# Correlated GBM and OU processes

I want to model two different stochastic processes, such that:

$$X_t , V_t$$ are correlated with coefficient $$\rho$$. Where:

$$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$$ and $$dV_t=\theta(\mu_2-V_t)dt+\sigma_2 dW_{2,t}$$.

Is there any source(book or paper) to address this problem?

• Could you please elaborate what the purpose of the Model is? Is it to drive some analytical formula, simulate something, or study the behaviour of some functions of these variables? – Magic is in the chain Oct 7 '18 at 14:18
• what correlation do you want? do you want the correlation of returns? of log normal returns? of the brownian driving processes? – will Oct 7 '18 at 15:08
• My omission. Between the Brownian Motion $dW_{2,t}$ and $dW_{1,t}$ – alexbougias Oct 8 '18 at 18:12
• I'm not sure if this is what you're looking for? if you want correlation $\rho$ between two Brownian motions, it's modeled as $$W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t,$$ – numerairX Oct 9 '18 at 13:55