Exercises 2.5 It^o's isometry Let us consider a deterministic function f : R ! R.

1. Prove that EP[

Z t 0

f(s)dWs] = 0 EP[

Z t 0

f(s)dWs

2

] =

Z t 0

f(s)2ds Hint: Assume that f(s) can be approximated by simple functions (s) =

i1si

These expressions can be extended to the case where f(s; !) is an adapted process. We have the It^o isometry formula:

EP[

Z t 0

f(s; !)dWs] = 0 EP[

Z t 0

f(s; !)dWs

2

] =

Z t 0

EP[f(s; !)2]ds 2. Let Z 2 N(0; 2) be a normal r.v. with zero mean and variance equal to 2. Prove that E[eZ] = e

2 2

3. Deduce that EP[e R t 0 f(s)dWs ] = e 1

2 R t 0 f(s)2ds