Stats 219  Stochastic Processes
Homework Set 7, Fall 2009
1. Exercise 4.2.4
(a) This can be easily proved by replacing
t
+
h
with
t
and
t
with s in the identity as given in hints.
Note that
A
t
is a deterministic process and hence comes out of the conditioning.
(b) Guassian ensures square integrable, independent increments come from the fact that
Cov
(
M
t
+
h

M
t
, M
t
) = 0, by martingale property.
(c) This is readily seen from (a) and (b)
2. Exercise 4.2.5
. Let
G
t
denote the canonical filtration of a Brownian motion
W
t
.
(a)Show that for any
λ
∈
IR, the S.P.
M
t
(
λ
) = exp(
λW
t

λ
2
t/
2), is a continuous time martingale with
respect to
G
t
.
(b) Explain why
d
k
dλ
k
M
t
(
λ
) are also martingales with respect to
G
t
.
(c) Compute the first three derivatives in
λ
of
M
t
(
λ
) at
λ
= 0 and deduce that the S.P.
W
2
t

t
and
W
3
t

3
tW
t
are also MGs.
ANS:
(a) let
Y
t
=
e
λW
t

λ
2
(
t/
2)
.
Then,
E

Y
t

=
e

λ
2
(
t/
2)
E
[
e
λW
t
] which since
W
t
is a Gaussian ran
dom variable, we know to be finite.
Also,
E
e
λ
(
W
t
+
h

W
t
)
=
e
λ
2
h/
2
yielding the identity
E
[
Y
t
+
h
F
t
] =
e

λ
2
(
t/
2)+
λW
t
=
Y
t
so
Y
t
is a martingale.
(b) The filtration in question does not involve
λ
and differential operator and expectation operator is
clearly exchangeable due to the nice nature of exponential function,i.e.
d
d
λ
M
s
(
λ
) =
d
d
λ
E
[
M
t
(
λ
)
F
s
] =
E
[
d
d
λ
M
t
(
λ
)
F
s
]. Hence
d
d
λ
M
t
(
λ
) is a martingale. Likewise, we repetitively apply differential operator to
prove that all the derivatives are still martingales.
(c)
M
0
t
(0) = exp(
λW
t

λ
2
t/
2)(
W
t

tλ
)

0
=
W
t
M
00
t
(0) = exp(
λW
t

λ
2
t/
2)(
W
t

tλ
)
2
+ exp(
λW
t

λ
2
t/
2)(

t
)

0
=
W
2
t

t
M
000
t
(0) = exp(
λW
t

λ
2
t/
2)(
W
t

tλ
)
3
+exp(
λW
t

λ
2
t/
2)(

2)(
W
t

tλ
)
λ
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 Spring '09
 Brownian Motion, WT, Martingale, Stopping time, optional stopping theorem

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