# MGF for two correlated wiener processes

#### MathCrash

Heads-up this forms part of an assignment question so I'm really just looking for a foothold on how to go about solving it.

Given two standard Brownian Motions such that
$image=http://latex.codecogs.com/png.latex?\large&space;\newline&space;Cov(Bt,Ws)&space;=&space;\rho&space;min(t,s),&space;where\&space;\rho&space;\ne&space;0&hash=e29c4eec54c14eeb3cb1f52e243ee402$

Find the MGF:
$image=http://latex.codecogs.com/png.latex?\large&space;\phi&space;(u)&space;=&space;E\begin{pmatrix}&space;e^{uB_tW_s}&space;\end{pmatrix},\&space;for\&space;0&space;<&space;s&space;<&space;t&hash=c7db77e116795eed5ebf3c75e5876d28$

My thoughts...
I know the distribution for Bt & Ws, so my first thought is to at least define the double integral (based on the joint pdf for a bivariate normal distribution, given I know the mean/var for both the Bt & Ws processes, as well as their correlation coeff.). I can then try to solve that

Is there an alternate/more-sane approach? Perhaps involving application of Ito's Lemma/stoch.calc rules?

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#### chiro

MHF Helper
Hey MathCrash.

I think you should do it the way you intended. There are results for MGF's involving multi-variate normal distributions which means if you get stuck, just look it up and follow the proof of how the results were obtained.

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