# Thread: Expectation with Brownian motion

1. ## Expectation with Brownian motion

I need to find $\mathbb{E}\left[e^{W(1)+W(2)}\right]$ , where $W$ is a standard Brownian motion.

I tried the following.

$W(1)+W(2)=(W(2)-W(1))+2W(1)$

Brownian motion has independent increments, and I should be able to write the above sum as two independent standard normal variables, i.e.

$(W(2)-W(1))+2W(1)=G_{1}+2G_{2}$, where $G_1$ and $G_2$ are independet standard normal variables.

Thus...

$\mathbb{E}\left[e^{W(1)+W(2)}\right]= \mathbb{E}\left[e^{G_{1}+2G_2}\right]=\mathbb{E}\left[e^{G_1}\right] \cdot \mathbb{E}\left[e^{2G_2}\right] = e^{1/2} \cdot e^{2} = e^{\frac{5}{2}}$

In the last line I used the fact that independent random variables have factoring moment-generating function and also the fact that $\mathbb{E}\left[e^{uG}\right] = e^{\frac{1}{2}u^2}$.

Is the above reasoning correct, if not, what is wrong? Thanks!

2. your reasoning up to the last line is fine.

I have not seen the result you use on the last line so i can comment on that.