M3508 Exercises 8
November 2008
1. Let 0 = t0 < t1 < · · · < tN = 1 is a partition of [0, 1], and let W (t) be
Brownian motion.
(a) Prove that
N −1
E W (ti ) W (ti+1 ) − W (ti )) = 0.
i=1
(b) What is
N −1
E W (ti+1 )(W (ti+1 ) − W (ti )) ?
i=1
2. (a) Prove directly from the definitions that
T T
tdW (t) = T W (T ) − W (t)dt.
0 0
T
[Hint: Fix a partition π, and write down the approximations for 0 tdW (t)
T
and 0 W (t)dt with respect to π. Can you rearrange anything?]