7CCMFMStochastic Analysis04 (CMFM04)

In all of the following questions, let W = (W(t) : t > 0) be a real-valued Brow-
nian motion on a probability space (
;A ; P) and let fFtgt>0 be the augmented
ltration generated by W.
1. (a) State the de nition of a simple stochastic process on an interval [0; T] and
its stochastic integral with respect to W. [30%]
(b) Let A 2 F0. Which of the following stochastic processes are simple on
the interval [0; 2]: [10%]
(1) H(t)(!) = t 1A(!) + t 1fjW(1)j61g(!)
(2) H(t)(!) = 1[0;1](t) 1fjW(1)j61g(!) + 1(1;2](t) 1A(!)
(3) H(t)(!) = 1[0;1](t) 1A(!) + 1(1;2](t) jW(1)(!)j
(4) H(t)(!) = 1[0;1](t) 1A(!) + 1(1;2](t) 1fjW(t)j61g(!)
(c) Consider the stochastic integral [30%]
I(t) :=
Z t
0
s dW(s) for t > 0:
(1) Calculate E[I(3)] and Var[I(3)] .
(3) Calculate Cov(I(2); I(3)).
(d) Prove for a simple stochastic process

H(t) : t 2 [0; T]

that
Z t
0
H(s) dW(s) : t 2 [0; T]


de nes a martingale with respect to fFtgt2[0;T]. [30%]

2. For a constant x 2 R de ne a stochastic process (X(t) : t > 0) by
X(t) = x +W(t) for all t > 0;
and for some constants a < x < b de ne the stopping time
:= infft > 0 : X(t) = 2 (a; b)g:
(a) State the de nition of a Brownian motion. [20%]
(b) Prove that (X(t) : t > 0) is a martingale with respect to fFtgt>0. [15%]
(c) Prove that

(X(t) x)2
t : t > 0) is a martingale with respect to
fFtgt>0. [15%]
(d) Prove that [25%]
P(X( ) = a) = b x
b a
and P(X( ) = b) = x a
b a
:
(e) Calculate E[ ]. You may use the fact E[ ] < 1 without proof. [25%]

3. Consider the stochastic di erential equation in R2
:

dX(t)
dY (t)

=

1
2X(t)
1
2Y (t)

dt +

Y (t)
X(t)

dW(t) for t 2 [0; T];

X(0)
Y (0)

=

1
0

:
(1)
and de ne stochastic processes

X(t) : t 2 [0; T]

and

Y (t) : t 2 [0; T]

by
X(t) = cos

W(t)

; Y (t) = sin

W(t)

for all t 2 [0; T]:
(a) Show that there exists a unique solution of (1). [25%]
(b) Show that the unique solution of (1) is given by

(X(t); Y (t)) : t 2 [0; T]

.
[25%]
(c) Determine the stochastic di erential equation which is satis ed by (Z(t) :
t 2 [0; T]), where Z(t) = X(t)Y (t) for all t 2 [0; T]. [25%]
(d) Determine Cov

X(t); Y (t)

for t 2 [0; T]. [25%] EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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