L2loc(W ) such that E(h · W ) is a martingale, the process W¯
defined by
W¯t =Wt −
∫ t
0
hs ds
is a Q-martingale, where Q = E(h ·W )TP. For such an h we have
St = S0 +
∫ t
0
Ss(bs − rs + σshs) ds+
∫ t
0
Ssσs dWs.
We have thus proved the following
4.4.1. Proposition. Assume that there exists an Rm-valued predictable process h ∈ L2loc(W ) such that
b−r+σh = 0 and E E(h ·W )T = 1. Then the measure Q := E(h ·W )TP is an equivalent local martingale
measure for S, hence S satisfies the NFLVR condition.
A kind of converse result can be formulate under an additional assumption on the filtration.
4.4.2. Proposition. Assume that F is the (completed) filtration generated by the Wiener process W . If
S satisfies the NFLVR condition, then there exists an Rm-valued predictable process h ∈ L2loc(W ) such
that b−r+σh = 0 and E E(h ·W )T = 1. In this case, Q := E(h ·W )TP is an equivalent local martingale
measure for S.
Simple sufficient conditions for completeness can also be stated.
4.4.3. Proposition. Assume that
(i) F is the (completed) filtration generated by the Wiener process W ;
(ii) the Brownian market model S satisfies the NFLVR property;
(iii) d = m and σ is invertible P⊗ Leb-almost everywhere.
Then the market is complete.
4.5. EXERCISES 39
Proof. There exists an equivalent probability measure Q and a Q-Wiener process W¯ such that
S = S0 + (Sσ) · W¯ .
Recalling that σ is invertible and S is strictly positive, this implies W¯ = (σ−1S−1) · S. Let X be a
bounded contingent claim. Then, by the martingale representation theorem, there exists a predictable
process K such that
X = EQX + (K · W¯ )T = EQX +
(
K · ((σ−1S−1) · S))
T
= EQX +
(
(Kσ−1S−1) · S)
T
,
i.e. X = EQX+(H ·S)T with H := Kσ−1S−1. To conclude, let us show that H is admissible: we have
H · S = K ·W by construction, hence (H · S)t = EQ[X|Ft] − EQX by the martingale representation
theorem, hence
|(H · S)t| 6 EQ[|X||Ft] + EQ |X| 6 2EQ |X|.
4.4.4. Remark. If d < m then the market is, in general, incomplete, by elementary linear algebra. If
d > m the market remains complete, but it may fail to satisfy the NFLVR assumption, because it might
be impossible to find equivalent local martingale measures.
The case of constant coefficients is particularly simple. Let σ be a real d× d matrix, so that, under
the measure Q,
Sit = S
i
s exp
(
−1
2
‖σi‖2(t− s) + σi(W¯t − W¯s)
)
for any 0 6 s < t 6 T . In other words, there exists a function Ψ : R+ × Rd × Rd → Rd such that
St = Ψ
(
t−s, Ss, σ(W¯t−W¯s)
)
. LetX = g(ST ) be a contingent claim, with g : R
d → R a Borel-measurable
function bounded from below. Then one has
Vt = EQ[g(ST )|Ft] = EQ
[
g ◦Ψ(T − t, St, σ(W¯T − W¯t))∣∣Ft]
= EQ
[
g ◦Ψ(T − t, St,
√
T − t σZ)]
=
∫
Rm
g ◦Ψ(T − t, St,
√
T − t σz)φ(z) dz,
where Z is a standard Gaussian random variable in Rm and φ is its density. Therefore, setting
F (t, x) :=
∫
Rm
g ◦Ψ(T − t, x,√T − t σz)φ(z) dz, t < T,
and F (T, x) = g(x), we have Vt = EQ[X|Ft] = F (t, St). In view of Proposition ??, we obtain an explicit
characterization of the hedging strategy.
The Black-Scholes’ formula follows as a special case, taking m = d = 1 and g(x) = (x−K)+. The
calculations leading to the explicit formula have already been carried out in a previous section.
4.5. Exercises
4.5.1. Exercise. Let X and Y be two Itô processes defined as
Xt = X0 +
∫ t
0
fs ds+
∫ t
0
αs dWs,
Yt = Y0 +
∫ t
0
gs ds+
∫ t
0
βs dWs.
(a) Obtain a representation of X2 as an Itô’s process;
(b) obtain a representation of (X + Y )2 as an Itô’s process;
(c) show that
XtYt = X0Y0 +
∫ t
0
(
Xsgs + Ysfs + αsβs
)
ds+
∫ t
0
(
Xsβs + Ysαs
)
dWs.
4.5.2. Exercise. Let (Ω,F ,F,P) be a filtered probability space where F is the natural filtration associ-
ated to the standard Wiener process W . LetM = (Mt)t∈[0,T ] be an (F,P)-martingale such thatMt > 0
P-a.s. for all t ∈ [0, T ].
40 4. FINANCE IN CONTINUOUS TIME
(a) Explain why there exists a predictable process q such that Mt =M0 +
∫ t
0
qs dWs;
(b) write stochastic integral equations satisfied by M and Y := logM ;
(c) show that there exists a predictable process ϕ such that
Mt =M0 exp
(∫ t
0
ϕs dWs − 1
2
∫ t
0
ϕ2s ds
)
and identify the process ϕ.
4.5.3. Exercise. Let pia a family of probability measures on Z+, indexed by a parameter a > 0, defined
as
pia(k) = e
−a a
k
k!
∀k ∈ Z+,
where, as usual, 0! := 1. Assume that there exists a probability space (Ω,F ,P) supporting a stochastic
process X : R+ × Ω → Z+ such that X0 = 0 and X has independent increments, where, for any
t > s > 0, the law of Xt −Xs is piβ(t−s), with β > 0 fixed. Let F be the natural filtration associated to
X.
(a) Prove that t 7→ Yt := Xt − βt is an F-martingale.
(b) Find a (non-random) function φ : R+ → R such that t 7→ Y 2t − φ(t) is an F-martingale.
(c) Let α > 0 be a constant. Find a (non-random) function ψ : R+ → R such that t 7→ ψ(t) exp(−αXt)
is an F-martingale.
4.5.4. Exercise. Let W be a Wiener process, and set
Xt =
∫ t
0
sin 2s dWs
for all t > 0.
Show that
(a) Xt is a well-defined random variable for all t > 0;
(b) X is a Gaussian process, and compute its expectation, its covariance function
(t, s) 7→ EXtXs, s, t ∈ R+
and its conditional expectation E[Xt|FWs ], 0 6 s 6 t;
(c) one has, for any t > 0,
Xt =Wt sin 2t− 2
∫ t
0
cos 2s dWs
4.5.5. Exercise. In the setting of a Black-Scholes market in discounted terms with one risky asset S
and one driving Wiener process, consider the contingent claim X = g(ST ), where
g(x) =
{
C, if K0 6 x 6 K1,
0, otherwise,
with C > 0 and 0 < K0 < K1.
(a) Determine whether the claim is attainable. If it is, prove that there exists a function F ∈
C1,2((0, T )× (0,∞)) such that Vt = F (t, St) for all 0 < t < T .
(b) Determine a self-financing replicating strategy of X.
(c) Assume that (European) call and put options with arbitrary strike price and with the same ma-
turity of X are traded. Prove (or disprove) that X cannot be replicated by a time-independent
portfolio H consisting of a finite combination of the underlying assets and the above-mentioned
European options.
4.5.6. Exercise. In the setting of a Black-Scholes market with one risky asset S and one driving Wiener
process, consider the contingent claim X = g(ST ), where g : R+ → R is a measurable function with
polynomial growth and T > 0. For any t ∈ [0, T ], let pit denote the price of X at time t.
Show that
4.5. EXERCISES 41
(a) the claim X is attainable and that pit = F (t, St), where F : [0, T ]× R+ → R can be written as
F (t, x) = e−r(T−t)
∫
R
g
(
x exp
(
σy
√
T − t+ (r − σ2/2)(T − t)))φ(y) dy,
where φ is the density function of the standard Gaussian measure on R;
(b) the function F can also be written as
F (t, x) = e−r(T−t)
∫
R+
g(z)ψ(x, z) dz,
where
ψ(x, z) =
1
zσ
√
2pi(T − t)e
−d(x,z)2/2, d(x, z) =
log x/z − σ2(T − t)/2
σ
√
T − t ;
(c) if g is convex, then x 7→ F (t, x) is convex.

CHAPTER 5
Portfolio optimization
5.1. Utility functions
The concept of utility function is fundamental in all of economics, and should be familiar from previous
courses. Here we need to make some (mostly technical) assumptions and define the “dual” of a utility
function.
5.1.1. Definition. A function U ∈ C1(]0,+∞[,R) is called a utility function if it is strictly increasing,
strictly concave, and
lim
x→+∞
U ′(x) = 0, lim
x↓0
U ′(x) = +∞.
The conjugate function U˜ :]0,+∞[→ R is defined by
U˜(y) = max
0(
U(x)− xy).
5.2. Maximization of the expected utility of final wealth
In the setting of an arbitrage-free complete market with 1+d assets, we are interested in the optimization
problem faced by an investor who wishes to maximize the expected utility from wealth at time T , having
an initial endowment x ∈ R at time zero. In other words, we deal with the problem
sup
H∈H
E U
(
x+ (H · S)T
)
, (5.2.1)
where H is the set of all predictable portfolio strategies.
The following simple result is the cornerstone of the whole argument allowing to obtain a complete
solution to the problem.
5.2.1. Proposition. Let X denote the set of random variables X : Ω→ R+ such that EQX = x. There
exists a random variable X∗ ∈ X such that
sup
X∈X
E U(X) = E U(X∗), (5.2.2)
and its replicating portfolio solves the optimization problem (5.2.1).
Proof. Note that X is a closed convex set of RN (recall that |Ω| = N), and X 7→ E U(X) is easily seen
to be a strictly concave function. Hence X∗ exists and is unique.1 Since the market is arbitrage-free
and complete, the law of one price holds true and we have
X∗ = x+ (H∗ · S)T
for some H∗ ∈ H. Then a straightforward argument by contradiction shows that H∗ is a maximizer for
problem (5.2.1).
This simple result clearly indicates a strategy to solve our optimization problem: first find an X∗
as above, then find its replicating portfolio, which will be the desired solution.
1This follows from the fact that a convex lower semi-continuous proper function on a closed convex set of a reflexive
Banach space attains its minimum, and that the minimizer is unique if the function is strictly convex. This result is
obviously an overkill in this situation, but very good to know nonetheless.
43
44 5. PORTFOLIO OPTIMIZATION
Let us concentrate on the constrained maximization problem (5.2.2). Introducing a Lagrange
multiplier y > 0, we can write
L(X, y) = E U(X) + y(x− EQX)
= E
[
U(X)− y dQ
dP
X
]
+ xy
6 E U˜
(
y
dQ
dP
)
+ xy (5.2.3)
Note that, since U is strictly increasing and continuously differentiable, the function U ′ is invertible,
and we set I = (U ′)←, i.e. U ′(x) = y if and only if x = I(y). One immediately verifies that I is a
strictly decreasing map from ]0,+∞[ into itself with
lim
y↓0
I(y) = lim
y↓0
U ′(y) = +∞, lim
y→+∞
I(y) = lim
y→+∞
U ′(y) = 0,
and
U˜(y) = U(I(y))− yI(y), 0 < y < +∞.
Therefore, choosing
X = I
(
y
dQ
dP
)
,
we get an equality in (5.2.3). In order to determine y, we just impose the budget constraint EQX = x,
i.e. we set
EQ I
(
y
dQ
dP
)
=: I(y) = x.
Note that the function I maps ]0,∞[ onto itself and it is strictly decreasing, hence invertible. In
particular, there exists y > 0 satisfying the above equation, and the problem is thus completely solved.
5.2.2. Example (Logarithmic utility). Let U(x) = log x. Then I(y) = 1/y and I(y) = 1/y, hence
X∗ = x dP/dQ.
5.2.3. Example (Power utility). Let U(x) = xα/α, 0 < α < 1. Then
I(y) = y−
1
1−α , I(y) = a y− 11−α ,
with a = E
(
dQ
dP
)− α
1−α . A simple computation yields
X∗ = x
1
a
(dQ
dP
)− 1
1−α
.
5.3. Optimization and no-arbitrage
In this subsection we assume that T = 1. The following important result says that the problem of
portfolio optimization in a one-period model is well-posed if and only if the market is free of arbitrage.
5.3.1. Theorem. Suppose that the utility function U satisfies the hypotheses of the previous subsection.
Then the optimization problem (5.2.1) admits a maximizer H∗ if and only if the market model is free
of arbitrage.
Proof. Let us assume that H∗ is a maximizer for problem (5.2.1), and that, by contradiction, that
arbitrage opportunities exist. Then we can find H ∈ H such that (H · S)T (ωi) > 0 for all i = 1, . . . , N ,
and (H ·S)T (ωi) > 0 for at least one i = 1, . . . , N . Without loss of generality, we can take (H ·S)T (ω1) >
0. Then we have, taking into account that U is strictly increasing and P({ωi}) > 0 for all i = 1, . . . , N ,
E U
(
x+ (H∗ · S)T
)
=
N∑
i=1
U
(
x+ (H∗ · S)T (ωi)
)
P({ωi})
<
N∑
i=1
U
(
x+ ((H∗ +H) · S)T (ωi)
)
P({ωi})
= E U
(
x+ ((H∗ +H) · S)T
)
5.4. A CLOSER LOOK 45
which contradicts the optimality of H∗. The model is thus free of arbitrage.
The proof of the converse implication is slightly more difficult and we omit it. Let us just briefly
mention that one proves that the set of strategies H is compact, and that the function H 7→ E U(x+
(H · S)T
)
is continuous, hence it attains its maximum on H.
5.4. A closer look
First we recall a theorem on convex optimization. We recall that R := R ∪ {+∞}. Let V be a vector
space, f , g1, . . . , gr be proper convex functions on V , and gr+1, . . . , gm affine functions on V , all with
values in R. Let C := D(f), and assume that C ⊆ D(gi) for every i = 1, . . . ,m. Let us also introduce
the convex sets
Ci := {x ∈ V : gi(x) 6 0}, i = 1, . . . , r,
Ci := {x ∈ V : gi(x) = 0}, i = r + 1, . . . ,m,
C0 := C ∩ C1 ∩ · · ·Cm.
The problem
inf
C0
f = inf
x∈C0
f(x)
will be called problem (P ).
5.4.1. Definition. A set of m numbers (λ1, . . . , λm) ⊂ R is called an m-tuple of Kuhn-Tucker multi-
pliers for problem (P ) if λi > 0 for all i = 1, . . . , r and
inf
V
(
f + λ1g1 + · · ·+ λmgm
)
= inf
C0
f.
5.4.2. Theorem. Let λ1, . . . , λm be Kuhn-Tucker multipliers for problem (P ), and
I := {i : 1 6 i 6 r, λi = 0},
J := {1, . . . ,m} \ I.
Let h : V → R be the proper convex function defined by
h = f + λ1g1 + · · ·+ λmgm,
and D ⊂ V be its set of minimizers. Then x ∈ V is a minimizer of f on C0, i.e. it is an optimum point
for problem (P ), if and only if x ∈ D with gi(x) = 0 for every i ∈ J and gi(x) 6 0 for every i ∈ I.
Proof. For any x ∈ C0 we have λigi(x) 6 0 for every i = 1, . . . ,m, hence also h(x) 6 f(x), with equality
if and only if x is such that λigi(x) = 0 for every i = 1, . . . ,m. Since infV h = infC0 f by assumption,
the set of minimum points of f is contained in the set of minimum points of h, and it must coincide
with D0.
5.4.3. Corollary. Let λ1, . . . , λm be Kuhn-Tucker multipliers for problem (P ) and assume that the
functions f , (gi) are lower semicontinuous. If h admits a unique minimum point x∗, then x∗ is the
unique optimum point of problem (P ).
Proof. See [10, p. 276].
We shall use convex conjugation, but not in the "canonical" setting. So let us spell out the detail
for future reference: if U is concave, then
U˜(y) := sup
x
(
U(x)− 〈x, y〉) = sup
x
(
U(−x)− 〈−x, y〉) = sup
x
(〈x, y〉 − (−U(−x))),
i.e. we have proved that U˜ is the convex conjugate of the convex function x 7→ −U(−x). Then we have,
taking the convex conjugate again,
−U(−x) = sup
y
(〈x, y〉 − U˜(y)),
hence
U(−x) = − sup
y
(〈x, y〉 − U˜(y)) = inf
y
(
U˜(y)− 〈x, y〉)
46 5. PORTFOLIO OPTIMIZATION
and finally
U(x) = inf
y
(
U˜(y) + 〈x, y〉).
We recall also the basic formula ∂f∗ = (∂f)←. We apply this to f = x 7→ −U(−x) and we get
U˜ ′ = (−U(−·)′)← = −(U ′)←.
The preparations from convex analysis finish here.
Now we can make rigorous the above approach in the case of a complete market, and we can also
say something more. In fact, let
L(X, y) = EU(X) + y(x− EQX) = E
[
U(X)− y dQ
dP
X
]
+ xy
By definition of convex conjugate (in fact we have to "adapt" to the current case) we have
sup
X
E
[
U(X)− y dQ
dP
X
]
+ xy = E U˜
(
y
dQ
dP
)
+ xy,
and the unique maximum point of this problem is
X∗(y) = I
(
y
dQ
dP
)
, I := (U ′)←.
In order for y to be a Kuhn-Tucker multiplier it suffices that EQX∗(y) = x. Denote the (unique)
solution to this by y∗(x).
Then we apply the theorem: let y∗(x) be as just defined, and
X∗(x) := I
(
y∗(x)
dQ
dP
)
.
Then X∗(x) is the optimum point for our problem.
Set
u(x) := sup
X∈X (x)
EU(X), v(y) := E U˜
(
y
dQ
dP
)
.
We have just proved that
u(x) = EU(X∗(x)) = E U˜(X∗(x)) + xy∗(x).
We note that, by definition, v is a convex function, hence we can take its conjugate again (adapting
again to the current non-canonical situation). We have the following result:
u(x) = inf
y>0
(
v(y) + xy
)
.
In fact, the point of minimum of y 7→ v(y) + xy is obtained by v′(y) = −x, which is equivalent to
E
dQ
dP
U˜ ′
(
y
dQ
dP
)
= E U˜ ′
(
y
dQ
dP
)
= −x,
where −U˜ ′ = (U ′)← = I, hence the point of minimum is exactly y∗(x), unique solution to EQ I(y) = x.
Then we have
inf
y>0
(
v(y) + xy
)
= v(y∗(x)) + xy∗(x) = E U˜
(
y∗(x)
dQ
dP
)
+ xy∗(x) = u(x).
We have proved the following facts:
• u and v are conjugate functions, i.e. u˜ = v;
• u has “the same qualitative properties” (in terms of being a utility function with all the corre-
sponding additional conditions);
• the maximum point for the optimization of expected utility of final wealth is given by
X∗(x) = I
(
y∗(x)
dQ
dP
)
,
where u′(x) = y∗(x), or equivalently by conjugation u˜′(y∗(x)) = −x. This is also equivalent to the
budget constraint EQ I(y∗(x)dQdP ) = x.
5.5. THE CASE OF INCOMPLETE MARKETS 47
• One has the (obvious) formulas
u′(x) = EU ′(X∗(x)), v′(y) = EQ U˜ ′(y
dQ
dP
).
5.5. The case of incomplete markets
We are going to use the fact that M a is a compact (bounded and closed) convex polytope of RN ,
as it follows by |Ω| = N . In particular, M a is the convex hull of its finitely many extreme points
{Q1, . . . ,Qm}. We consider the optimization problem
u(x) := sup
X∈X (x)
EU(X),
where X (x) is the set of contingent claims X ∈ L0(FT ) such that EQX 6 x for all Q ∈ M a. For any
λ1, . . . , λm ∈ R+ and X ∈ X (x) we have
EU(X) 6 EU(X) +
m∑
k=1
λk(x− EQk X)
= E
[
U(X)−
m∑
k=1
λk
dQk
dP
X
]
+ x
m∑
k=1
λk.
Setting y :=
∑
λk, as (λ1, . . . , λk) varies in R
m
+ , the measure
∑ λk
y Q
k varies in M a, hence we can write
EU(X) 6 E
[
U(X)− y dQ
dP
X
]
+ xy = EU(X) + y(x− EQX)
6 E U˜
(
y
dQ
dP
)
+ xy
for all (X, y,Q) ∈ X (x)× R+ ×M a, therefore also
EU(X) 6 inf
Q∈Ma
E U˜
(
y
dQ
dP
)
+ xy
for all (X, y) ∈ X (x)× R+. Let us now introduce the dual value function
v(y) := inf
Q∈Ma
E U˜
(
y
dQ
dP
)
.
We claim that there exists a minimizer Q∗ = Q∗(y) and that it belongs to M e. In fact, the function
Q 7→ E U˜
(
y dQdP
)
is continuous on the compact set M a, hence it attains its minimum, and the minimizer
is unique by strict convexity of U˜ . Moreover, if, by contradiction, qn∗ = 0 for some n, take any Q ∈ M e
and set Qε := εQ+(1−ε)Q∗, so that Qε ∈ M e, and note that U˜(yqn∗ /pn) < U˜(yqn/pn) = U˜(0) because
U˜ ′(0) = −∞, hence U˜ is strictly decreasing in a neighborhood of zero. Then, for sufficiently small ε,
we have a contradiction to Q∗ being a minimizer. Hence Q∗ must be in M e. Then we have proved that
for every y > 0 there exists Q∗ = Q∗(y) such that
v(y) = E U˜
(
y
dQ∗
dP
)
We can now proceed in analogy to the case of complete markets: we define y∗(x) as the unique solution
to v′(y) = −x. Let us first compute (formally):
v(y) = V (y,Q∗), V (y,Q) := E U˜
(
y
dQ
dP
)
.
Then
v′(y) =
∂V
∂y
(y,Q∗(y)) +
∂V
∂Q
(y,Q∗(y))Q′∗(y) =
∂V
∂y
(y,Q∗(y))
because Q∗ is a minimizer, hence ∂V∂Q (y,Q∗(y)) = 0. Therefore v
′(y) = −x is equivalent to
E
dQ∗
dP
U˜ ′
(
y
dQ∗
dP
)
= −EQ∗ I
(
y
dQ∗
dP
)
= −x,
48 5. PORTFOLIO OPTIMIZATION
and the optimizer for our problem is then
X∗(x) = I
(
y∗(x)
dQ∗
dP
)
.
The following statements about the case of incomplete markets are true:
• v = u˜, i.e. u and v are conjugate in the sense of convex analysis:
u(x) = inf
y>0
(v(y) + xy).
• u is a utility function with the same properties of U .
• The optimizers X∗(x) and Q∗(y) exists and are unique and are related by:
X∗(x) = I
(
y
dQ∗(y)
dP
)
, y
dQ∗(y)
dP
= U ′(X∗(x)),
u′(x) = y, v′(y) = −x.
5.6. Notes
The approach to portfolio optimization outlined in this chapter is the so-called convex duality (or mar-
tingale) approach to utility maximization. Several excellent references on this topic are available, among
which [4, 7, 8, 9, 11]. Properties of utility functions can be found in any textbook on microeconomics.
5.7. Exercises
5.7.1. Exercise. Let α > 0 be a constant, and define the function U : R ∋ x 7→ 1− e−αx.
(a) Show that U is a utility function, and that the inverse of its derivative I := (U ′)← is given by
I(y) = − 1
α
log
y
α
.
Consider the problem of an agent, with initial endowment x ∈ R+, who seeks to optimize his utility
(given by U) of final wealth. Using a Brownian model S of a financial market, which is assumed to be
free of arbitrage, complete, and with a finite horizon,
(b) show that the optimal wealth profile X∗ is
X∗ = − 1
α
log
dQ
dP
+ x+
1
α
E
[
dQ
dP
log
dQ
dP
]
,
where Q denotes the unique equivalent martingale measure;
(c) write the portfolio optimization problem faced by the agent and solve it.
5.7.2. Exercise. In the setting of a financial market in discounted terms in discrete time on a finite
probability space, consider the optimization problem
sup
H∈H
U
(
x+ (H · S)T
)
,
where H is the set of predictable processes, U is a utility function, and x > 0.
(a) Prove that if the above optimization problem admits a maximizer H∗, then the market is free of
arbitrage.
Assume that the market has one period only (i.e. T = 1) and that the optimization problem admits a
maximizer H∗.
(b) Show that E
[
U ′(x+ (H∗ · S)1
)
∆S1
]
= 0.
(c) Show that there exists a (positive) constant Z such that
F ∋ A 7→ Q(A) := 1
Z
∫
A
U ′
(
x+ (H∗ · S)1
)
dP
defines an equivalent risk-neutral measure. Determine the constant Z.
Bibliography
1. T. Björk, Arbitrage theory in continuous time, Oxford UP, 2004.
2. H. Brézis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
MR 2759829 (2012a:35002)
3. F. Delbaen and W. Schachermayer, The mathematics of arbitrage, Springer Verlag, Berlin, 2006. MR MR2200584
(2007a:91001)
4. H. Föllmer and A. Schied, Stochastic finance, Walter de Gruyter & Co., Berlin, 2004. MR MR2169807 (2006d:91002)
5. J. Hull, Options, futures, and other derivatives, Prentice Hall, 2008.
6. Yu. M. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer, Statistics and control of stochastic processes
(Moscow, 1995/1996), World Sci. Publ., River Edge, NJ, 1997, pp. 191–203. MR 1647282
7. I. Karatzas, Lectures on the mathematics of finance, American Mathematical Society, Providence, RI, 1997.
MR MR1421066 (98h:90001)
8. I. Karatzas and S. E. Shreve, Methods of mathematical finance, Springer Verlag, New York, 1998. MR MR1640352
(2000e:91076)
9. S. R. Pliska, Introduction to mathematical finance: Discrete time models, Blackwell, 1997.
10. R. T. Rockafellar, Convex analysis, Princeton UP, Princeton, NJ, 1997, Reprint of the 1970 original. MR 97m:49001
11. W. Schachermayer, Utility maximisation in incomplete markets, Lecture Notes in Math., vol. 1856, Springer, Berlin,
2004, pp. 255–293. MR MR2113724 (2005k:91173)
49
Notes on
Stochastic Calculus
Carlo Marinelli
This version: 9.III.2021
These lecture notes are very preliminary and will be corrected and/or expanded when needed.
They are meant to serve as a summary of the topics covered in class, and should not be redistributed.
Please report any error or misprint to the author.
Readers who would like to have more information on stopping times and martingales can consult,
for instance, [7, Chapter 1] and/or [8, Chapter 3].
CHAPTER 1
Preliminaries
This chapter may appear of technical nature, at certain points, and the reader should not be discouraged
if he finds it, at first sight, arid and difficult to understand. A possible advice is to skip it on first reading,
and to return to it when needed.
Let (Ω,F ,P) be a probability space. Recall that a set A ⊂ Ω is (F ,P)-negligible if there exists
A′ ∈ F such that P(A′) = 0 and A ⊂ A′. If F contains all negligible sets, the probability space is said
to be complete.
1.1. On the notion of stochastic process
Given a probability space (Ω,F ,P) and a measurable space (E, E), a stochastic process indexed by a
set I is a collection (Xi)i∈I of E-valued random variables, i.e. Xi : Ω→ E is (F , E)-measurable for all
i ∈ I. Given ω ∈ Ω, the map i 7→ Xi(ω) of I to E is called the trajectory, or the path, of X associated
to ω. If I and E are topological spaces, X is called continuous if i 7→ Xi(ω) is continuous for P-a.a.
ω ∈ Ω. Entirely analogous definitions hold for left-continuous or right-continuous processes.
We shall use as index set R+ := [0,+∞[ (sometimes also R+ := R+ ∪ {+∞}), and we shall view
stochastic processes as maps from Ω×R+ to E. It is then natural to say that a process X is measurable
if (ω, t) 7→ Xt(ω) is (B(R+)⊗F , E)-measurable.
We shall need the following equivalence relations on the set of E-valued stochastic processes defined
on the same probability space:
(a) X and Y are a modification of each other if the set {ω ∈ Ω : Xt(ω) 6= Yt(ω)} is negligible for
every t ∈ R+;
(b) X and Y are indistinguishable if the set
⋃
t∈R+
{
ω ∈ Ω : Xt(ω) 6= Yt(ω)
}
is negligible.
If the probability space is complete, X is a modification of Y if
P
({ω ∈ Ω : Xt(ω) = Yt(ω)}) = 1 ∀t ∈ R+,
while X and Y are indistinguishable if
P
({ω ∈ Ω : Xt(ω) = Yt(ω) ∀t ∈ R+}) = 1.
If E = R, a process that is indistinguishable from the zero process is called evanescent, and a subset of
Ω× R+ is called evanescent if its indicator function is an evanescent process.
It is obvious that two indistinguishable processes are a modification of each other, but the converse
is in general not true. On the other hand, the following very useful result holds.
1.1.1. Theorem. Let E be a separated topological space.1 Assume that X and Y are left- or right-
continuous and a modification of each other. Then X and Y are indistinguishable.
Proof. Let N ⊂ Ω be a negligible set such that, for all ω ∈ Ω \ N , the paths X(·, ω) and Y (·, ω) are
right-continuous and Xt(ω) = Yt(ω) for all t ∈ Q+. Keeping ω fixed and passing to the limit shows
that Xt(ω) = Yt(ω) for all t ∈ R+.
As a general convention, processes that are indistinguishable will be considered equal. Similarly,
inequalities and analogous relations between two processes will be meant to hold outside an evanescent
set. Indistinguishability is an equivalence relation on the set of stochastic processes, therefore this
convention amounts to saying that we shall identify a stochastic process with its equivalence class, or
1Readers that are not familiar with the basics of topology can safely assume that E = Rd.
3
4 1. PRELIMINARIES
that we shall consider stochastic processes as elements of the quotient set of all stochastic processes
with respect to this equivalence relation.
1.2. Filtrations
Let (Ω,F ,P) be a probability space. A filtration F = (Ft)t∈[0,∞] on it is an increasing family of
σ-algebras contained in F . Moreover, we set
F0− := F0, F∞− := σ
(⋃
t∈[0,∞[
Ft
)
The filtration F is said to satisfy the “usual conditions” if it is right-continuous, i.e.
Ft = F+t :=
⋂
u>t
Fu,
and F0 containes the (F ,P)-negligible sets of F∞. We shall always assume that the reference filtration
satisfies the usual conditions.
This hypothesis is needed, for example, to deduce that if X and Y are random variables such that
X is Ft-measurable and X = Y P-a.s., then Y is also Ft-measurable, or or to guarantee that any
martingale admits a right-continuous version.
1.2.1. Definition. A process X is called adapted to the filtration F = (Ft)t∈R+ if Xt is Ft-measurable
for every t ∈ R+.
The smallest filtration with respect to which X is adapted is clearly the one defined by FXt :=
σ((Xs)s∈[0,t]) for every t ∈ R+. In figurative language one may say that a process X is adapted if its
evolution is non-anticipative, i.e. it does not depend on future information.
Since we view stochastic processes as maps on Ω × R+, it is natural to introduce measurability
properties with respect to σ-algebras on this space. In particular, we shall always assume that stochastic
processes are measurable with respect to the product σ-algebra F ⊗ B(R+), where B(R+) stands for
the Borel σ-algebra of R+. Note that the adaptedness of a process does not imply measurability.
A central notion of measurability for processes is predictability, that we define next.
1.2.2. Definition. The σ-algebra P ⊂ F ⊗ B(R+) generated by the set of all left-continuous adapted
real-valued processes is called the predictable σ-algebra. A stochastic process X that is measurable with
respect to P is called predictable.
We introduce a set of predictable processes that will play a prominent role in the construction of
the stochastic integral.
1.2.3. Definition. The vector space E of elementary processes is the set of real-valued processes H of
the type
H = h010 +
n−1∑
k=1
hj1]tk,tk+1],
where n ∈ N, (tk) ⊂ R+ is an increasing sequence, and hk ∈ L∞(Ftk) for every k = 0, . . . , n− 1.
1.2.4. Proposition. The predictable σ-algebra P is generated by each one of the following families of
sets:
(a) the set E of elementary processes;
(b) the set of continuous adapted real-valued processes.
Proof. Let us denote by P1 and P2 the σ-algebras generated by continuous processes and left-continuous
processes, respectively. Then evidently P1 ⊆ P2 and P ⊆ P2.
1.4. MARTINGALES 5
1.3. Stopping times
1.3.1. Definition. A random variable T : Ω→ R+ ∪ {+∞} is called stopping time if{
ω ∈ Ω : T (ω) ≤ t} ∈ Ft ∀t ∈ R+.
Note that the definition depends on the reference filtration. A stopping time T is called finite if
P(T <∞) = 1.
1.3.2. Proposition. Let S and T be stopping times and (Tn)n∈N be a sequence of stopping times (with
respect to the same filtration). Then
(a) S + T , S ∧ T , and S ∨ T are stopping times;
(b) supn Tn and infn Tn are stopping times;
(c) lim supn Tn and lim infn Tn are stopping times.
The proof is left as an exercise. Note that all results about sequences of stopping times in (b) and
(c), except for the one about the supremum, depend on the right-continuity of the reference filtration.
The following simple approximation result is very useful.
1.3.3. Lemma. Let T a stopping time. For any n ∈ N, the random variable Tn : Ω → R+ ∪ {+∞}
defined by
Tn(ω) :=
n2n∑
k=1
k
2n
1{ k−1
2n
2n
} +∞1{T (ω)>n}
is a stopping time taking only a finite number of values and dominating T . Moreover, Tn converges
pointwise to T as n→∞.
1.3.4. Definition. Given two stopping times S and T , the stochastic interval [[S, T ]] is defined as
[[S, T ]] :=
{
(ω, t) ∈ Ω× R+ : S(ω) ≤ t ≤ T (ω)
}
.
Other stochastic intervals (open or closed on the left or the right) are defined in en entirely analogous
way. Note that a stochastic interval is always a subset of Ω×R+, even when the stopping times defining
are infinite on an event of strictly positive probability.
If the stopping time T takes only a finite number of values, let us say {t1, . . . , tn,+∞}, then a
simple verification shows that, setting t0 = 0,
(1.3.1) 1[[0,T ]] =
n∑
k=1
1{T>tk−1} 1]tk−1,tk](t) + 1{T>tn} 1]tn,∞[(t)
1.3.5. Definition. Let T be a stopping time. The σ-algebra FT is
FT := {A ∈ F∞ : A ∩ {T ≤ t} ∈ Ft ∀t ∈ R+}.
1.4. Martingales
1.4.1. Definition. Let E be a normed vector space. An E-valued adapted stochastic process M is a
martingale if E‖Mt‖ < ∞ for all t ≥ 0 and Ms = E[Mt|Fs] for all s ≤ t. If E = R and the equality
sign is replaced by ≤ or ≥, then M is called a submartingale or a supermartingale, respectively.
Note that the definition depends both on the filtration and on the probability measure.
Jensen’s inequality immediately implies the following: let f : R→ R is a convex (concave) function
and M is a martingale such that f(Mt) ∈ L1(P) for all t. Then f(M) is a submartingale (supermartin-
gale). Similarly, if M is a submartingale and f is a convex increasing function such that f(Mt) ∈ L1(P)
for all t, then f(M) is a submartingale. In particular, if M is a martingale, then |M | and M+ are
submartingales. For any p ≥ 1, if Mt ∈ Lp(P) for all t, the same holds for |M |p.
1.4.2. Theorem. Let M be a submartingale. If the function t 7→ EMt is right-continuous, then M
admits a càdlàg2 version.
2continu à droite avec des limites à gauche, i.e. right-continuous with left limits (RCLL).
6 1. PRELIMINARIES
In particular, a martingale always has a càdlàg version. (The proof is omitted.)
1.4.3. Theorem (Doob). Let M be a submartingale such that supt∈R+ EM
+
t < ∞. Then there exists
M∞ := limt→∞Mt.
1.4.4. Theorem. A martingale M is a uniformly integrable martingale if and only if there exists a
random variable M∞ ∈ L1(P) such that Mt = E[M∞|cFt].
1.4.5. Theorem (Optional sampling). Let M be a submartingale and T ≤ S stopping times. One has
(1.4.1) MT ≤ E[MS |FT ]
under any one of the following hypotheses:
(a) S and T are bounded;
(b) M is uniformly integrable.
If M is a martingale, then (1.4.1) holds with equality sign under any one of the above hypotheses. If M
is a supermartingale bounded from below, then MT ≥ E[MS |FT ] for arbitrary stopping times T ≤ S.
Recall that X∗t := sups∈[0,t]|Xs| and ‖X‖p := ‖X‖Lp(P).
1.4.6. Theorem (Doob’s inequality). Let M be a positive submartingale or a martingale. One has
P(M∗∞ ≥ λ) ≤ sup
t∈R+
E|Mt|p
λp
∀p ≥ 1, λ > 0
and ∥∥M∗∞∥∥p ≤ pp− 1 supt∈R+∥∥Mt∥∥p ∀p ∈ ]1,∞[.
The following fundamental result, which we recall without proof, is usually called the Doob-Meyer
decomposition.
1.4.7. Theorem (Meyer). Let X be a submartingale such that the family (XT ), T finite stopping time,
is uniformly integrable. Then there exists a unique predictable increasing process A with A0 = 0 and
A∞ ∈ L1(P) such that X −A is a uniformly integrable martingale.
Let H2 denote the space of martingales bounded in L2. If M ∈ H2, then M is uniformly integrable,
hence there exists a random variable M ∈ L2(F∞−) such that
Mt = E
[
M∞
∣∣F∞−] ∀t ∈ R+.
This establishes an (algebraic) isomorphism of H2 and L2(F∞−), with which one can transport the
Hilbert space structure of L2(F∞−) to H2 in a natural way, i.e. defining the inner product〈
M,N
〉
H2
:=
〈
M∞, N∞
〉
L2
= EM∞N∞,
hence, in particular ‖M‖H2 = ‖M∞‖L2 .
The following consequences of theorem 1.4.7 will play a crucial role.
1.4.8. Corollary. Let M , N ∈ H2. Then there exists a unique predictable process of integrable
variation 〈M,N〉 such that MN − 〈M,N〉 is a uniformly integrable martingale.
1.4.9. Definition. An adapted process M is a local martingale if there exists an increasing sequence of
stopping times (Tn)n∈N with Tn →∞ (called localizing sequence) such that (M−M0)Tn is a martingale
for every n ∈ N.
More generally, let C be a class of processes (e.g., bounded processes). A process X with X0 = 0 is
said to belong to Cloc if there exists a localizing sequence (Tn) such that XTn ∈ C for every n ∈ N.
1.4.10. Proposition. (a) A martingale is a local martingale.
(b) If M is a continuous local martingale, then M ∈M2loc.
(c) A local martingale M is a martingale if and only if {MT : T bounded s.t.} is uniformly integrable.
1.4.11. Corollary. Let M , N ∈ H2loc. Then there exists a unique predictable process 〈M,N〉 with
locally finite variation such that MN − 〈M,N〉 is a local martingale.
1.6. NOTES 7
The predictable covariation enjoys the following properties:
(i) (M,N) 7→ 〈M,N〉 is bilinear and symmetric;
(ii) 〈M,N〉 = 〈M −M0, N −N0〉;
(iii) 〈M,N〉 = 14
(〈M +N,M +N〉 − 〈M −N,M −N〉);
(iv) 〈M,M〉 is an increasing process;
(v) for any stopping time T , 〈M,N〉T = 〈MT , N〉;
(vi) 〈Mn,Mn〉∞ → 0 in probability implies (Mn −Mn0 )∗∞ → 0 in probability.
1.5. Semimartingales
1.5.1. Definition. An adapted càdlàg process S is a (one-dimensional) semimartingale if there exist
an F0-measurable random variable S0, a local martingale M and a finite-variation process A with
M0 = A0 = 0 such that
S = S0 +M +A.
If the process A is predictable, then the martingale is called special. In this case the decomposition
is unique, and it is called canonical. One can show that a semimartingale with bounded jumps is special.
1.5.2. Remark. Semimartingales constitute an interesting class of processes for many reasons, among
which the following two: (i) The Bichteler-Dellacherie-Mokobodzki theorem says that semimartingales
are the largest class of “good” integrators. Roughly speaking, this means that if we want to define
a stochastic integral that satisfies a minimal set of properties that we expect from an integral, then
the integrator is necessarily a semimartingale (see, e.g., [2, 11]). (ii) If S describes a discounted price
process and the market satisfies a suitable no-arbitrage property with respect to elementary strategies,
then S is a semimartingale.
1.6. Notes
Properties of stopping times are extensively studied in [1]. An encyclopedic treatment of the theory of
martingales in continuous time can be found in [3].

CHAPTER 2
Stochastic integration
The goal is to construct the integral of a suitable class of processes with respect to semimartingales.
The whole construction will require several steps.
As the whole construction requires several steps (all of which, however, are conceptually simple),
we provide a brief description of the developments to follow. Let S = S0 +M + A a decomposition
of a semimartingale S as in the above definition. It is natural to expect that we should have, for an
integrable process H, H · S = H ·M + H · A. Since the integral with respect to the finite-variation
process A can be done pathwise, the issue is to give a meaning to H ·M . This task can be reduced to
defining the integral for M bounded in L2. Under this assumption, we start with H elementary and
with the “natural” definition of H ·M in this case, and we prove a fundamental isometry result.
2.1. Stochastic integration against square-integrable martingales
Let E be the vector space of processes H of the form
H = h01[[0]] +
n∑
k=1
hk1]]tk,tk+1]],
with hk ∈ L∞(Ftk). For any process X, the stochastic integral H ·X can be defined as
H ·X :=
n∑
k=1
hk
(
Xtk+1 −Xtk),
that is,
(H ·X)t :=
n∑
k=1
hk
(
Xt∧tk+1 −Xt∧tk
) ∀t ∈ R+.
2.1.1. Lemma. If M ∈ H2 and H ∈ E, then the process H ·M belongs to H2 and
〈H ·M,H ·M〉 = H2 · 〈M,M〉,
therefore
∥∥H ·M∥∥
H2
= E
(
H2 · 〈M,M〉)
∞
.
Proof. One has
(H ·M)2 =
(∑
hk(M
tk+1 −M tk)
)2
=
∑
h2k(M
tk+1 −M tk)2 +
∑
j 6=k
hkhj(M
tk+1 −M tk)(M tj+1 −M tj ),
where the second term on the right-hand side is a martingale by an application of the tower property
of conditional expectation, and
(M tk+1 −M tk)2 = (M tk+1)2 − (M tk)2 − 2M tk(M tk+1 −M tk).
Therefore the first term on the right-hand side of the above expression for (H ·M)2 can be written as∑
k
h2k
(
(M tk+1)2 − (M tk)2)− 2∑
k
h2kM
tk(M tk+1 −M tk),
9
10 2. STOCHASTIC INTEGRATION
where the second term is again a martingale, i.e., collecting terms,
(H ·M)2 = N +
∑
k
h2k
(
(M tk+1)2 − (M tk)2),
with N a martingale. Since
H2 · 〈M,M〉 =
∑
k
h2k
(〈M,M〉tk+1 − 〈M,M〉tk),
we deduce that
(H ·M)2 −H2 · 〈M,M〉 = N +
∑
k
h2k
(
(M tk+1)2 − 〈M,M〉tk+1)
−
∑
k
h2k
(
(M tk)2 − 〈M,M〉tk),
where, by definition of predictable quadratic variation, the right-hand side is again a martingale. Since
H2 · 〈M,M〉 is predictable, the Doob-Meyer decomposition theorem implies that it is the compensator
of H ·M . It then follows immediately by corollary 1.4.8 that∥∥H ·M∥∥
H2
= E(H ·M)2∞ = E
(
H2 · 〈M,M〉)
∞
.
2.1.2. Lemma. The vector space L2(M) of predictable processes H such that (H2 · 〈M,M〉)∞ ∈ L1,
endowed with the scalar product 〈
H,K
〉
L2(M)
:= E
(
(HK) · 〈M,M〉)
∞
,
is a Hilbert space. Furthermore, E is dense in L2(M).
Proof. In fact, defining the finite positive measure µ on P as µ(A) = E(1A ·〈M,M〉)∞, the space L2(M)
coincides with L2(Ω× R+,P, P ). The proof of the second statement is omitted.
The previous lemma then says that the linear map
IM : E −→ H2
H 7−→ H ·M.
is an isometry from E ⊂ L2(M) to H2, which can hence be uniquely extended to a linear isometric map
from the closure of E in L2(M), which coincides with the whole Hilbert space L2(M). We have thus
shown the
2.1.3. Theorem. Let M ∈ H2. The map IM admits a unique extension, denoted by the same symbol,
to a continuous linear isometric map from L2(M) to H2.
For any H ∈ L2(M), we shall write H ·M to mean IM (H), and it will be called the stochastic
integral of H with respect to M .
We can look at the stochastic integral also as the bilinear map
I : E ×H2 −→ H2
(H,M) 7−→ H ·M.
We have already shown that, for any M ∈ H2, H 7→ I(H,M) = H ·M is continuous in the topology of
L2(M). The map I is continuous also in its second argument, in the sense of the
2.1.4. Lemma. If H ∈ L∞, then M 7→ H ·M is a continuous endomorphism of H2.
Proof. If Mn →M in H2, then, by bilinearity of I,∥∥H ·Mn −H ·M∥∥2
H2
=
∥∥H · (Mn −M)∥∥2
H2
= E
(
H2 · 〈Mn −M,Mn −M〉)
∞
≤ ‖H‖2L∞E〈Mn −M,Mn −M〉∞,
where E〈Mn−M,Mn−M〉∞ = E|Mn∞−M∞|2, which converges to zero as n→∞ by assumption.
2.2. STOCHASTIC INTEGRATION AGAINST LOCALLY SQUARE-INTEGRABLE LOCAL MARTINGALES 11
The most important properties of the stochastic integral are the following ones.
2.1.5. Proposition. Let M ∈ H2 and H ∈ L2(M).
(a) If Hn → H in L2(M), then Hn ·M → H ·M in H2, hence, in particular, (Hn ·M −H ·M)∗∞ → 0
in L2.
(b) 〈H ·M,H ·M〉 = H2 · 〈M,M〉.
(c) (H ·M)0 = 0 and H · (M −M0) = H ·M .
(d) ∆(H ·M) = H∆M .
(e) Let K ∈ L2(H ·M). Then K · (H ·M) = (KH) ·M .
(f) For every stopping time T ,
MT = M0 + 1[[0,T ]] ·M
and
(H ·M)T = H ·MT = (H1[[0,T ]]) ·M.
Proof. If Hn → H in L2(M), then Hn ·M → H ·M in H2, hence (a) follows by Doob’s inequality.
(b) Let H ∈ L2(M). Then there exists a sequence (Hn) ⊂ E converging to H in L2(M), and
(Hn ·M)2 − (Hn)2 · 〈M,M〉 is a martingale by lemma 2.1.1. The convergence of Hn ·M to H ·M in
H2 implies that (Hn ·M)2t converges to (H ·M)2t in L1 for all t ∈ R+, and the convergence of Hn to
H in L2(M) implies that ((Hn)
2 · 〈M,M〉)t converges to (H2 · 〈M,M〉)t in L1 for all t ∈ R+, therefore
〈H ·M,H ·M〉 −H2 · 〈M,M〉 is a martingale.
(c) is evident by construction of the stochastic integral.
(d) Omitted.
(e) If H,K ∈ E , simple algebra shows that the claim is true. Let (Hn) ⊂ E converge to H in L2(M).
Then Hn ·M converges to H ·M in H2, hence lemma 2.1.4 yields K ·(Hn ·M) = (KHn)·M → K ·(H ·M)
in H2. Since K is bounded, KHn → KH in L2(M), hence (KHn) ·M → (KH) ·M in H2, therefore, by
uniqueness of the limit, K · (H ·M) = (KH) ·M . Now let (Kn) ⊂ E converge to K in L2(H ·M). Then
Kn · (H ·M) = (KnH) ·M → K · (H ·M), hence, by the associativity property of Lebesgue-Stieltjes
integrals, ∥∥KnH −KH∥∥2
L2(M)
= E
(
(Kn −K)2H2 · 〈M,M〉)
∞
= E
(
(Kn −K)2 · (H2 · 〈M,M〉))
∞
= E
(
(Kn −K)2 · 〈H ·M,H ·M〉))
∞
=
∥∥Kn −K∥∥2
L2(H·M)
,
where the last term converges to zero by assumption. Then (KnH) ·M → (KH) ·M , so that, by
uniqueness of the limit, K · (H ·M) = (KH) ·M .
(f) Let T be a stopping time, and (Tn) a sequence of stopping times taking a finite number of
values and converging pointwise to T from above. It follows from (1.3.1) that 1[[0,Tn]] ∈ E and that
M0 + 1[[0,Tn]] ·M = MTn . Since M is right-continuous and Tn converges to T from above, we have
MTn → MT a.s., and 1[[0,Tn]] → 1[[0,T ]] in L2(M) by dominated convergence, hence also 1[[0,Tn]] ·M →
1[[0,T ]] ·M in H2. We have thus proved that M0 + 1[[0,T ]] ·M = MT . This implies, using (d) and (e),
that for any H ∈ L2(M),
(H ·M)T = 1[[0,T ]] · (H ·M) = H · (1[[0,T ]] ·M) = H ·MT ,
we well as
(H ·M)T = 1[[0,T ]] · (H ·M) = (H1[[0,T ]]) ·M = (H1[[0,T ]]) ·MT .
2.2. Stochastic integration against locally square-integrable local martingales
By means of localization techniques, we are now going to further extend the class of integrators as well
as of integrands.
Let us introduce the vector space H2loc of local martingales M that are locally square-integrable, i.e.
such that there exists a localizing sequence (Tn) such that M
Tn ∈ H2 for every n. Given M ∈ H2loc, the
12 2. STOCHASTIC INTEGRATION
vector space L2loc(M) is made of predictable processes H for which there exists a sequence of stopping
times (Tn) increasing to infinity such that H1[[0,Tn]] ∈ L2(M) for every n.
2.2.1. Theorem. Let M ∈ H2loc. The map IM : E → H2loc can be extended to a linear map on L2loc(M),
denoted by the same symbol, which is unique among those that satisfy the condition
(H ·M)Tn = H1[[0,Tn]] ·MTn ∀n ≥ 1
for every sequence (Tn) of localizing stopping times for M and H.
Proof. Let H ∈ L2loc(M), and (Tn) be a localizing sequence for M such that H1[[0,Tn]] ∈ L2(MTn) for
every n. Then H1[[0,Tn]] ·MTn is well defined for every n, and, by Proposition 2.1.5(f),(
H1[[0,Tn+1]] ·MTn+1
)Tn
= H1[[0,Tn]] ·MTn .
This consistency property implies that there exists a process H ·M such that
(H ·M)Tn = H1[[0,Tn]] ·MTn .
In order to show that H · M does not depend on the localizing sequence (Tn), let (Sj) be another
localizing sequence (in the same sense as (Tn) is), and H ·M the corresponding global process. Then,
again by Proposition 2.1.5(f),
(H ·M)Sj∧Tn = (H ·MSj)Tn = H ·MSj∧Tn = (H ·MTn)Sj = (H ·M)Sj∧Tn ,
that is, H ·M and H ·M coincide on [[0, Sj ∧Tn]] for all j and n. Letting j tend to infinity, we conclude
that they coincide also on [[0, Tn]] for all n, hence they are equal. The asserted uniqueness of H ·M is an
obvious consequence of its construction. To conclude, we only have to show that H 7→ H ·M is linear.
To this purpose, let H1, H2 ∈ L2loc(M), and (Tn) a localizing sequence for M , H1, and H2. Then (Tn)
is localizing also for H1 +H2, hence(
(H1 +H2) ·M
)Tn
= (H1 +H2)1[[0,Tn]] ·MTn
= H11[[0,Tn]] ·MTn +H21[[0,Tn]] ·MTn
= (H1 ·M)Tn + (H2 ·M)Tn
= (H1 ·M +H2 ·M)Tn .
This means that (H1 +H2) ·M coincides with H1 ·M +H2 ·M on [[0, Tn]] for all n ≥ 1, thus also on
Ω× R+. A completely similar argument shows that (aH) ·M = a(H ·M) for any a ∈ R.
The stochastic integral enjoys the following properties.
2.2.2. Proposition. Let M ∈ H2loc and H ∈ L2loc(M).
(a) H ·M ∈ H2 if and only if M ∈ H2 and H ∈ L2(M).
(b) If Hn → H pointwise and there exists K ∈ L2loc(M) such that |Hn| ≤ K, then Hn ·M → H ·M
in the ucp topology, i.e. (Hn ·M −H ·M)∗t → 0 in probability for every t ∈ R+.
(c) 〈H ·M,H ·M〉 = H2 · 〈M,M〉.
(d) (H ·M)0 = 0 and H · (M −M0) = H ·M .
(e) ∆(H ·M) = H∆M .
(f) Let K ∈ L2loc(H ·M). Then K · (H ·M) = (KH) ·M .
(g) For every stopping time T ,
MT = M0 + 1[[0,T ]] ·M
and
(H ·M)T = H ·MT = (H1[[0,T ]]) ·M = (H1[[0,T ]]) ·MT .
Proof. Only (a) needs a proof, all other properties being consequences of the corresponding properties
proved in the previous section. If H ·M ∈ H2, corollary 1.4.8 implies that 〈H ·M,H ·M〉∞ ∈ L1, hence
also, by (c), (H2 · 〈M,M〉)∞ ∈ L1, that is, H ∈ L2(M). Conversely, assume that H ∈ L2(M) and let
2.3. STOCHASTIC INTEGRATION AGAINST SEMIMARTINGALES 13
(Tn) be a localizing sequence of stopping times such that (H ·M)Tn ∈ H2. The monotone convergence
theorem yields
‖H ·M‖H2 ≤
∥∥(H ·M)∗∞∥∥L2 = limn→∞∥∥(H ·M)∗Tn∥∥L2 ,
where, by Doob’s inequality,∥∥(H ·M)∗Tn∥∥L2 . ∥∥(H ·M)Tn∥∥L2 = E(H2 · 〈M,M〉)Tn ≤ E(H2 · 〈M,M〉)∞ <∞.
2.2.3. Corollary. Let (Mn) ⊂ H2loc and Hn ∈ L2loc(Mn) for every n ≥ 1. If H2n ·〈Mn,Mn〉 → 0 locally,
then (Hn ·Mn)→ 0 in the ucp topology.
Proof. Exercise.
One has the following characterization of the stochastic integral, which we report without proof.
2.2.4. Proposition. H ·M is the unique process X ∈ H2loc such that 〈X,N〉 = H · 〈M,N〉 for all
N ∈ H2loc.
2.3. Stochastic integration against semimartingales
In order to integrate with respect to semimartingales, we rely on the following decomposition result for
local martingales, the proof of which is omitted (it requires fine properties of stopping times).
2.3.1. Lemma. Let M be a local martingale. Then there exist an FV local martingale M ′ and a local
martingale M ′′ with bounded jumps (in particular, M ′′ ∈ H2loc) such that M = M0 +M ′ +M ′′.
Given S = S0 +M +A and H ∈ L2loc(M ′′) ∩ L(A) ∩ L(M ′), we can thus set
H · S := H · (A+M ′) +H ·M ′′.
For any decomposition of M and any decomposition of S, every predictable locally bounded process is
integrable with respect to S. Unless otherwise stated, only predictable locally bounded processes will
be integrated with respect to semimartingales.
In order to show that H · S is well-defined, we need to show that it does not depend on the
decomposition of S. Assume that
S = S0 +M +A = S0 +M
′ +A′,
where A, A′ are FV processes and M , M ′ ∈ H2loc. The identity M − M ′ = A′ − A implies that
A′ − A ∈ H2loc and M − M ′ is FV, hence, by uniqueness of the stochastic integral with respect to
martingales in H2loc and of the pathwise Lebesgue-Stieltjes integral,
H · (M −M ′) = H · (A′ −A),
where the integral can be interpreted as the stochastic integral or the pathwise Lebesgue-Stieltjes
integral. Recalling that the stochastic integral and the pathwise Lebesgue-Stieltjes integral coincide if
the integrand is predictable locally bounded and the integrator is an FV martingale,1 bilinearity yields
H ·M −H ·M ′ = H ·A′ −H ·A,
which is equivalent to the independence of H · S on the decomposition of S.
The main result is thus the
2.3.2. Theorem. Let S be a semimartingale. The linear map IM : H 7→ H · S defined on E admits a
unique linear extension to the vector space of locally bounded predictable processes, with values in the
vector space of semimartingales, denoted by the same symbol, such that the following continuity property
is satisfied: if a sequence of locally bounded predictable processes (Hn) converges pointwise to H and
there exists a locally bounded predictable process K such that |Hn| ≤ K for all n, then Hn · S → H · S
in the ucp topology.
The stochastic integral of locally bounded predictable processes with respect to semimartingales
enjoys the following properties.
1This is clear for elementary processes, hence also for all bounded predictable processes by passage to the limit, and
for all locally bounded predictable processes by localization.
14 2. STOCHASTIC INTEGRATION
2.3.3. Proposition. Let H be a locally bounded predictable process and S a semimartingale.
(1) If S is a local martingale, then H · S is a local martingale;
(2) if S is FV, then H · S is FV; Lebesgue-Stieltjes integral;
(3) (H · S)0 = 0 and H · S = H · (S − S0);
(4) ∆(H · S) = H∆S;
(5) For any stopping time T one has ST = S0 + 1[[0,T ]] · S and
(H · S)T = (H1[[0,T ]]) · S = (H1[[0,T ]]) · ST = H · ST .
(6) Let K be a locally bounded predictable process. Then the “associative” property holds: K · (H ·S) =
(KH) · S.
It is possible to further extend the stochastic integral in several directions, one of which is essential
for the needs of general arbitrage theory in continuous time. In particular, one can define the stochastic
integral of Rd-valued unbounded predictable processes with respect to Rd-valued semimartingales. We
shall not carry out this further extension, which we shall call the stochastic integral in the sense of
semimartingales, but we shall just mention some of its properties.
The stochastic integral (H,M) 7→ H · M in the sense of semimartingales does not satisfy the
property: M local martingale, then H ·M local martingale. The following criterion is therefore very
useful.
2.3.4. Lemma (Ansel and Stricker). Let M be a local martingale, H integrable with respect to M in the
sense of semimartingales. If there exist a localizing sequence (Tn) such that E
(
(H ·M)−)∗
Tn
< ∞ for
every n, then H ·M is a local martingale.
2.4. Quadratic variation and Itô’s formula
We begin with an approximation result for stochastic integrals of left-continuous processes.
Let τ be sequence of stopping times (Tn)n∈N with T0 = 0 and Tn < Tn+1 on {Tn >∞}. Set
τ(H · S)t :=
∑
n
HTn
(
S
Tn+1
t − STnt
)
,
and
|τ |t := sup
n∈N
∣∣Tn+1 ∧ t− Tn ∧ t∣∣.
Let (τn) be a sequence of such sequences of stopping times. We call it a Riemann sequence if |τn| → 0
on [0, t] for all t > 0.
2.4.1. Proposition. Let H be càdlàg or càglàd. Then τn(H ·S)→ H− ·S as n→∞ in the ucp topology.
Proof. Let
Hn :=
∞∑
m=1
HTn,m1]]Tn,m,Tn,m+1]].
Then Hn is left-continuous, hence predictable, and Hn → H−. Note that H∗ is adapted and left-
continuous, hence locally bounded. Moreover |Hn| ≤ H∗, hence we can use the dominated convergence
theorem: Hn · S → H · S ucp.
Let τ be a subdivision as before, and set
Sτ (X,Y ) :=
∑
n
(
XTn+1 −XTn)(Y Tn+1 − Y Tn).
2.4.2. Theorem. Let X and Y be semimartingales. There exists a process [X,Y ] ∈ V such that
Sτn(X,Y )→ [X,Y ] ucp. Moreover, the following integration by parts formula holds:
XY = X0Y0 +X− · Y + Y− ·X + [X,Y ].
2.4. QUADRATIC VARIATION AND ITÔ’S FORMULA 15
We start with a little algebraic observation: assume that F (·, ·) and G(·, ·) are bilinear maps that
coincide on the diagonal, i.e. F (x, x) = G(x, x) for all x. Then F and G are equal. In fact, bilinearity
implies the polarization formula
F (x, y) =
1
4
(
F (x+ y, x+ y)− F (x− y, x− y)).
Proof. By the observation, we can consider X = Y . Then
Sτ (X,X) =
∑
n
(
XTn+1 −XTn)2.
Now use (a− b)2 = a2 − b2 − 2b(a− b), so
Sτ (X,X) =
∑
n
((
XTn+1
)2 − (XTn)2)− 2∑
n
XTn
(
XTn+1 −XTn)
= X2 −X20 − 2τ(X ·X),
where, by the previous proposition, τ(X ·X)→ X− ·X ucp. Then we have proved
Sτn(X,X)→ [X,X] := X2 −X20 − 2X− ·X.
The general case follows by the algebraic observation.
First properties:
• The map (X,Y ) 7→ [X,Y ] is bilinear and symmetric. (Trivial)
• [X,Y ] = [X −X0, Y − Y0] (direct calculation)
• [X,X] is increasing, hence [X,Y ] is FV
• let H be a predictable locally bounded process. Then [H ·X,Y ] = H · [X,Y ]. (compute)
• [X,Y ]T = [XT , Y ] for every stopping time T ; (compute)
In fact, Sτ (X,X) is increasing, hence so is its limit [X,X] ∈ V+. Then by the polarization formula
[X,Y ] ∈ V.
• ∆[X,Y ] = (∆X)(∆Y ).
In fact, remember that ∆(H ·X) = H ·∆X. Then
∆[X,X] = ∆(X2)− 2X−∆X = X2 −X2− − 2X−∆X = (X− +∆X)2 −X2− − 2X−∆X = (∆X)2.
Now use the observation about bilinearity.
Other properties: let X semimartingale and Y ∈ V.
(a) [X,Y ] = ∆X · Y , which implies XY = Y− ·X +X · Y (the implication is easy).
(b) If Y is also predictable, then [X,Y ] = ∆Y ·X, which implies XY = X− · Y + Y ·X. (Note that
Y ∈ V predictable implies Y locally bounded, the proof is not immediate.)
(c) If X is local martingale and Y is predictable, then [X,Y ] is a local martingale.
Proof. By (b), [X,Y ] = ∆Y ·X, which is a local martingale because ∆Y is predictable and X is
a local martingale.
(d) If X or Y is continuous, then [X,Y ] = 0.
Proof. By (a) or (b), since continuity implies predictability, [X,Y ] = ∆X · Y or [X,Y ] = ∆Y ·X,
hence the integrand is zero.
Now we look at the square brackets of local martingales: Let X and Y be local martingales.
(a) XY −X0Y0 − [X,Y ] is local martingale. Proof: it is equal to X− · Y + Y− ·X.
(b) If X, Y ∈ H2loc, then [X,Y ] ∈ Aloc and [X,Y ] − 〈X,Y 〉 is local martingale, i.e. 〈X,Y 〉 is the
compensator of [X,Y ].
(c) X ∈ H2 (H2loc) if and only if [X,X] ∈ A (Aloc) and X0 ∈ L2.
16 2. STOCHASTIC INTEGRATION
Proof. We prove (b) and (c). We take X = Y , then use polarization. Take X ∈ H2 by localization.
Then clearly X0 ∈ L2. And X2 − X20 − [X,X] is local martingale by the integration by parts
formula, and X2 − X20 − [X,X] is local martingale by the Doob-Meyer decomposition. Then
[X,X]− 〈X,X〉 is local martingale and in V, and by general results Mloc ∩ V ⊂ Aloc. Moreover,
E[X,X]∞ = E〈X,X〉∞ <∞ because X ∈ H2. Hence [X,X] ∈ A, which yields [X,X]− 〈X,X〉 is
UI martingale. Since X2−〈X,X〉 is UI martingale, by difference X2− [X,X] is UI martingale.
(d) X = X0 if and only if [X,X] = 0.
Proof. Enough to show “if”: X2 − X20 local martingale implies X − X0 orthogonal to itself (by
direct calculation). Then we recall the lemma below: X −X0 local martingale orthogonal to itself
implies X = X0.
2.4.3. Lemma. M local martingale orthogonal to itself. Then M = M0.
Proof. M , M2 UI martingale by localization. Then EMt = EM0, EM
2
t = EM
2
0 implies E(Mt−M0)2 =
0.
2.4.4. Exercise. Let M be a continous local martingale. Then [M,M ] = 〈M,M〉.
To prepare for Itô’s formula, we need a further decomposition for semimartingales.
2.4.5. Theorem. Let M be local martingale. Then there exists a unique decomposition
M = M0 +M
c +Md,
M c continuous local martingale.
We do not prove it, although it is not really hard. It follows by the other decomposition, that we
also did not prove.
2.4.6. Corollary. Let S be semimartingale. Then there exists a unique continuous martingale Sc such
that
S = S0 + S
c +A+Md.
Uniqueness follows by: intersection of V and continuous local martingales are constants.
Let f ∈ C2(Rd). Recall that, for any x0 ∈ Rd, one has Df(x0) ∈ L(Rd;R) ≃ Rd and D2f(x0) ∈
L2(Rd;R) ≃ Md×d. In other words, the (Fréchet, or total) derivative of f at any point x0, which is
a linear functional on Rd, can be represented by the gradient of f at x0, which is an element of R
d,
and, similarly, the second (Fréchet) derivative of f at x0, which is a bilinear function on R
d, can be
represented by its Hessian matrix at x0.
In the following, for any FV process A, we shall denote its continuous component by Ac. Moreover, if
S is an Rd-valued semimartingale, the L2(Rd;R)-values process [[S, S]] is defined as [[S, S]]ij := [Si, Sj ].2
2.4.7. Theorem. Let S be an Rd-valued semimartingale and f ∈ C2(Rd). Then f(S) is a semimartin-
gale and
f(S) = f(S0) +Df(S−) · S + 1
2
TrD2f(S−) · [[S, S]]c +
∑
s≤·
(
f(S)− f(S−)−Df(S−)∆S
)
In other words,
f(St) = f(S0) +
d∑
i=1
∫ t
0
∂if(Ss−) dS
i
s +
1
2
d∑
i,j=1
∫ t
0
∂2ijf(Ss−) d[S
i, Sj ]cs
+
∑
s≤t
(
f(Ss)− f(Ss−)−∇f(Ss−)∆Ss
)
.
2There is of course no connection with stochastic intervals, although, admittedly, the notation is unfortunate.
2.5. MARTINGALE REPRESENTATION THEOREMS 17
Proof. The conclusion is certainly true if f = ‖·‖2, as it follows by the formula of integration by parts.
The same formula implies that Itô’s formula holds true for x 7→ xf(x) if it holds for f . In particular, it
holds for polynomials of any order. By localization, given any compact set K ⊂ Rd, we can reduce to
the case where St ∈ K P-a.s. for all t ≥ 0. Since polynomials are dense in C2(K) (in the topology of
the latter space), by dominated convergence one infers that Itô’s formula is true for any f ∈ C2(K).
An important application is the stochastic exponential: let S be a semimartingale with S0 = 0. Its
stochastic exponential E(S) is the solution X to
X = 1 +X− · S.
It follows by Itô’s formula that
X = exp
(
S − 1
2
[S, S]c
)∏
(1 + ∆S)e−∆S .
2.5. Martingale representation theorems
The goal of this section is to show that the stochastic integral IW , with W a Wiener process, is a
surjective map from L2(W ) to H2, if the reference filtration is the natural one generated by W . In
particular, this proves the IW is an isometric isomorphism of the Hilbert spaces L2(W ) and H2(FW ).
The core of the argument is a density result that we prove next. For simplicity we assume that W
is a standard real Wiener process. Let F be the set of functions f : R+ → R of the type
f =
∑
k
λk1]tk,tk+1],
where the sum is finite, (λk) ∈ R, (tk) is an increasing (finite) sequence in R+.
2.5.1. Lemma. The vector space generated by
(
(E(f ·W ))∞
)
f∈F
is dense in L2(Ω,FW∞ ,P).
Proof. By a well-known corollary of the Hahn-Banach separation theorem, it suffices to show that if
Y ∈ L2(Ω,FW∞ ,P) is such that EY E(f ·W )∞ = 0 for every f ∈ F , then Y = 0. Let
f =
n∑
k=1
λk1]tk,tk+1],
and define the function φ : Cn → C by
φ(z1, . . . , zn) := EY exp
(∑
k
zk(Wtk+1 −Wtk)
)
.
Then φ is analytic (the proof of this fact can be “copied” from the proof of analyticity of the moment
generating function of Gaussian random vectors), and by assumption φ is zero on Rn. The analytic
continuation theorem for holomorphic functions then implies that φ = 0 on the whole Cn, in particular,
setting δWk := Wtk+1 −Wtk for convenience of notation,
0 = φ(i(λ1, · · · , λn)) = EY exp
(∑
k
iλkδWk
)
,
where,
EY exp
(∑
k
iλkδWk
)
=
∫
Ω
exp
(∑
k
iλkδWk
)
d(Y P) =
∫
Rn
exp(i〈λ, x〉) d((Y P) ◦ (δW )←) = 0,
i.e. the characteristic function of the measure (Y P) ◦ (δW )←, hence the measure itself, is zero. Since
f was arbitrary, this implies that the measure Y P is zero on the sub-σ-algebra of F generated by the
increments of W , which coincides with the σ-algebra FW∞ . This in turn implies that Y = 0.
2.5.2. Theorem. Let X ∈ L2(Ω,FW∞ ,P). Then there exists a unique predictable process H ∈ L2(W )
such that X = EX + (H ·W )∞.
18 2. STOCHASTIC INTEGRATION
Proof. Let K be the sub-vector space of L2 := L2(Ω,FW∞ ,P) consisting of random variables X for which
there exists H ∈ L2(W ) such that X = EX + (H ·W )∞. We are going to show that K is closed: let
(Xn) ⊂ K, Xn = EXn + (Hn ·W )∞, be a sequence converging in L2. Then, by the Cauchy-Schwarz
inequality,
|EXn − EX| ≤ E|Xn −X| ≤ ‖Xn −X‖L2 → 0.
Therefore we only have to show that there exists H ∈ L2(W ) such that (Hn ·W )∞ → (H ·W )∞ in L2.
Since IW is an isomorphism of L2(W ) to its image, which is hence closed, there exists indeedH ∈ L2(W )
such that Hn ·W → H ·W in H2. Since M 7→ M∞ is an isomorphism of H2 and L2(Ω,FW∞ ,P), we
conclude that (Hn ·W )∞ → (H ·W )∞ in L2 as desired. We have thus proved that K is closed in L2.
The existence proof is completed if we show that the vector space generated by
(
(E(f ·W ))∞
)
f∈F
is
contained in K. To this purpose, note that M := E(f ·W ) is the unique solution to M = 1+(Mf) ·W .
Since M has finite moments of all orders and f is bounded, we have Mf ∈ L2(W ), hence M ∈ K, as
claimed. Uniqueness is an immediate consequence of the injectivity of IW and of H2 being isomorphic
to L2(Ω,FW∞ ,P).
2.6. Changes of measure
Let P and Q be probability measures on FT and denote their restrictions to Ft by Pt and Qt, respectively.
If Q is absolutely continuous with respect to P, there exists ρT ∈ L1+(FT ,P) with EρT = 1 such that
Q = ρTP. Moreover, setting ρt := E[ρT |Ft], one has Qt = ρtPt for all t ∈ [0, T ], as it follows by the
trival identities
Qt(A) = Eρ1A = EE[ρ|Ft]1A = ρtPt(A) ∀A ∈ Ft.
It goes without saying that the process ρ is a (uniformly integrable) positive P-martingale. If P and
Q are equivalent, then Pt and Qt are obviously equivalent for all t ∈ [0, T ], and the previous identity
implies that ρ is a strictly positive process. Slightly more precisely, Pt = ηtQt for all t ∈ [0, T ], with η
a uniformly integrable strictly positive Q-martingale and ηρ = 1.
A minor extension of these reasonings yields the
2.6.1. Proposition. Let P and Q be equivalent probability measures on FT . An adapted càdlàg process
M is a Q-martingale if and only if Mρ is a P-martingale.
Proof. M is a Q-martingale if and only if, for any 0 ≤ t ≤ s and A ∈ Ft,
EQ1AMt = EQ1AMs,
where, by well-known properties of conditional expectation,
EQ1AMt = EρT 1AMt = EE[ρT |Ft]1AMt = E1AρtMt
and, completely analogously,
EQ1AMs = EρT 1AMs = EE[ρT |Fs]1AMs = E1AρsMs,
hence E1AρtMt = E1AρsMs, i.e. Mtρt = E[Msρs|Ft].
By localization one obtains the
2.6.2. Corollary. M is a Q-local martingale if and only if Mρ is a P-local martingale.
Proof. Let (Tn) be a localizing sequence for M , so that M
Tn is a Q-martingale, hence MTnρ is a P-
martingale by the previous proposition, for every n ∈ N. Then (Mρ)Tn = MTnρ − MTn(ρ − ρTn),
where
MTn(ρ− ρTn) = MTn(ρ− ρTn)1[[Tn,∞[[
is a P-martingale as well, for every n ∈ N. By equivalence of P and Q, the sequence of stopping times
(Tn) tends to infinity P-almost surely, hence Mρ is a P-local martingale.
Conversely, let (Tn) be a localizing sequence for Mρ, so that (Mρ)
Tn is a P-martingale, and
MTnρ = (Mρ)Tn +MTn(ρ− ρTn)1[[Tn,∞[[,
2.6. CHANGES OF MEASURE 19
where the second term on the right-hand side is also a P-martingale. Then, for every n ∈ N, MTnρ is
a P-martingale, hence, by the previous proposition, MTn is a Q-martingale. As before equivalence of P
and Q implies that (Tn) tends to infinity, hence M is a Q-local martingale.
An immediate consequence of the previous proposition is the so-called Bayes’ formula: for any
X ∈ L∞(FT ) or X ∈ L0+(FT ),
EQ[X|Ft] = 1
ρt
E[XρT |Ft] = E
[
X
ρT
ρt
∣∣∣Ft].
We can now prove a (simple) version of Girsanov’s theorem.
2.6.3. Theorem. Let M be a P-local martingale. Then
M − 1
ρ
· [ρ,M ]
is a Q-local martingale.
Proof. The integration by parts formula yields
Mρ = ρ− ·M +M− · ρ+ [ρ,M ],
hence, multiplying both sides by 1/ρ,
M − 1
ρ
[ρ,M ] =
1
ρ
(
ρ− ·M +M− · ρ
)
=: N
Since both M and ρ are P-local martingales, Nρ = ρ− ·M +M− · ρ is a P-local martingale, hence N is
a Q-local martingale by Corollary 2.6.2. We also have, thank to the ingration by parts formula,
1
ρ
[ρ,M ] = (1/ρ−) · [ρ,M ] + [ρ,M ]− · (1/ρ) +
[
1/ρ, [ρ,M ]
]
.
Note that
1
ρt
= EFtQ
dP
dQ
∀t ∈ R+
implies that 1/ρ is a Q-martingale, hence N ′ := [ρ,M ]− · (1/ρ) is a Q-local martingale. Moreover,[
1/ρ, [ρ,M ]
]
=
∑
∆(1/ρ)∆[ρ,M ],
hence
(1/ρ−) · [ρ,M ] +
[
1/ρ, [ρ,M ]
]
= (1/ρ) · [ρ,M ],
therefore
1
ρ
· [ρ,M ] = 1
ρ
[ρ,M ]−N ′,
thus also
M − 1
ρ
· [ρ,M ] = M − 1
ρ
[ρ,M ] +N ′ = N +N ′,
which concludes the proof.
As a corollary we get a fundamental invariance result.
2.6.4. Theorem. Let P and Q be equivalent probability measures. Let S be a P-semimartingale with
decomposition S = S0+M+A. Then S is also a Q-semimartingale with decomposition S = S0+N+B,
with
N = M − 1
ρ
· [ρ,M ].
2.6.5. Remark. In fact a more general invariance result is true: let Q be a probability measure locally
absolutely continuous with respect to P (i.e., Q is absolutely continuous with respect to P on Ft for all
t > 0) and S be a P-semimartingale. Then S is also a Q-semimartingale. Note also that [S, S] does not
depend on the probability measure, hence it is the same for both P and Q.
20 2. STOCHASTIC INTEGRATION
The classical Girsanov theorem for Wiener processes can easily be deduced from Theorem 2.6.3.
We state it and prove it by elementary means nonetheless.
2.6.6. Theorem (Girsanov). Let W be an Rd-valued Wiener process with unit covariance operator and
h : Ω× [0, T ]→ Rd ≃ L(Rd,R) be predictable and bounded. Let L be the exponential martingale defined
by
Lt := E(h ·W )t = exp
(∫ t
0
hs dWs − 1
2
∫ t
0
‖hs‖2 ds
)
, t ≥ 0,
and Q be the probability measure on F defined by Q = LTP, i.e.
F ∋ A 7→ Q(A) :=
∫
A
LT dP.
Then the process
W˜t := Wt −
∫ t
0
hs ds
is a Brownian motion on (Ω,F ,F,Q).
Proof. Let us assume for simplicity that d = 1. We are going to show that, under the measure Q, W˜
is a continuous process with independent increments such that the law of W˜t − W˜s, 0 ≤ s ≤ t ≤ T , is
centered Gaussian with variance t− s. The pathwise continuity of W˜ is obvious.
(a) Let us show that
EQ exp
(
iξ(W˜t − W˜s)
)
= ELT exp
(
iξ(W˜t − W˜s)
)
= exp
(−ξ2(t− s)/2)
for any ξ ∈ R. Since L is a martingale with respect to P, it is easily seen that
ELT exp
(
iξ(W˜t − W˜s)
)
= E exp(−iξW˜s)E
[
exp(iξW˜t)LT
∣∣Fs]
= E exp(−iξW˜s)E
[
E[exp(iξW˜t)LT |Ft]
∣∣Fs]
= E exp(−iξW˜s)E[exp(iξW˜t)Lt|Fs],
where, by the integration by parts formula,
eiξW˜tLt = 1 +
∫ t
0
eiξW˜s dLs +
∫ t
0
Lsd
(
eiξW˜s
)
+
[
L, eiξW˜
]
t
.
Itô’s formula yields
eiξW˜t = 1 +
∫ t
0
eiξW˜siξ dWs −
∫ t
0
eiξW˜s
(
iξhs +
1
2
ξ2
)
ds,
hence, recalling that Lt = 1 +
∫ t
0
hsLs dWs,[
L, eiξW˜
]
t
= iξ
∫ t
0
eiξW˜shsLs ds
as well as
eiξW˜tLt = 1 +
∫ t
0
eiξW˜sLs(iξ + hs) dWs − ξ
2
2
∫ t
0
eiξW˜sLs ds.
Another (but easier) application of the integration by parts formula shows that t 7→ Lt exp
(
iξW˜ t+tξ2/2
)
is a martingale, i.e.
E
[
etξ
2/2eiξW˜tLt|Fs
]
= esξ
2/2eiξW˜sLs,
which is equivalent to
E
[
eiξW˜tLt|Fs
]
= e−(t−s)ξ
2/2eiξW˜sLs.
This in turn implies
ELT exp
(
iξ(W˜t − W˜s)
)
= Ee−iξW˜sE
[
eiξW˜tLt|Fs
]
= e−(t−s)ξ
2/2,
thus proving the claim about the distribution of the increments of W˜ .
(b) The proof that W˜ has independent increments under Q is similar, thus omitted.
2.7. COMPLEMENTS 21
2.7. Complements
Let S be an Rd-valued continuous semimartingale, and define the Md×d-valued process [[S, S]] by
[[S, S]]ij := [Si, Sj ]. Proposition 4.2.23 immediately yields [[S, S]] = [[M,M ]]. In particular, ifM = K ·W ,
one has
[[M,M ]]t =
∫ t
0
(
KsQ
1/2
)(
KsQ
1/2
)∗
ds,
where Q is the covariance operator of W . In fact, if the covariance operator of W is the identity, then
[[K ·W,K ·W ]]ijt =
[
(K ·W )i, (K ·W )j]
t
=
[∑
k
Kik ·W k,
∑
k
Kjk ·W k
]
t
=
∫ t
0
∑
k
Kiks K
jk
s ds =
∫ t
0
∑
k
Kiks (K
∗
s )
kj ds
=
(∫ t
0
KsK
∗
s ds
)ij
.
For the case of a general covariance operator Q, it is enough to recall that Q−1/2W is a Wiener process
with covariance operator equal to the identity, and to write K ·W = (KQ1/2) · (Q−1/2W ).

CHAPTER 3
Stochastic differential equations
3.1. Existence and uniqueness of strong solutions
Let (Ω,F ,P) be a probability space endowed with a filtration F = (Ft)t∈R+ satisfying the usual con-
ditions. If f : Ω × R+ × R → R is a function (satisfying suitable measurability conditions) and Y is a
stochastic process, we shall write g(Y ) to denote the process (ω, t) 7→ g(ω, t, Yt(ω)).
3.1.1. Theorem. Let S be a semimartingale with S0 = 0, H be an adapted càdlàg process, and f : Ω×
R+ × R→ R be a function such that
(a) x 7→ f(ω, t, x) is Lipschitz continuous with Lipschitz constant C for every (ω, t) ∈ Ω× R+;
(b) ω 7→ f(ω, t, x) is Fs-measurable for every (t, x) ∈ R+ × R;
(c) t 7→ f(ω, t, x) is càdlàg for every (ω, x) ∈ Ω× R.
Then there exists a unique càdlàg adapted process X such that
(3.1.1) X = H + f(X−) · S.
The following lemma shows that the stochastic integral in (3.1.1) is well defined, i.e. that the
equation is meaningful.
3.1.2. Lemma. Let f be a function as in the previous theorem. If Y is an adapted càdlàg process, then
f(Y−) is an adapted càglàd process, hence, in particular, predictable and locally bounded.
Proof. For any s ∈ R+ fixed, since (ω, x) 7→ f(ω, s, x) is Fs ⊗ B(R)-measurable, ω 7→ f(ω, s, Ys−(ω))
is Fs-measurable by composition, i.e. f(Y−) is adapted. Let us show that f(Y−) admits right limits:
denoting the limit from the right of s 7→ f(ω, s, x) as s→ t by f(ω, t+, x), one has, for s > t,∣∣f(ω, s, Ys−(ω))− f(ω, t+, Yt(ω))∣∣ ≤ ∣∣f(ω, s, Ys−(ω))− f(ω, s, Yt(ω))∣∣
+
∣∣f(ω, s, Yt(ω))− f(ω, t+, Yt(ω))∣∣,
where the first term on the right-hand side is dominated by C|Ys−(ω)−Yt(ω)|, which converges to zero
as s tends to t from above. Similarly, the second term on the right-hand side tends to zero by definition
of f(ω, t+, x). The proof of continuity on the left of f(Y−) is entirely similar, hence omitted.
The proof of theorem 3.1.1 is based on a series of lemmata that reduce the problem to an easier
one.
3.1.3. Lemma. Assume that the hypotheses of theorem 3.1.1 are satisfied. The existence and uniqueness
of a solution to (3.1.1) holds if and only if it holds under the following additional assumption: there
exists a positive constant a, possibly depending on the Lipschitz constant C, such that |∆S| ≤ a.
Proof. It obviously suffices to prove the sufficient condition. Let (Tk) the sequence of stopping times
exhausting the jumps of S larger than a in absolute value, and set S1 := ST1−, so that |∆S1| ≤ a.
Setting H1 := HT1−, by hypothesis there exists a unique càdlàg adapted process X1 such that X1 =
H1+ f(X1−) ·S1. Therefore X1 is the unique solution to (3.1.1) on [[0, T1[[. Setting X := X11[[0,T1[[, the
process X can be extended to a solution to (3.1.1) on [[0, T1]], in a unique way, setting
∆XT1 := ∆HT1 + f(·, T1, XT1−)∆ST1 .
Considering now (3.1.1) on [[T1, T2]], we can repeat the same argument, thus extending X to the unique
solution to (3.1.1) on [[0, T2]]. By iteration, the proof is concluded.
23
24 3. STOCHASTIC DIFFERENTIAL EQUATIONS
3.1.4. Lemma. Assume that the hypotheses of theorem 3.1.1 are satisfied. The existence and uniqueness
of a solution to (3.1.1) holds if and only if it holds under the following additional assumption: there
exists a constant b ≤ 1, possibly depending on C, such that S = M +V , where M is a square-integrable
martingale with M0 = 0 and [M,M ]∞ ≤ b, and V is a predictable process with finite variation and
V0 = 0 such that
∫∞
0
|dV | ≤ b.
Proof. Thanks to the previous lemma we can assume that |∆S| ≤ a := b/4. Since a semimartingale
with bounded jumps is special, one has S = M + V , with V a predictable process such that V0 = 0.
Moreover, one can show1 that the jumps of M and V are bounded (in absolute value) by b/2. Let (Tn)
be the increasing sequence of stopping times defined as follows: T0 := 0,
T 1n := inf
{
t > Tn−1 : [M,M ]t − [M,M ]Tn−1 ≥ b/2
}
,
T 2n := inf
{
t > Tn−1 :
∫ t
Tn−1
|dV | ≥ b/2},
Tn := T
1
n ∧ T 2n .
Then one has, for every n ∈ N,
[N,N ]Tn − [N,N ]Tn−1 ≤ b/2 + ∆[N,N ]Tn ≤ b/2 + (b/2)2 ≤ b/2 + b/2 = b,∫ Tn
Tn−1
|dV | ≤ b/2 + |∆VTn−1 | ≤ b/2 + b/2 = b,
therefore, setting S1 := ST1 and H1 = HT1 , by hypothesis there exists a unique càdlàg adapted process
X1 such that X1 = H1 + f(X1−) · S1. In particular, the process X1 solves (3.1.1) on [[0, T1]]. A unique
solution to (3.1.1) can thus be obtained by collating solutions constructed on [[Tn−1, Tn]], n ≥ 1.
3.1.5. Lemma. Assume that the hypotheses of theorem 3.1.1 are satisfied. The existence and uniqueness
of a solution to (3.1.1) holds if and only if it holds under the additional assumption that H = 0.
Proof. Introducing the function
f˜ : Ω× R+ × R −→ R
(ω, t, x) 7−→ f(ω, t, x+Ht(ω)),
X solves (3.1.1) if and only if X − H solves X˜ = f˜(widetildeX−) · S. Since f˜ satisfies the same
assumptions of f (in particular, it has the same Lipschitz constant), the latter equation admits a
unique solution by assumption.
3.1.6. Lemma. Assume that the hypotheses of theorem 3.1.1 are satisfied. The existence and uniqueness
of a solution to (3.1.1) holds if and only if it holds under the additional assumption that the function
f(·, ·, 0) is bounded.
Proof. Thanks to the previous lemma, we can assume H = 0. Introducing the increasing sequence of
stopping times (Tn) defined as
Tn := inf
{
t ≥ 0 : |f(·, t, 0)| ≥ n},
the function fn(ω, t, x) = f(ω, t, x)1]0,Tn(ω)](t) is bounded by n, thanks to the continuity on the left
of t 7→ f(ω, t, x) for every (ω, x). Therefore, setting Sn := STn , there exists a unique solution to
Xn = fn(Xn−) · Sn. The solution to (3.1.1) is obtained by defining a process X equal to Xn on [[0, Tn]]
for each n ∈ N.
We are now ready to prove the existence and uniqueness of solutions.
1For the moment we do not it as it requires some fine properties of stopping times
3.1. EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS 25
Proof of Theorem 3.1.1. In view of the above lemmata, we are going to assume that f , in addition to
(a), (b) and (c), is such that H = 0, f(·, ·, 0) < c for a constant c > 0, and S = M + V , where
[M,M ]∞ ≤ b,
∫ ∞
0
|dV | ≤ b, b < 1 ∧ 1
3C2
.
Let S be the Banach space of adapted càdlàg processes X with X0 = 0 such that X
∗
∞ ∈ L2(P), endowed
with the norm ∥∥X∥∥
S
:=
∥∥X∗∞∥∥2.
Let Φ be the map on S defined as
Φ: X −→ F (X−) · S.
We are going to show that Φ is a contracting endomorphism of S, i.e. that Φ(X) belongs to S for every
X ∈ S, and that there exists a constant α ∈ ]0, 1[ such that ‖Φ(X)−Φ(Y )‖S ≤ α‖X−Y ‖S. Equivalently,
we shall show that Φ(0) ∈ S and that Φ is contracting. In fact, Φ(0) = f(0) ·M + f(0) · V =: N + A,
where N is a local martingale with
[N,N ]∞ =
(
f(0)2 · [M,M ])
∞
≤ c2b,
hence, by Doob’s inequality, ∥∥N∥∥
S
=
∥∥N∗∞∥∥2 ≤ 2c√b.
Moreover,
A∗∞ ≤
∫ ∞
0
|f(0)| |dV | ≤ cb,
hence ‖A‖S ≤ cb, so that Φ(0) ∈ S. Similarly,
Φ(X)− Φ(Y ) = (f(X−)− f(Y−)) ·M + (f(X−)− f(Y−)) · V =: N +A,
where
[N,N ]∞ =
(
(f(X−)− f(Y−))2 · [M,M ]
)
∞
≤ bC2(X − Y )∗2∞,
hence, by Doob’s inequality,∥∥N∥∥
S
=
∥∥N∗∞∥∥2 ≤ 2∥∥[N,N ]1/2∞ ∥∥2 ≤ 2C√b∥∥X − Y ∥∥S.
Moreover,
A∗∞ ≤ C
∫ ∞
0
|X − Y | |dV | ≤ bC(X − Y )∗∞,
hence ‖A‖S ≤ bC‖X − Y ‖S, so that∥∥Φ(X)− Φ(Y )∥∥
S
≤ (2
√
b+ b)C
∥∥X − Y ∥∥
S
.
Choosing b such that α := (2
√
b+ b)C < 1, the Banach-Caccioppoli contraction principle implies that
Φ has a unique fixed point in S, that is, (3.1.1) has a unique solution in S. As a final step, we need to
show that there is no other adapted càdlàg solution (not necessarily belonging to S). To this purpose,
assuming that Y is a solution, i.e. Y = f(Y−) · S, the proof is completed if we show that Y ∈ S. Let
n ∈ N and T be the stopping time defined as
T := inf
{
t ≥ 0 : |Yt| ≥ n
}
,
so that |Y | ≤ n on [[0, T [[, and
|YT | ≤ |XT−|+ |f(·, T,XT−)|+ |∆ST |,
where |XT−| ≤ n and, by the Lipschitz continuity of f ,
|f(·, T,XT−)| ≤ |f(·, T, 0)|+ C|XT−| ≤ c+ Cn.
Moreover,
|∆ST | ≤ |∆MT |+ |∆VT | ≤ [M,M ]1/2∞ +
∫ ∞
0
|dV | ≤
√
b+ b,
which implies that Y T is bounded, in particular it belongs to S. Then Y T is such that Y T = f(Y T− ) ·ST .
Since ST satisfies the same assumptions on S, by the existence and uniqueness in S proved above, it
26 3. STOCHASTIC DIFFERENTIAL EQUATIONS
follows that X = Y on [[0, T ]]. Since this holds for every n ∈ N and T → ∞ as n → ∞, we conclude
that X = Y .
The existence and uniqueness result can be extended to systems of equations. If f : Ω×R+×Rd →
L(Rm,Rd) is a function such that f ij := 〈fei, ej〉 satisfies the measurability and Lipschitz continuity
assumptions of the previous theorem for every i = 1, . . . ,m and j = 1, . . . , d, and S = (S1, . . . , Sm)
is a vector of real semimartingales, then, for any Rd-valued càdlàg adapted process H, the equation
X = H + f(X−) · S admits a unique solution X. In coordinates, this can be written as
Xi = Hi +
m∑
j=1
f ij(X−) · Sj , i = 1, . . . , d.
The existence and uniqueness of solutions to “classical” equations of the type
Xt = X0 +
∫ t
0
b(s,Xs) ds+
∫ t
0
σ(s,Xs) dWs
is just a special case. Here X can also be Rd-valued and W can be Rm-valued.
CHAPTER 4
Da sistemare?
There is another decomposition of semimartingales that is very useful. We begin with local martingales.
4.0.1. Theorem. Let M be a local martingale. Then M = M0 +M
c +Md, where M c is a continuous
local martingale, and MdN is a local martingale for every continuous local martingale N .
The condition on Md is expressed saying that Md is a purely discontinuous local martingale. These
are defined by the property of being orthogonal to every continuous local martingale. Orthogonality
here has a special meaning (Cercare di dare interpretazione algebrica). Then we have
S = S0 +A+M
c +Md.
This decomposition is not unique, but M c is unique, hence it is usually denoted Sc and called the
continuous martingale component of S.
Recall Doob-Meyer: if X is local submartingale, there exists a unique increasing predictable process
A such that X − A is a local martingale. Now consider Xc, which is a continuous local martingale,
hence (Xc)2 is a local submartingale, hence there exists 〈Xc, Xc〉 such that (Xc)2 − 〈Xc, Xc〉 is a local
martingale.
Let M be a square integrable martingale, so that 〈M,M〉 exists (and is unique). One has
E
∣∣∣∣∫ t
0
Hs dMs
∣∣∣∣2 = E ∫ ∞
0
H2s d〈M,M〉s,
i.e. H 7→ H ·M is an isometry from L2(M) to L2(P). This holds for H elementary, but can be extended
by continuity and density to all processes in L2(M) that are predictable. If M is continuous, H ·M is
itself continuous.
Let A be a proces with finite variation, and define L(A) as the set of optional process H such that∫ t
0
|Hs(ω)| |dAs(ω)| <∞ ∀ω, t
Then H ·A can be defined pathwise and it adapted. If A and H are predictable, then H ·A is predictable.
This integral has the following properties:
• H ∈ L1(A) implies H ·A ∈ V;
• (H,A) 7→ H ·A is bilinear in the appropriate domains;
• ∥∥(H ·A)∞∥∥BV = ∫∞0 |H| |dA|;• Let D ∈ P. Then 1D · (H ·A) = (H1D) ·A;
• ∆(H ·A) = H∆A;
• Hn := H1|H|≤n, then Hn ·A→ H ·A in ucp.
Let M be local martingale and H simple:
H(t, ω) := Y0(ω)10(t) +
∑
k
Yk(ω)1]tk,tk+1](t)
(finite linear combinations of elementary functions): here Yk is Ftk -measurable. Then we set
(H ·M)t :=
∑
k
hk
(
Mt∧Tk+1 −Mt∧Tk
)
.
27
28 4. DA SISTEMARE?
Then H ·M is a local martingale and
[H ·M,H ·M ] = H2 · [M,M ].
Define L1(M) as the set of predictable processes H such that(
H2 · [M,M ])1/2 ∈ L1(P),
with the obvious identification of processes H and K.
4.0.2. Lemma. Simple processes are dense in L1(M).
4.0.3. Theorem (Burkholder-Davis-Gundy). Let M be a local martingale, T a stopping time, and
p ∈ [1,∞[. One has ∥∥M∗T∥∥Lp(P) hp ∥∥[M,M ]T∥∥Lp(P).
We can now localize, defining L1loc(M) as the set of predictable processes such that there exists a
increasing sequence of stopping times (Tn) such that H1[0,Tn] ∈ L1(M). Then the construction satisfies
the “patching” condition (
H1[0,Tn+1] ·M
)Tn
= H1[0,Tn] ·M.
We define the process H ·M by stipulating that it coincides with H1[0,Tn] ·M on [0, Tn] for all n. One
can show that H ·M does not depend on the localizing sequence, and is thus well defined.
Properties of this integral:
• L1loc(M) is a vector space.
• H 7→ H ·M is a linear map from L1loc(M) to the set of local martingales.
• (H,M) 7→ H · M is bilinear in the appropriate domains. For this one has to show that H ∈
L1loc(M1) ∩ L1loc(M2) implies that H ∈ L1(α1M1 + α2M2).
• (H ·M)T = H ·MT = H1[0,T ] ·M ;
• If D ∈ P, the associative property holds:
1D · (H ·M) = (H1D) ·M ;
• Hn ·M → H ·M in H1 (Hn is H cut at n);
• ∆(H ·M) = H∆M .
We can now look at the general case of integration against S = S0 + A +M , where A has locally
finite variation and M is a local martingale. We define L(S) as the intersection of L(A) and L1loc(M),
and
H · S = H ·A+H ·M.
The set L(S) is not empty because it contains all locally bounded processes, in particular all left-
continuous processes. The integral H · S does not depende on the decomposition.
If S is at the same time a local martingale and a process with finite variation, one should check that
H · S in the Stieltjes sense and in the sense of stochastic integration with respect to local martingales
coincide. This is in fact the case.
The stochastic integral has quite unexpected features: it is not true that H ∈ L1loc(M), M local
martingale implies that H · M is a local martingale. Similarly, one can find A finite variation and
H ∈ L(A) such thatH ·A is not of finite variation. All these “pathologies” involve unbounded integrands.
If the integrand H is locally bounded, we revert to the “expected” behavior.
4.0.4. Proposition. The mapping H 7→ H ·S, defined on the set of predictable locally bounded processes,
has the following properties:
1) H · S is a semimartingale;
2) H 7→ H · S is linear;
3) if S is a local martingale, then H · S is a local martingale;
4) if S has finite variation, then H · S is a Stieltjes integral;
5) (H · S)0 = 0 and H · S = H · (S − S0);
6) ∆(H · S) = H∆S;
4. DA SISTEMARE? 29
7) If K is predictable and locally bounded, then K · (H · S) = (KH) · S;
8) “passage to the limit under integral sign”: if Hn → H pointwise, |Hn| ≤ K, K predictable locally
bounded, then Hn · S converges in ucp to H · S, i.e.
sup
s∈[0,t]
|(Hn · S)s − (H · S)s| → 0
in P-measure as n→∞.
30 4. DA SISTEMARE?
4.1. Pezzi da sistemare
4.1.1. Quadratic variation
Let E be a normed space. An E-valued process X is of finite quadratic variation if there exists a finite
real-valued process [X,X] such that∑
pi
‖Xtk+1 −Xtk‖2E
|pi|→0−−−−→ [X,X]t
in probability for all t ≥ 0 and every partition pi of [0, t].
4.1.2. Loc mg
Local martingales have the following properties:
• M loc mart, T stopping time implies MT local martingale;
• Tn reduces M , (Sn) increasing (to ∞) seq of stopp times. Then T ∧ Sn is localizing sequence.
• Loc mg form a vector space.
4.1.3. H2
H2: cont mg bounded in L2, i.e. supt≥0 EM
2
t <∞. Modulo indisguishability.
H2 is Hilbert space with scalar product
〈M,N〉 := EM∞N∞ = E[M∞, N∞].
Let M ∈ H2. Set
L2(M) := L2(Ω× R+,R, dP⊗ d[M,M ]).
This is a Hilbert space.
4.1.4. Loc mg
Let M be a continuous local martingale. There exists a sequence (Tn) such that M
Tn is continuous
square-integrable martingale.
Sn := n ∧ inf
{
t ≥ 0 : (H2 · [M,M ])t ≥ n
}
.
Take Tn := Sn ∧Rn. Then MTn is cont sq int mg and H1[0,Tn] ∈ L2(MTn). Then
(H ·M)t(ω) := H1[0,Tn] ·MTn
on [[0, Tn]] :=
{
(ω, t) : 0 ≤ t ≤ Tn(ω)
}
. The definition is well posed, i.e. it depends only on H and M .
4.1.5. Loc mg
If M is a continuous local martingale of finite variation, then M = 0 a.s.
4.1.1. Theorem. Let M and N be continuous local martingales. Then there exists a P-a.s. unique
process [M,N ] of locally finite variation such that MN − [M,N ] is a local martingale. In particular,
every local martingale is of finite quadratic variation and its quadratic variation is increasing. The map
(M,N) 7→ [M,N ] is symmetric and bilinear. One has
[M,N ]T = [MT , N ] = [M,NT ]
for every stopping time T .
4.1.2. Proposition. Let (Mn) be a sequence of continuous local martingales. supt≥0|Mnt | → 0 in
probability if and only if [Mn,Mn]∞ → 0 in probability.
4.1.3. Theorem (Burkholder, Davis, and Gundy). Let M be a continuous local martingale. For every
p > 0, one has
E sup
t≥0
|Mt|p hp E[M,M ]p/2∞ .
Question: when is a local martingale a martingale?
4.1.4. Corollary. Let M be continuous local martingale such that E[M,M ]
1/2
∞ < ∞. Then M is a
uniformly integrable martingale.
4.1. PEZZI DA SISTEMARE 31
4.1.6. Stoch int
After the construction of stochatic integrals:
4.1.5. Proposition. Mn cont loc mg, Hn ∈ L2loc(Mn).
(Hn ·Mn)∗∞ → 0 if and only if ((Hn)2 · [Mn,Mn])∞ → 0
in probability.
Prove that the stochastic integral is associative.
4.1.7. Class (DL)
Let M denote the set of all stopping times.
4.1.6. Definition. A measurable stochastic process X is said to be
• of class (D) if the set {XT : T ∈M, T <∞} is uniformly integrable;
• of class (DL) is the set {XT : T ∈M, T ≤ t} is uniformly integrable for every t ≥ 0.
4.1.7. Theorem. A local martingale is a martingale if and only if it is of class (DL).
Proof. It is enough to prove sufficiency. Let M be a local martingale and (Tn) a sequence of localizing
stopping times. For any s ≤ t one has
MTns = E
[
MTnt
∣∣Fs] = E[Mt∧Tn ∣∣Fs],
and MTns → Ms, Mt∧Tn → Mt a.s. as n → ∞. Since (Mt∧Tn)n is uniformly integrable by hypothesis,
we also have (conditional) convergence in L1, hence
Ms = E
[
lim
n
Mt∧Tn
∣∣Fs] = E[Mt|Fs].

4.1.8. Terminology
A process X is square-integrable if Xt ∈ L2(P) for all t ≥ 0.
4.1.9. Esercizio sq integrable mg
4.1.8. Exercise. Let M be a square-integrable martingale. Then, for any s ≤ t,
E
[
(Mt −Ms)2
∣∣Fs] = E[M2t −M2s ∣∣Fs].
32 4. DA SISTEMARE?
4.2. Filtrations, stopping times, and martingales
This section is of technical nature and the reader should not be discouraged if he finds it, at first sight,
arid and difficult to understand. A possible advice is to skip it, on first reading, and to return to it
when needed.
4.2.1. On the notion of stochastic process
Given a probability space (Ω,F ,P) and a measurable space (E, E), usare insieme degli indici di tempo
generale a stochastic process is a collection (Xt)t∈R+ of E-valued random variables, i.e. Xt : Ω→ E is
F/E-measurable for all t ∈ R+. Given ω ∈ Ω, the map t 7→ Xt(ω) of R+ → E is called the trajectory,
or a path, of X associated to ω.
Non c’è nulla di specifico sulla “continuità”, e detto male. If E is a topological space, X is called
continuous if its trajectories are continuous for P-a.a. ω ∈ Ω. Right- and left-continuous processes are
defined completely analogously. We shall mostly view stochastic processes as maps of R+ × Ω to E,
and we shall say that X is measurable if (t, ω)→ Xt(ω) is B(R+)⊗F/E-measurable.
We shall need the following equivalence relations on the set of E-valued stochastic processes defined
on the same probability space:
(a) X and Y are a modification of eachother if the set {ω ∈ Ω : Xt(ω) 6= Yt(ω)} is negligible bisogna
dire cosa significa negligible for all t ∈ R+;
(b) X and Y are indistinguishable if the set
⋃
t∈R+
{
ω ∈ Ω : Xt(ω) 6= Yt(ω)
}
is negligible.
If E = R, a process that is indistinguishable from the zero process is called evanescent. It is obvious
that two indistinguishable processes are a modification of eachother, but the converse is in general not
true (find a counterexample!). On the other hand, the following very useful result holds.
4.2.1. Theorem. Let E be a Hausdorff topological space. Assume that X and Y are right-continuous
and a modification of eachother. Then X and Y are indistinguishable.
Proof. Let N ⊂ Ω be a negligible set such that, for all ω ∈ Ω \ N , the paths X(·, ω) and Y (·, ω) are
right-continuous and X(t, ω) = Y (t, ω) for all t ∈ Q+. Keeping ω fixed and passing to the limit shows
that X(t, ω) = Y (t, ω) for all t ∈ R+.
La dimostrazione precedente è troppo stringata. Ne faremo una molto analoga più oltre.
Da aggiungere: (vedi [DM]) Si può vedere proc stoc anche come variabile aleatoria in spazio di
funzioni indicizzate da t.
4.2.2. Filtrations and stopping times
4.2.2. Definition. Definire filtration.
Let us define
Ft− :=
∨
sFs, Ft− :=
⋂
s>t
Fs, F∞ :=
∨
t≥0
Ft,
with F0− := F0
A filtration F = (Ft)t∈[0,∞] is right-continuous if Ft = Ft+ for all t ∈ R+. It is clear that the
filtration (Ft+)t∈[0,∞] is right-continuous.
4.2.3. Definition. A stopping time (with respect to the filtration F) is a positive random variable
T : Ω→ [0,∞] such that {T ≤ t} ∈ Ft for all t ≥ 0.
4.2.4. Proposition. Assume that F is right-continuous. The following assertions are equivalent:
(a) T is a stopping time;
(b) {T < t} ∈ Ft for all t ∈ [0,∞];
(c) 1]0,T ] is adapted Controllare, poco chiaro
For an F-stopping time T , we shall also need the σ-algebra FT defined as
FT :=
{
A ∈ F : A ∩ {T ≤ t} ∈ Ft ∀t ≥ 0
}
.
4.2. FILTRATIONS, STOPPING TIMES, AND MARTINGALES 33
4.2.5. Proposition. Let E be a metric space, A ⊆ E closed, X coordinate process on C(R+, E),
DA(ω) := inf
{
t ≥ 0 : Xt(ω) ∈ A
}
.
Then DA is a stopping time with respect to σ(Xs;x ≤ t).
DA is called the entry time of A. On the other hand, the hitting time TA is stopping time only
with respect to F+.
Define, on the subset of {T <∞} ⊆ Ω,
XT (ω) := XT (ω)(ω).
Natural questions: is XT 1T<∞ a random variable? Is XT FT -measurable?
Given a filtered probability space (Ω,F ,F,P), we say that the filtration F = (Ft)t∈[0,∞] satisfies the
“usual conditions” if
(i) it is right-continuous, i.e. Ft = ∩u>tFu;
(ii) F0 containes the (F ,P)-negligible sets of F∞.
Recall that an event Γ is (F ,P)-negligible if there exists Γ′ ∈ F such that P(Γ′) = 0 and Γ ⊂ Γ′.
From now on, when we deal with continuous-time processes, we shall always assume that the reference
filtration satisfies the usual conditions. This hypothesis is needed, for example, to deduce that Y is
Ft-measurable if X = Y P-a.s. and X is Ft-measurable, or to guarantee that any martingale admits a
right-continuous version.
The filtration generated by a stochastic process X is not complete, in general. The filtration
obtained by completion is called the natural filtration of X, and denoted by FX .
4.2.6. Definition. A stochastic process X is F-adapted if Xt is Ft-measurable for all t ≥ 0.
In figurative words, one could say that an adapted process is non-anticipative. Clearly any process
is adapted to its natural filtration.
We shall require other notions of measurability for stochastic processes, that are more restrictive
than adaptedness.
4.2.7. Definition. A stochastic process X is progressively measurable if the map Ω × [0, t] ∋ (ω, s) 7→
Xs(ω) is Ft ⊗ B([0, t])-measurable for all t ≥ 0.
It is immediate that a progressively measurable process is adapted, while it is less obvious, although
not difficult, that an adapted process with values in a metric space and with left- or right-continuous
paths is progressively measurable.
4.2.8. Definition. The σ-algebra P ⊂ F ⊗ B(R+) generated by the set of all left-continuous adapted
real-valued processes is called the predictable σ-algebra. A stochastic process X that is measurable with
respect to P is called predictable.
Stopping times enjoy many useful properties, some of which are collected in the following Proposi-
tion.
4.2.9. Proposition. Let S and T be stopping times.
(a) T is FT -measurable;
(b) if X is a continuous adapted process, then XT is FT -measurable;
(c) S ∧ T is a stopping time;
(d) if S ≤ T P-a.s., then FS ⊆ FT .
The proof, though not difficult, is omitted.
34 4. DA SISTEMARE?
4.2.3. Martingales, local martingales and semimartingales
4.2.10. Definition. An F-adapted stochastic process M is a (F,P)-martingale if E|Mt| < ∞ for all
t ≥ 0 and Ms = E[Mt|Fs] for all s ≤ t. If the equality sign is replaced by ≤ or ≥, then M is called a
submartingale or a supermartingale, respectively.
4.2.11. Example. (i) Let ξ ∈ L1(P), and set Xt := E[ξ|Ft]. Then X is a martingale.
(ii) Let X˜ be a process with independent increments such that X˜t ∈ L1(P) for all t, and set Xt :=
X˜t − EX˜t. Then X is a martingale.
(iii) With X˜ as before, if X˜t ∈ L2(P) for all t, the process Y defined by Yt := X2t −EX2t is a martingale.
(iv) With X˜ as before and θ ∈ R, if exp(θX˜t) ∈ L1(P) for all t, then Z defined by
Zt :=
exp(θX˜t)
E exp(θX˜t)
is a martingale. In particular, if X˜ is a standard Wiener process, then it is a martingale, as well as Y
and Z defined as
Yt = W
2
t − t, Zt = exp
(
θWt − 1
2
θ2t
)
.
Jensen’s inequality immediately implies the following.
4.2.12. Proposition. Let f : R → R is a convex (concave) function and M is a martingale such
that f(Mt) ∈ L1(P) for all t. Then f(M) is a submartingale (supermartingale). Similarly, if M is a
submartingale and f is a convex increasing function such that f(Mt) ∈ L1(P) for all t, then f(M) is a
submartingale.
In particular, if M is a martingale, then |M | and M+ are submartingales. For any p ≥ 1, if
Mt ∈ Lp(P) for all t, the same holds for |M |p.
Given a stopping time T and a stochastic process X, we shall denote by XT the process defined by
XTt := XT∧t for all t ∈ R+.
4.2.13. Definition. An adapted process M is a local martingale if there exists a sequence of stopping
times (Tn)n∈N with Tn →∞ a.s. such that MTn is a martingale for every n ∈ N.
It is worth noting that there exist local martingales that are not martingales, for instance the
process X = 1/‖W‖, where W is an R3-valued Wiener process starting at W0 = (1, 0, 0).
The following simple lemma is very useful in connection with the no-arbitrage property.
4.2.14. Lemma. Let M be a local martingale such that M ≥ −α, with α ≥ 0 a fixed real number. Then
M is a supermartingale.
Proof. Exercise (apply Fatou’s lemma).
4.2.15. Definition. A stochastic process S is a semimartingale if there exist a local martingale M and
a process with integrable variation A such that S = M +A.
Throughout these notes we shall always assume that both M and A have continuous paths, i.e. we
shall deal only with continuous semimartingales.
4.2.4. Wiener processes
4.2.16. Definition. An Rd-valued Wiener process (or Brownian motion) is a stochastic process W with
continuous paths and with independent increments such that W0 = 0 and Wt−Ws ∼ N(0, (t− s)Q) for
any t ≥ s ≥ 0, where Q is a positive-definite symmetric d× d matrix.
The existence of Wiener processes is far from trivial, but we just accept it as a fact, and we limit
ourselves to noting that there exist by now many proofs.
The matrix Q (which can of course be identified with a linear endomorphism of Rd) is called the
covariance matrix (or operator) of the Wiener process W . If Q = I, then W is called the standard
d-dimensional Wiener process. Moreover, in this case one immediately sees that W 1,W 2, . . . ,W d are
one-dimensional independent standard Wiener processes.
4.2. FILTRATIONS, STOPPING TIMES, AND MARTINGALES 35
We are now going to prove that W (ω, ·) has infinite variation for P-a.a. ω ∈ Ω, but has finite
quadratic variation on any compact (time) interval. Before doing that, we need some preparations.
Recall that a decomposition pi of the interval [0, T ] is a set of real numbers {t0, t1, . . . , tn} such that
0 = t0 < t1 < · · · < tn−1 < tn = T . Let |pi| := maxk |tk+1 − tk|. Denoting by Π the set of all
decompositions of [0, T ], setting
Vpi(f) :=
n−1∑
k=0
∣∣f(tk+1)− f(tk)∣∣
for any f : [0, T ]→ R, the (first) variation of f is defined by
V (f) ≡
∫ T
0
|df | := sup
pi∈Π
Vpi(f).
The function f is said to be of bounded variation (BV), in symbols f ∈ BV (0, T ), if V (f) < ∞.
Analogously, we set
Jpi(f) :=
n−1∑
k=0
∣∣f(tk+1)− f(tk)∣∣2, [f, f ] := sup
pi∈Π
Jpi(f) = lim
|pi|→0
Jpi(f),
where [f, f ] is called the quadratic variation of f (over the interval [0, T ]). The following elementary
lemma shows that any continuous function with non-zero quadratic variation cannot have bounded
variation.
4.2.17. Lemma. Let f ∈ C([0, T ]) ∩BV (0, T ). Then [f, f ] = 0.
Proof. Since f is continuous on a compact set, then it is uniformly continuous, i.e. for any ε > 0 there
exists δ > 0 such that |t − s| < δ implies |f(t) − f(s)| < ε. Taking |pi| < δ implies Jpi(f) < εVpi(f),
hence [f, f ] = lim|pi|→0 Jpi(f) = 0.
4.2.18. Proposition. One has lim|pi|→0 Jpi(B) = T in L
2(Ω,F ,P). In particular, B(ω, ·) 6∈ BV (0, T )
P-a.s. for all T > 0.
Proof. Since B(tk+1)−B(tk) d= N
(
0, (tk+1−tk)1/2
)
, one has, writing Jpi in place of Jpi(B) for simplicity,
EJpi =
∑
k
E
∣∣f(tk+1)− f(tk)∣∣2 = T,
hence E|Jpi − T |2 = EJ2pi − T 2, where
EJ2pi = E
∣∣∣∑
k
∣∣B(tk+1)−B(tk)∣∣2∣∣∣2
=
∑
k
E
∣∣B(tk+1)−B(tk)∣∣4 +∑
j 6=k
E
∣∣B(tj+1)−B(tj)∣∣2∣∣B(tk+1)−B(tk)∣∣2.
By well-known properties of Gaussian laws we have E
∣∣B(tk+1) − B(tk)∣∣4 = 3(tk+1 − tk)2, and by
independence of increments
E
∣∣B(tj+1)−B(tj)∣∣2∣∣B(tk+1)−B(tk)∣∣2 = (tj+1 − tj)(tk+1 − tk),
therefore
EJ2pi = 2
∑
k
(tk+1 − tk)2 +
∑
k
(tk+1 − tk)2 +
∑
j 6=k
(tj+1 − tj)(tk+1 − tk)
= 2
∑
k
(tk+1 − tk)2 +
(∑
k
(tk+1 − tk)2
)2
= 2
∑
k
(tk+1 − tk)2 + T 2,
which immediately implies that E|Jpi−T |2 → 0 as |pi| → 0. The a.s. statement follows by an elementary
reasoning, recalling that convergence in L2(P) implies convergence P-a.s. along a subsequence.
36 4. DA SISTEMARE?
4.2.5. Stochastic integration with respect to a Wiener process
4.2.6
Since a continuous local martingale has BV paths if and only if it is a constant, it is evident that
pathwise integration (in the Lebesgue-Stieltjes sense) against local martingales cannot be performed.
4.2.19. Definition. A predictable process H : Ω×R+ → Rd with H0 = 0 is called a simple strategy if
there exist stopping times 0 ≤ T0 ≤ T1 ≤ · · · ≤ Tn < ∞ and random variables f0, f1, . . . , fn such that
fk is FTk -measurable for each k = 0, 1, . . . , n, and
H =
n−1∑
k=0
fk1]Tk,Tk+1].
A simple strategy is bounded if fk ∈ L∞ for all k. A simple strategy has bounded support if there exists
t ∈ R+ such that Tk ≤ t for all k.
If H is a simple strategy we can define
(H · S)∞ ≡
∫ ∞
0
Ht dSt :=
n−1∑
k=0
〈
fk, STk+1 − STk
〉
Rd
.
In order to extend this construction to general predictable integrands that are not simple, we shall
concentrate on the case that the integrator is a Wiener process W . The fundamental observation, due
to Itô, is that the following isometry holds for any simple bounded strategy H with bounded support:
‖(H ·W )∞‖L2(Ω,F,P) = ‖H‖L2(Ω×R+,P,P⊗Leb),
i.e.
E
(∫ ∞
0
Ht dWt
)2
= E
∫ ∞
0
‖Ht‖2 dt.
Now one can extend the isometry to the closure of the set of simple integrands in L2(P ⊗ Leb). In
particular, the closure of the set of simple bounded integrands with bounded support in L2(Ω×R+,P,P⊗
Leb) coincides with the whole space L2(Ω×R+,P,P⊗Leb). Moreover, the closure in L2(Ω,F ,P) of the
set of integrals (H ·W )∞, with H simple bounded with bounded support, is the set
{
f ∈ L2(Ω,F ,P) :
Ef = 0
}
.
The next step is to define the process R+ ∋ t 7→ (H · W )t. For this we need Doob’s maximal
inequality.
4.2.20.Theorem (Doob). For any local martingaleM withM0 = 0, one has ‖M∗∞‖L2(P) ≤ 2‖M∞‖L2(P),
that is
E sup
t∈R+
|Mt|2 ≤ 4E|M∞|2.
Let H ∈ L2(Ω×R+,P,P⊗Leb), and (Hn)n a sequence of bounded simple integrands with bounded
support such that
‖Hn −H‖L2(Ω×R+) ≤
1
4
1
4n
.
Then ∥∥∥ sup
t∈R+
(
(Hn −Hm) ·W )
t
∥∥∥
L2(Ω)
≤ 2∥∥((Hn −Hm) ·W )
∞
∥∥
L2(Ω)
= 2
∥∥Hn −Hm∥∥
L2(Ω×R+)
,
hence, by Chebyshev’s inequality,
P
(
sup
t∈R+
(
(Hn −Hm) ·W )
t
>
1
2n
)
<
1
2n
.
This implies, by Borel-Cantelli’s lemma, that t 7→ (Hn ·W )t converges uniformly P-a.s. as a sequence
of continuous functions. The (continuous!) limit process is denoted by H ·W .
The following remarkable results asserts that any local martingale with respect to the Brownian
filtration FW can be written as a stochastic integral with respect to W .
4.2. FILTRATIONS, STOPPING TIMES, AND MARTINGALES 37
4.2.21. Theorem (Martingale representation). Let M be an FW -local martingale. Then there exist a
predictable Rd-valued process H with
∫ t
0
‖Hs‖2 ds <∞ P-a.s. for all t ≥ 0 such that
Mt = M0 + (H ·W )t ∀t ≥ 0.
In particular, M has a continuous version.
Recalling the martingale representation theorem, if M = K ·W , then we set H ·M = (HK) ·W .
Moreover, if S is a continuous semimartingale with decomposition S = K ·W +A, we set
H · S := (HK) ·W +H ·A.
From now on we shall consider only continuous semimartingales of the form S = H ·W + A, where A
is an integral with respect to Lebsegue measure. Therefore, with some abuse of terminology, we shall
refer to such processes simply as continuous semimartingales.
4.2.7. Quadratic variation
Given a subdivision pi = {0 = t0, t1, . . . , tn = a} of the interval [0, a], we shall set |pi| := maxk |tk+1−tk|.
4.2.22. Definition. A real-valued process X has finite quadratic variation if there exists a finite process
[X,X] such that, for any t > 0 and any sequence (pin)n∈N of subdivisions of [0, t],
lim
|pin|→0
n−1∑
i=0
∣∣Xti+1 −Xti ∣∣2 = [X,X]t
in probability.
The quadratic covariation of two processes X and Y is defined similarly, requiring that there exists
a finite process [X,Y ] such that
lim
|pin|→0
n−1∑
i=0
(
Xti+1 −Xti
)(
Yti+1 − Yti
)
= [X,Y ]t
in probability. Note that it immediately follows by the definition that (X,Y ) 7→ [X,Y ] is a symmetric
and bilinear operation. In particular, the following polarization formula holds:
[X,Y ] =
1
4
(
[X + Y,X + Y ]− [X − Y,X − Y ]).
We shall need the following properties of quadratic (co)variation of continuous semimartingales.
4.2.23. Proposition. Let M , N ∈ Mcloc and A be a finite-variation process with continuous paths.
Then
(i) [M,N ]t is finite P-a.s. for all t and [M,N ] is continuous and of finite variation;
(ii) MN − [M,N ] ∈Mcloc;
(iii) if M and N are independent, then [M,N ] = 0;
(iv) [A,A] = 0 and [M,A] = 0.
(v) if M = K ·W , then [M,M ]t =
∫ t
0
K2s ds for all t ∈ R+.
Let S be an Rd-valued continuous semimartingale, and define the Md×d-valued process [[S, S]] by
[[S, S]]ij := [Si, Sj ]. Proposition 4.2.23 immediately yields [[S, S]] = [[M,M ]]. In particular, ifM = K ·W ,
one has
[[M,M ]]t =
∫ t
0
(
KsQ
1/2
)(
KsQ
1/2
)∗
ds.
38 4. DA SISTEMARE?
In fact, if the covariance operator of W is the identity, then
[[K ·W,K ·W ]]ijt =
[
(K ·W )i, (K ·W )j]
t
=
[∑
k
Kik ·W k,
∑
k
Kjk ·W k
]
t
=
∫ t
0
∑
k
Kiks K
jk
s ds =
∫ t
0
∑
k
Kiks (K
∗
s )
kj ds
=
(∫ t
0
KsK
∗
s ds
)ij
.
For the case of a general covariance operator Q, it is enough to recall that Q−1/2W is a Wiener process
with covariance operator equal to the identity, and to write K ·W = (KQ1/2) · (Q−1/2W ).
4.2.8. Itô’s formula
4.2.24. Proposition (Integration by parts). Let X, Y be continuous semimartingales. Then
XtYt = X0Y0 + (X · Y )t + (Y ·X)t + [X,Y ]t.
In particular,
X2t = X
2
0 + 2(X ·X)t + [X,X]t.
Proof. It is enough to prove the second assertion, as the first one follows by the second via polarization.
For a subdivision pi of [0, t], one has∑
k
(
Xtk+1 −Xtk
)2
=
∑
k
(
X2tk+1 + 2X
2
tk
−X2tk − 2XtkXtk+1
)
=
∑
k
(
X2tk+1 −X2tk
)− 2∑
k
Xtk
(
Xtk+1 −Xtk
)
= X2t −X20 − 2
∑
k
Xtk
(
Xtk+1 −Xtk
)
.
Since the left-hand side converges to [X,X]t in probability as |pi| → 0, the proof is completed if we show
that ∑
k
Xtk
(
Xtk+1 −Xtk
) P−→ (X ·X)t
as |pi| → 0. Note that∑
k
Xtk
(
Xtk+1 −Xtk
)
=
∑
k
Xtk
(
Atk+1 −Atk
)
+
∑
k
Xtk
(
(H ·W )tk+1 − (H ·W )tk
)
,
where the first term on the right-hand side converges to (X ·A)t because A has integrable variation and
X is continuous. Moreover, since
(H ·W )tk+1 − (H ·W )tk =
∫ tk+1
tk
Hs dWs,
one has ∑
k
Xtk
(
(H ·W )tk+1 − (H ·W )tk
)
=
∑
k
∫ tk+1
tk
XtkHs dWs =
∫ t
0
XpisHs dWs,
where Xpi is the discretization of X corresponding to pi. If XH ∈ L2(Ω × R+,P,P ⊗ Leb), since
XpiH → XH in L2(Ω× R+,P,P⊗ Leb) as |pi| → 0, we conclude∑
k
Xtk
(
Xtk+1 −Xtk
) P−→ ((XH) ·W )
t
+ (X ·A)t = (X ·X)t.
If XH 6∈ L2(Ω×R+,P,P⊗Leb), then one can proceed by localization, as X is continuous. The details
are omitted.
4.2. FILTRATIONS, STOPPING TIMES, AND MARTINGALES 39
Let f ∈ C2(Rd). Recall that, for any x0 ∈ Rd, one has Df(x0) ∈ L(Rd → R) ≃ Rd and D2f(x0) ∈
L2(Rd → R) ≃ Md×d. In other words, the (Fréchet, or total) derivative of f at any point x0, which
is a linear functional on Rd, can be represented by the gradient of f at x0, which is an element of R
d,
and, similarly, the second (Fréchet) derivative of f at x0, which is a bilinear function on R
d, can be
represented by its Hessian matrix at x0.
4.2.25. Theorem (Itô’s formula). Let S be an Rd-valued continuous semimartingale and f ∈ C2(Rd).
Then f(S) is a real continuous semimartingale and
f(St) = f(S0) +
∫ t
0
Df(Ss) dSs +
1
2
Tr
∫ t
0
D2f(Ss) d[[S, S]]s
≡ f(S0) +
d∑
i=1
∫ t
0
∂f
∂xi
f(Ss) dS
i
s +
1
2
d∑
i,j=1
∫ t
0
∂2f
∂xi ∂xj
f(Ss) d[S
i, Sj ]s
We shall restrict our attention to a special class of continuous semimartingales that often go under
the name of Itô processes, i.e. processes of the type
(4.2.1) St = x+
∫ t
0
Fs ds+
∫ t
0
Gs dWs ∀t ≥ 0,
or equivalently, using differential notation,
dSt = Ft dt+Gt dWt, X0 = x,
where F is an adapted Rd-valued process, G is a predictable L(Rm,Rd)-valued process, and W is an
Rm-valued Wiener process. Moreover, we assume that the following integrability conditions are satisfied:∫ ∞
0
(‖Fs‖Rd + ‖Gs‖2L(Rm,Rd)) ds <∞ P-a.s.
The processes F and G are often called the drift and the diffusion coefficients, respectively, of the
process S.
If S is an Itô process, then one immediately verifies that Itô’s formula can be written as follows:
f(St) = f(S0) +
∫ t
0
L˜Ss f(Ss) ds+
∫ t
0
Df(Ss)Gs dWs,
where L˜Ss is the (random) operator defined by
L˜Ss : C
2
b (R
d) ∋ φ 7→ DφFs + 1
2
Tr
[
D2φ
(
GsQ
1/2
)(
GsQ
1/2
)∗]
.
In the special case that
St = S0 +
∫ t
0
b(Ss) ds+
∫ t
0
σ(Ss) dWs,
where b : R+×Rd → Rd and σ : R+×Rd → L(Rm,Rd) are deterministic functions, Itô’s formula yields
f(St) = f(S0) +
∫ t
0
Lsf(Ss) ds+
∫ t
0
Df(Ss)σ(Ss) dWs,
where Ls is the deterministic differential operator on C
2
b (R
d) defined as
[Ltφ](x) := Dφ(x)bt(x) +
1
2
Tr
[
D2φ(x)
(
σt(x)Q
1/2
)(
σt(x)Q
1/2
)∗]
.
The time-dependent differential operator Lt is called the Kolmogorov operator associated to the diffusion
S and plays an essential role in the connection between second-order parabolic partial differential
equations and diffusion processes.
4.2.26. Exercise. Using Itô’s formula show that the following hold true:
(a) Mt := B
2
t − t is a martingale.
40 4. DA SISTEMARE?
(b) Let M be a continuous local martingale, and set
Zt := exp
(
Mt − 1
2
[M,M ]t
)
.
Then Z is a strictly positive local martingale.
The local martingale Z is called the stochastic exponential of M and is often denoted by E(M). If
M is a Brownian martingale, i.e. M = H ·W for some predictable process H, then one has that
Zt := exp
(∫ t
0
Hs dWs − 1
2
∫ t
0
‖HsQ1/2‖2 ds
)
is a local martingale. The following result, which gives conditions onM guaranteeing that the stochastic
exponential E(M) is a martingale (rather than just a local martingale), is very important (the proof is
omitted).
4.2.27.Theorem (Novikov). LetM be a continuous local martingale withM0 = 0 and E exp([M,M ]∞) <
∞. Then E(M) as a martingale.
Note that, if M = H ·W and H : [0, T ] × Ω → Rd, then Novikov’s criterion is satisfied if there
exists N > 0 such that ‖Ht(ω)‖ < N for all (t, ω).
4.2.28. Exercise. Assume that the Itô processes X1, X2 are defined by
X1t = x
1 +
∫ t
0
f1s ds+
∫ t
0
g1s dW
1
s ,
X2t = x
2 +
∫ t
0
f2s ds+
∫ t
0
g2s dW
2
s .
Obtain an expression for X1tX
2
t assuming that (a) W
1 ≡ W 2, (b) W 1, W 2 are independent, and (c)
W 1, W 2 are dependent.
4.2.29. Exercise. Let dSit = S
i
t(r dt + σi dW
i
t ), i = 1, 2, with W
1, W 2 independent. Setting S :=
(S1S2)1/2, show that
dSt = St
(
r − (σ21 + σ22)/8
)
dt+
1
2
(
σ1 dW
1 + σ2 dW
2
)
.
4.2.30. Exercise. Let
dXt = a(b−Xt) dt+ γYt dW 1t ,
dYt = c(δ −Xt) dt+ η
√
Yt dW
2
t .
Determine the Kolmogorov operator L associated to the process Z := (X,Y ).
Let us see how one can “remove the drift” in (4.2.1) using Girsanov’s theorem: assuming that Gt is
invertible P-a.s. for all t ∈ R+, one can write
dSt = Gt
(
dWt +G
−1
t Ft dt),
therefore, setting ht := −G−1t Ft and assuming enough integrability, one has that
W¯t := Wt +
∫ t
0
G−1s Fs ds
is a Q-Wiener process, hence
St = S0 +
∫ t
0
Gt dW¯t.
Note that using Girsanov’s theorem one can change the drift, but not the diffusion coefficient. Also
note that S is a Q-martingale if
EQ
∫ ∞
0
∥∥GsQ1/2∥∥2 ds = ELT ∫ ∞
0
∥∥GsQ1/2∥∥2 ds <∞.
4.2. FILTRATIONS, STOPPING TIMES, AND MARTINGALES 41
4.2.9. Connection with PDEs
Assume that X is such that
(4.2.2) Xt = x+
∫ t
0
b(s,Xs) ds+
∫ t
0
σ(s,Xs) dWs,
where
b : R+ × Rd → Rd, σ : R+ × Rd → L(Rm,Rd)
are deterministic functions and x ∈ Rd. One also says that X is a solution to the stochastic differential
equation (4.2.2).
Denoting by Lt the Kolmogorov operator associated to X, one observes that, at least on a purely
formal level, freely interchanging the order of differentiation and integration, the function u(t, x) :=
Ef(Xt), with f ∈ C2b (Rd), is expected to satisfy Kolmogorov’s partial differential equation
∂u
∂t
= Ltu, u(0, x) = f(x).
Let us now establish the representation formula by Feynman and Kac: assume that v ∈ C1,2([0, T ]×
Rd) satisfies the terminal value problem
(4.2.3)
∂v
∂t
+ Ltv = 0, v(T, x) = g(x).
Itô’s formula yields
g(XT ) = v(T,XT ) = v(t,Xt) +
∫ T
t
(∂sv + Lsv)(s,Xs) ds+
∫ T
t
Dv(s,Xs)σs(Xs) dWs.
Denoting by Et,x expectation conditional on Xt = x, we see that
Et,xg(XT ) = v(t, x).
We summarize stating a more general version (the proof is omitted, as it is a rather simple generalization
of the above argument).
4.2.31. Theorem (Feynmac-Kac). Let f , g, h be continuous functions. Assume that the terminal value
problem
∂tv + Ltv − hv + f = 0, v(T, x) = g(x),
admits a solution v ∈ C1,2([0, T [×Rd). Then
v(t, x) = Et,x
[
g(XT ) exp
(
−
∫ T
t
h(s,Xs) ds
)
+
∫ T
t
f(u,Xu) exp
(
−
∫ u
t
h(s,Xs) ds
)
du
]
It should be noted that we have not proved that the function u defined as u(t, x) := Et,xg(XT )
solves the terminal value problem (4.2.3)!
Proof of Feynman-Kac, assuming f = 0 for simplicity: recall IBP (for continuous-time processes)
XY = X · Y + Y ·X + [X,Y ].
Let Y be the FV process defined by
Yt := exp
(
−
∫ t
0
h(s,Xs) ds
)
.
Then we have, taking into account that [X,Y ] = 0,
v(t,Xt)Yt = v(0, X0) +
∫ t
0
−h(s,Xs)Ysv(s,Xs) ds+
∫ t
0
Ys dv(s,Xs) ds,
where (Itô)
v(s,Xs) = v(0, X0) +
∫ t
0
(∂s + L)v(s,Xs) ds+
∫ t
0
∂xv(s,Xs)σs dWs,
42 4. DA SISTEMARE?
hence
v(t,Xt)Yt = v(0, X0) +
∫ t
0
Ys(∂s + L− h)v(s,Xs) ds+
∫ t
0
Ys∂xv(s,Xs)σs dWs.
If ∂sv +Lv − hv = 0, then v(·, X)Y is a local martingale, and a true martingale if the integrand in the
stochastic integral is L2(Ω× [0, T ]). In this case,
Ytv(t,Xt) = E[YT v(T,XT )|Ft],
hence
v(t,Xt) = E
[
g(XT ) exp
(
−
∫ T
t
h(s,Xs) ds
)∣∣∣∣Ft].
Bibliography
1. C. Dellacherie, Capacités et processus stochastiques, Springer Verlag, Berlin-New York, 1972. MR 0448504
2. , Un survol de la théorie de l’intégrale stochastique, Stochastic Process. Appl. 10 (1980), no. 2, 115–144.
MR 587420
3. C. Dellacherie and P.-A. Meyer, Probabilités et potentiel. Chapitres V à VIII, Hermann, Paris, 1980. MR MR566768
(82b:60001)
4. Sheng Wu He, Jia Gang Wang, and Jia An Yan, Semimartingale theory and stochastic calculus, Kexue Chubanshe
(Science Press), Beijing; CRC Press, Boca Raton, FL, 1992. MR 1219534
5. J. Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin,
1979. MR MR542115 (81e:60053)
6. J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, second ed., Springer Verlag, Berlin, 2003.
MR MR1943877 (2003j:60001)
7. I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, second ed., Springer Verlag, New York, 1991.
MR 92h:60127
8. J.-F. Le Gall, Brownian motion, martingales, and stochastic calculus, Springer, 2016. MR 3497465
9. M. Métivier, Semimartingales, Walter de Gruyter & Co., Berlin, 1982. MR MR688144 (84i:60002)
10. M. Métivier and J. Pellaumail, Stochastic integration, Academic Press, New York, 1980. MR MR578177 (82b:60060)
11. Ph. E. Protter, Stochastic integration and differential equations, second ed., Springer Verlag, Berlin, 2004.
MR MR2020294 (2005k:60008)
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