## Expectation of time integral

I've 2 questions:

let be $W_s$ a standard Brownian motion:

• using Ito's formula show that $\left( W_t,\int_0^t W_sds \right)$ has a normal distribution;
• and calculate $E\left[e^{W_t}e^{\int_0^t W_sds} \right] .$

For the first part, i know that $W_t$ and $\int_0^t W_sds$ have normal distribution with mean and variance respectively $(0,t)$ and $(0, t^3/3)$, but i need help with Ito's formula.

For the second part i've tried to solve $E\left[e^{W_t}e^{\int_0^t W_sds} \right]= \iint e^{W_t}e^{\int_0^t W_sds} \;\phi \left( W_t,\int_0^t W_sds \right)\: dW_t \int_0^t W_sds$...

Is these the only way?

P.S. sorry for my poor english

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Standard homework. – Didier Piau Jan 9 2011 at 11:29
Those are homeworks... i want only to know if i'm right, or if it's possible to do something better than this. – Sephi Jan 9 2011 at 21:08
MO explicitly bans homework questions, see the FAQ page mathoverflow.net/faq. Your question would belong better on math.stackexchange.com. – Didier Piau Jan 9 2011 at 22:01
Right this a thread for math.stackexchange, can a moderator transfer this to the Math.SE forum ? – The Bridge Jan 10 2011 at 21:26