Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Problemset 3.
Let x(t) be stochastic process on (, Ft , P ) such that x(t) is progressively measurable and
almost surely continuous. Moreover let
y (t) = x(t) x(0)
and
y 2 ( t)
t
0
b(s, x(s)ds
t
a(s, x(s)ds
0
be martingales with respect to (, Ft , P ).
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Take Home Final.
Due on Dec 18
Q1. Consider a population of xed size N that consists k types of size n1 (t), n2 (t), . . . , nk (t)
repectively in generation t. At generation t + 1 the population reproduces according to
the folowing rule. Each member of t
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Chapter 1
Brownian Motion
1.1
Stochastic Process
A stochastic process can be thought of in one of many equivalent ways. We
can begin with an underlying probability space (, , P ) and a real valued
stochastic process can be dened as a collection of random
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
4.3. RANDOM TIME CHANGE AND UNIQUENESS IN ONE DIMENSION.55
4.3
Random Time Change and Uniqueness in
One Dimension.
One of the properties of Martingales is Doobs stopping theorem. If M (t) is a
Martingale with respect to ( , Ft , P ) and 0 1 2 C are two bo
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
66
4.6
CHAPTER 4.
STOCHASTIC DIFFERENTIAL EQUATIONS.
Example of nonuniqueness.
If we try to construct a solution to the martingale problem in 1 dimension
coresponding to a(x) = x with 0 < < 1, it is easy to show nonuniqueness,
due to the nature of the
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Problemset 2.
Due Oct 16, 2000
Q1. Show that a martingale that is almost surely a continuous function of bounded variation, is a constant.
Q2. The Poisson process x(t), with rate 1, has the following properties.
1) x(t) t is a Martingale
2) (x(t) t)2 t is
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Problem Set 5.
Oct 30,2000
Let h 0 be given. Consider a Markov chain on R, with transition probability density
h (x, y ) =
1
1
exp[ (y x hb(x)2 ]
2h
2h
What is the Radon Nikodym derivative of this Markov chain with respect to the random
walk with transit
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
3.3. DIFFUSIONS AS STOCAHSTIC INTEGRALS
3.3
41
Diusions as Stocahstic Integrals
If ( , Ft , P ) is a probability space and () is a d dimensional Brownian Motion
relative to it, i.e. (t) is a diusion with parameters [0, I ] relative to ( , Ft , P ),
a stoc
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Due Oct 9, 2000
Q1. Let x(t) be a Gaussian process, with E [X (t)] = 0 and Cov[x(t)x(s)] = (s, t).
Let
(h) =
sup E [(x(t) x(s)2 ] =
0s,tT
tsh
If
sup [(t, t) + (s, s) 2(s, t)].
0s,tT
tsh
1
(h) C [log ]
h
for some C and > 1, show by the use of the GRR i
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Chapter 4
Stochastic Dierential
Equations.
4.1
Existence and Uniqueness.
Our goal in this chapter is to construct Markov Processes that are Diusions
in Rd corresponding to specied coecients a(t , x) = cfw_ai,j (t , x) and b(t , x) =
cfw_bi (t , x). Itos m
Advanced Topics in Numerical Analysis (Monte Carlo Methods)
MATH GA 2012

Fall 2012
Chapter 2
Diusion Processes
2.1
What is a Diusion Process?
When we want to model a stochastic process in continuous time it is almost
impossible to specify in some reasonable manner a consistent set of nite dimensional distributions. The one exception is