An example of stochastic volatility model:
$$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$
where $Z_t$ and $W_t$ are Brownian motions and $dZ_t dW_t = 0$.
Want to ask if $X_T$ and $g_T$ are uncorrelated for $T>t$? i.e. $E_t\left[ X_T g_T \right] = E_t\left[ X_T \right]E_t\left[ g_T \right] $ ?
Thanks a lot