Department of Computer Science, University College London
COMP0045 Probability Theory
and Stochastic Processes
Dr Guido Germano
Main summer assessment 2020–2021
Choose question 1 or 2 for further reading and write a brief report/essay of 3–4 pages.
Writing more will not necessarily be penalised, but sheer length does not automatically
lead to a higher mark; aim at quality. You are welcome to find more references than those
provided as a start.
You may use handwriting, LATEX, Pages, LibreOffice or other word processors. If you
choose to type, submit both the PDF and the source code (.tex, .bib, .docx, etc.).
The marking criteria are similar to those for the dissertation on your summer project,
that are described in detail on the Moodle pages of COMP0076/77, and for question 1 of
the coursework done in term 1.
Questions
1. Feller square-root, Bessel and Rayleigh processes
We have introduced in detail the Wiener and the Ornstein-Uhlenbeck processes, solving
the Fokker-Planck equation when the diffusion coefficient is constant and the drift term
is constant or linear. These processes are Gaussian and supported on the real numbers
R. In physics they are appropriate models e.g. for the coordinates and velocities of a
Brownian particle, which can be positive or negative.
In finance most quantities are positive (e.g. prices, interest rates, exchange rates,
volatilities, volumes) and thus processes to model them must be restricted to the semi-
infinite support R+ (interest rates may be slightly smaller than zero, but still have a lower
bound). A standard way of mapping to R+ a process X with support on R is to take
its exponential eX ; when X is the Wiener process or arithmetic Brownian motion W , the
result is geometric Brownian motion. Another possibility is to take the absolute value |X|
or the square X2; when X = W , the result is the Bessel process. The nomenclature is not
univocal in the literature: sometimes |W | is called the Bessel process and W 2 the squared
Bessel process, sometimes the Bessel process means W 2: you are welcome to include an
overview in your essay, explore why this process is called after Bessel, and recommend
which variant is more appropriate.
The generalised (squared) Bessel process is governed by the stochastic differential
equation
dY (t) = (a0 + a1Y (t))dt+ bY
β(t)dW (t)
that includes a linear drift and an exponent, for a total of four parameters. Depending
on the values of these parameters, special cases are arithmetic Brownian motion when
a1 = 0, β = 0; the (squared) Bessel process when a1 = 0 and β = 1/2; constant elasticity
UCL COMP0045 2020–2021 MSA 1 TURN OVER
of variance or Cox processes when a0 = 0, a1 > 0, that include geometric Brownian motion
when β = 0; the Ornstein-Uhlenbeck process when a1 < 0, β = 0; the Feller square-root
(FSR) process when a1 < 0, β = 1/2; and the Chan-Karolyi-Longstaff-Sanders (CKLS)
process when a1 < 0, β > 0. The FSR process is used in finance in the Cox-Ingersoll-
Ross model of the term structure of interest rates (1985) and for the volatility dynamics
in many stochastic volatility models starting from that of Heston (1993).
Explore this general picture and then focus on aspects of your choice, e.g. why the
process is called after Bessel, the derivation and properties of the Feller square-root pro-
cess, the Feller condition, the Rayleigh process that generalises |W | rather than W 2, etc.
A key reference is
William Feller, Annals of Mathematics 54 (1), 173-182, 1951, DOI 10.2307/1969318,
but you may use more modern ones instead. Other references can be found in Sections I
and II of
Edgar Martin, Ulrich Behn, Guido Germano, Physical Review 83 (5), 051115, 2011,
DOI 10.1103/PhysRevE.83.05111.
2. Feynman-Kac theorem
On page 104 of the lecture notes we have briefly seen the link between the Fokker-Planck
equation and stochastic differential equations, given by the Feynman-Kac theorem. Look
up a proof. If you find more proofs with the same or different conditions, you are welcome
to report them with a brief discussion and context. Simple and intuitive proofs are
preferred to excessively formal ones.
UCL COMP0045 2020–2021 MSA 2 END OF PAPER