# Thread: MGF for two correlated wiener processes

1. ## MGF for two correlated wiener processes

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 $\large&space;\newline&space;Cov(Bt,Ws)&space;=&space;\rho&space;min(t,s),&space;where\&space;\rho&space;\ne&space;0$

Find the MGF: $\large&space;\phi&space;(u)&space;=&space;E\begin{pmatrix}&space;e^{uB_tW_s}&space;\end{pmatrix},\&space;for\&space;0&space;<&space;s&space;<&space;t$

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?

2. ## Re: MGF for two correlated wiener processes

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.