1. ## stochastic processes

let be Brownian motion on R, . Put
a) Prove
for all

b) Use power series expansion of the exponential function on both sides, compare the terms with the same power of u and deduse that

c) Prove that

for all functions f such that the integral of the right converges.

2. Originally Posted by elenaas
let be Brownian motion on R, . Put
a) Prove
for all

b) Use power series expansion of the exponential function on both sides, compare the terms with the same power of u and deduse that

c) Prove that

for all functions f such that the integral of the right converges.
a) and c) are easy consequences of the fact that $\displaystyle B_t$ is a Gaussian random variable of mean 0 and variance $\displaystyle t$. As for b), you already have a hint given, you should specify what causes you difficulty.

3. I have a problem with a)
if is a random variable with density then for any (well behaved) function :

From that:

So I have a problem to calculate this integral
I tried with integration by parts and got more complicated integral.
So what do you think?

4. Originally Posted by elenaas
I have a problem with a)
if is a random variable with density then for any (well behaved) function :

From that:

So I have a problem to calculate this integral
I tried with integration by parts and got more complicated integral.
So what do you think?
I think your teacher assumes you already know the characteristic function of a Gaussian random variable: if $\displaystyle X\sim \mathcal{N}(\mu,\sigma^2)$, then $\displaystyle \Phi_X(t)=E[e^{itX}]=e^{it\mu-t^2\sigma^2/2}$. This is well-known but not quite easy to prove, you can't find it by direct computation. One way is to see that, if $\displaystyle X\sim \mathcal{N}(0,\sigma^2)$ and $\displaystyle f(t)=E[e^{itX}]$, then $\displaystyle f'(t)=-\sigma^2 t f(t)$ (you need to differentiate and then integrate by part), and $\displaystyle f(0)=1$, hence $\displaystyle f(t)=e^{-\sigma^2 t^2/2}$. The case $\displaystyle \mathcal{N}(\mu,\sigma^2)$ follows. But, once again: you can't be expected to invent that, this is the kind of thing one is supposed to know (not the proof, but the formula).

I solved a) and c) but now I have difficulty to prove that $E\left&space;[&space;B_{t}^{2k}&space;\right&space;]=\frac{\left&space;(&space;2k&space;\right&space;)!}{2^{k}k!}t^{k};k\in&space;N$
which implies b).
What do you think?

6. Originally Posted by elenaas
I solved a) and c) but now I have difficulty to prove that $E\left&space;[&space;B_{t}^{2k}&space;\right&space;]=\frac{\left&space;(&space;2k&space;\right&space;)!}{2^{k}k!}t^{k};k\in&space;N$
which implies b).
What do you think?
From a), you have $\displaystyle E[e^{iuB_t}]=e^{-\frac{u^2 t}{2}}$, hence $\displaystyle E[\sum_{n=0}^\infty \frac{(iuB_t)^n}{n!}]=\sum_{k=0}^\infty \frac{(-u^2 t)^k}{2^k k!}$ by using the power series expansion of the exponentials. Using the bounded convergence theorem, you can justify that $\displaystyle E[\sum_{n=0}^\infty \frac{(iuB_t)^n}{n!}]=\sum_{n=0}^\infty E [\frac{(iuB_t)^n}{n!}]$. So that you get, for all real $\displaystyle u$,

$\displaystyle \sum_{n=0}^\infty \frac{i^n E[B_t^n]}{n!}u^n = \sum_{k=0}^\infty \frac{(-1)^k t^k}{2^k k!}u^{2k}$.

Identifying coefficients gives $\displaystyle E[B_t^n]=0$ if $\displaystyle n$ is odd, and your formula if $\displaystyle n=2k$.

7. Thank you!