Math 285, Spring 2009
Homework 10, not to be handed in
1. Exercise 9.5, page 229 of the text.
2. Exercise 9.6, page 230 of the text.
3. Let Wt,t≥0, be standard one-dimensional Brownian motion, with W0= 0. Define, for t≥0
and k= 0,1,2,...,
mk(t) = E[W2k
t].
Use Itˆo’s formula (and then take expectations—the martingale terms have mean zero!) to show
that
mk(t) = t(2k−1)mk−1(t), t > 0, k = 1,2,3,....
Deduce by recursion that
mk(t) = 1 ·3·5···(2k−1)tk, t > 0,k = 1,2,....
4. Let Wt,t≥0, be standard one-dimensional Brownian motion, with W0= 0. Let fand gbe
(non-random) continuous functions on [0,1] and suppose there are constants Cand Dsuch that
C+Z1
0
f(s)dWs=D+Z1
0
g(s)dWs.
Show that C=Dand that f(s) = g(s) for all s∈[0,1].
[Hints: The martingale Mt=Rt
0[f(s)−g(s)] dWssatisfies M0= 0 and M1=D−C. Deduce from
this that C=D. Consequently, M1= 0. Now use the formula for the variance of a stochastic
integral to see that R1
0[f(s)−g(s)]2ds = 0.]
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