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?

  • $\begingroup$ 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? $\endgroup$ – Magic is in the chain Oct 7 '18 at 14:18
  • $\begingroup$ what correlation do you want? do you want the correlation of returns? of log normal returns? of the brownian driving processes? $\endgroup$ – will Oct 7 '18 at 15:08
  • $\begingroup$ My omission. Between the Brownian Motion $dW_{2,t} $ and $dW_{1,t}$ $\endgroup$ – Coxswaiiiin Oct 8 '18 at 18:12
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    $\begingroup$ 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 $$ W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t, $$ $\endgroup$ – numerairX Oct 9 '18 at 13:55

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