Geometric Brownian Motion

[Prometheus]

So i want to find the expectation of a geometric Brownian motion:

 

[math]E[e^{K^TW_t}][/math]

 

Where K is a constant vector and [math]W_t[/math] is a vector of normal Brownian motions, both of length n.

 

I assume that as [math]K^TW_t[/math] is a scalar I can just proceed in a similar fashion as the univariate case to get:

 

[math]e^{\frac{1}{2}K^TK}[/math]

 

But is it that simple or am i missing something as i suspect?

 

As always, help very much appreciated.

Edited December 23, 2014 by Prometheus

[Prometheus]

The above is nearly correct, just a small typo. The bit missing from my understanding was simply realising that [latex] K^TW_t = ||K||^2 [/latex]

 

Now stuck on a related question.

 

I now have [latex] e^{iK^TdW_t}[/latex], where [latex]dW_t[/latex] is an infinitesimal Brownian increment. I want to find the Taylor expansion of this - I would know how to proceed in the univariate case, but i don't understand the multivariate case. Does anyone have any hints or good links?