My problem is to model the dynamics of 6 state variables using GBM, in particular:- the exchange rate X - the foreign stock price S - simple compunded domestic forward rate d - simple compounded foreign forward rate f- the foreign ZCB Pf- the domestic ZCB Pdassuming we want also to model correlations between those variables do you suggest me to use a specific Browian motion for each variable or to consider an unique vector of Brownian motion for all varibles, allowing each variable to have specific sensitivities to each Brownian motion composing the vector?Note that I need to compute, through multiminesional Ito, dynamics of relative prices of such as (S*X), (Pf*X) and (Pd) to the domestic mmaRegards

I can be more clear. Imagine we have to model the dynamics of two quantities, for example the foreign stock price S and the exchange rate foreign/domestic. XThe simplest way to medel them is to write the SDE for S and for X each one following a geometric Brownian motionAllowing the Wiener process to be the same for the two processes, how can I model the correlation in this case? (would be too simplicistic?)What about the case of two different Wiener processes (let's say Ws and Wx)? What about the correlation between the two processes now?

"Different" BMs doesn't mean anything if you don't specify the joint distribution. On the other hand, using only one BM for the two SDEs would mean that there is a single risk factor in the market, which is too simplistic, so multiple BMs is a better idea.In fact you could either :- write two SDES with two BMs W^1_t and W^2_t that are correlated : E(dW^1 dW^2) = rho dt ;- or have a standard 2-dimensional BM : (B^1_t,B^2_t), that is with independent components, and write two SDEs for S_t and X_t, each of them containing terms in both B^1 and B^2.In fact both situations are essentially equivalent ; I'd choose the second one because I think standard BMs are easier to handle, e.g. w.r.t. Ito's formula, but this only my humble opinion.

Thank you for your clear explaination... just another questionIf I choose the second possibility (i.e. 2 dimendional BM) that mean I have two sources of risk in the market, and I want to model the correlaton between the two processes (in this case between the FX and the foreign stock), I will write dW*dW=rho *dt with dW a 2 dimensional BM... right?But what about BM with higher dimension (let's say n) so n risk factor in the market... the correlation shoud be as before dW*dW=rho*dt with rho a nxn matrix?Doing like this we don't have no more correlation between processes (S or X) but between risk factors... is that right?Thank you for your help!