Suppose we have the following set of differential equations:
$$ \left\{\begin{array}{ll} dr(t) = a(\theta(t) - r(t))dt + \sigma_r dW_1(t) \\ dS(t) = r(t)S(t)dt + \sigma_SS(t)dW_2(t) \end{array} \right.$$ where $W_1(t)$ and $W_2(t)$ are correlated Brownian motions with correlation coefficient $\rho$, the function $\theta(t)$ is deterministic, $a \in \mathbb{R}$, $\sigma_r > 0$ and $\sigma_S >0$ are given constants. I would like to find an expression for $\mathbb{E}[S(t)]$. I got that $$ r(t) = e^{-at}\left(r(0) + a\int_0^t\theta(s)e^{as}ds + \sigma_r\int_0^te^{as}dW_1(s) \right)$$ $$ S(t) = S(0)\exp\left\{ \int_0^t r(x)dx - \frac{\sigma_S^2t}{2} + \sigma_SW_2(t)\right\} $$ so I also calculated $$ \int_0^tr(x) dx = \frac{r(0)}{a}(1-e^{-at}) + \int_0^t \theta(s)(1-e^{a(s-t)})ds + \frac{\sigma_r}{a}\int_0^t(1-e^{a(s-t)})dW_1(s) $$