M5A42 APPLIED STOCHASTIC PROCESSES
PROBLEM SHEET 3
Term 1 2010-2011
1. Let W (t) be a standard one dimensional Brownian motion. Calculate the following expectations.
(a) EeiW (t) .
(b) Eei(W (t)+W (s) , t, s, (0, +).
(c) E(
(d) Ee
n
2
i=1 ci W (ti ) ,
Pn
1.
(i) Since X(t) and Y (t) are Gaussian, it is enough to show that the two processes have the same
mean and covariance as W (t). Clearly, we have EX(t) = 0, EY (t) = 0. Furthermore,
1
1
E(X(t)X(s) = E(W (ct)W (cs) = min(ct, cs) = min(t, s).
c
c
Similarly
M5A42 APPLIED STOCHASTIC PROCESSES
PROBLEM SHEET 1 SOLUTIONS
Term 1 2010-2011
1. Calculate the mean, variance and characteristic function of the following probability density functions.
(a) The exponential distribution with density
ex x > 0,
0
x < 0,
f (x
1.
(i) The pdf of W (t) is
g(x, t) =
1 x2 /2t
e
.
2t
The covariance function of W (t) is
R(t, s) = min(t, s).
[5] SEEN
(ii) Since g(x, t) is an even function of x and x3 is an odd function we immediately get that
E(W (t)3 = 0. To calculate the fourth mom
1.
(i) (a) We have
m
EXn =
aj =: b
j=1
and
m
Var(Xn ) =
2
a2 .
j
j=1
Now we calculate the covariance. Set k = k
m
ai ni+j
E[(Xn b)(Xn+v b)] = E
i=1
aj n+vi+j+1
j=1
2 (am amv
=
m
+ . . . av+1 a1 ),
0
if v m 1,
if v m.
Hence, the covariance between Xn
M5A42 APPLIED STOCHASTIC PROCESSES
PROBLEM SHEET 5 SOLUTIONS
Term 1 2010-2011
1. Consider the Brownian motion with D =
The Fokker-Planck equation is
1
2
on the interval [0,1] with reecting boundary conditions.
p
1 2p
=
,
t
2 x2
x p(0, t) = x p(1, t) = 0.
M5A42 APPLIED STOCHASTIC PROCESSES
PROBLEM SHEET 6 SOLUTIONS
Term 1 2010-2011
1. Let a, D be positive constants and let X(t) be the diffusion process on [0, 1] with periodic boundary
conditions and with drift and diffusion coefcients a(x) = a and b(x) = 2
M5A42 APPLIED STOCHASTIC PROCESSES
PROBLEM SHEET 4 SOLUTIONS
Term 1 2010-2011
1. Let Wt be a one dimensional Brownian motion and let , > 0 and dene
St = et+Wt .
(a) Calculate the mean and the variance of St .
(b) Calculate the probability density function