The University of Sydney
School of Mathematics and Statistics
Assignment 2: Stochastic Analysis
MATH4512: Semester 2, 2023
Lecturer: Anna Aksamit
Due on Wednesday 25nd October at 11:59pm on TurnitIn
1. Let (Wt,Ft) be a Brownian motion.
(a) Find f(x) such that f(Wt + t) is a martingale.
(b) Find an increasing process A such that f(Wt + t) = BAt for some Brownian motion B.
2. Consider a diffusion X with drift coefficient µ(x) = cx and diffusion coefficient σ(x) = 1. Give its
generator and show that X2t − 2c
∫ t
0
X2sds− t is a martingale.
3. Let (Wt) be a Brownian motion, (Lt) be its local time at 0 and (St) be its running supremum.
(a) Is the process Yt = StW
4
t a semimartingale? If yes, find its semimartingale decomposition.
(b) Show that the process Zt = L
3/2
t − 32L1/2t |Wt| is a local martingale.
4. Let W be a Brownian motion and (Ft) be its natural filtration on (Ω,F ,P).
(a) Find the predictable representation of W 51 with respect to W .
(b) Show that there is a unique solution X satisfying the following SDE with σ > 0{
dXs = σXsdWs
X0 = 1.
Then prove that X is a true martingale.
(c) Show that (Ft) has PRP with respect to X.
(d) Find the predictable representation of W 51 with respect to X.
(e) Define Q by dQdP |Ft = Xt. Does (Ft) have PRP under Q? If the answer is yes, provide a Q-martingale
having PRP for (Ft).
5. Question 35 in MRLN Let W be a one-dimensional standard Brownian motion defined on a filtered
probability space (Ω, (Ft),P). Consider the following SDEs
dXt = 3X
1
3 dt+ 3X
2
3 dWt, X0 = 0, (1)
and {
dY 1t = − 12Y 1t dt+ sgn(Y 2t )
√
1− (Y 1t )2dWt, Y 10 = 0
dY 2t = − 12Y 2t dt− sgn(Y 1t )
√
1− (Y 2t )2dWt, Y 20 = 1
(2)
Apply Itoˆ’s formula to answer the following questions.
(a) Show that the process Xt =W
3
t satisfies (1).
(b) Show that the process (Y 1t , Y
2
t ) = (sin(Wt), cos(Wt)) satisfies (2).
(c) Find the semimartingale representation of the process Zt = (Y
1
t )
2 and its quadratic variation ⟨Z⟩t.
(d) Find the local martingale part of the process Ut = exp(Xt) and the cross-variation between Z and
U .
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