## stochastic processes- Brownian motion

let $B_{t}$ be Brownian motion on R, $B_{0}&space;=&space;0$.
a) Prove that
$E\left&space;[&space;exp(iuB_{t})&space;\right&space;]&space;=&space;exp(&space;-&space;\frac&space;{1}{2}u^{2}t)&space;for&space;all&space;u\epsilon&space;R$
b) Use power series expansion of the exponential function on both sides, compare the terms with the same power of u and deduse that
$E\left&space;[&space;B_{t}^{4}&space;\right&space;]&space;=&space;3t^{2}$
c) Prove that
$E\left&space;[&space;f(B_{t})&space;\right&space;]&space;=&space;\frac&space;{1}{\sqrt&space;{2\pi&space;t}}\int&space;f\left&space;(&space;x&space;\right&space;)e^{&space;-&space;\frac&space;{x^{2}}{2t}}dx$
for all functions f such that the integral of the right converges.