# Changing GeometricBrownianMotionProcess function

Since the function GeometricBrownianMotionProcess is given by Mathematica I have some technical questions. If we consider the following example:

ListLogPlot[Table[RandomFunction[
GeometricBrownianMotionProcess[0.5"(*" σ"*)",".4(*" μ"*)",10"(*" Startingvalue"*)"],
{0"(*" StartingTime"*)",14"(*" EndingTime"*)",".005(*" ΔTime"*)"}]["Path"],{1}],Joined→True]


whereas S constitutes the starting value.

Does this given function consider the following dynamics: $dS(t+1)=μS(t)dt+ σS(t) dB(t)$

Whereas $S(t+1)$ is determined by $S(t+1)=S(t) e^{(r-1/2 σ^2 )t+σB(t)}$?

If yes, is it possible to change the given GeometricBrownianMotionProcess which is represented by Bt and split it up into two processes such as, for example,

$B_t= \sqrt{ ρ} W_{t0} + \sqrt{ (1-ρ)} W_{ti}$

Where $W_{t0}$ and $W_{ti}$ are independent standard Brownian motions

Here $ρ$ is, for example, the mutual dependence among two variables created by the latent variable $W_{t0}$.

So is it possible to change the GeometricBrownianMotionProcess function this way or do I have to generate two independent Brownian motions and merge them together into $B_t$?

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Could this problem be somehow solved by the LogMultinormalDistribution function? – Milan Ivica Jun 13 '13 at 12:23