Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process:
$$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$
Suppose $g \in L^2(\Omega \times [0,T])$. How, if it all, can one calculate $E\left[\int_0^T g(s) d\eta(s) \right]$?
My initial thought and there may be an error:
$E\left[\int_0^T g(s) d\eta(s) \right] = E\left[\int_0^T \lambda \eta(s) g(s) ds + \int_{0}^{T} \sigma g(s) dW_s\right] = E\left[\int_0^T \lambda \eta(s) g(s) ds \right] = \int_0^T E\left[\lambda \eta(s) g(s) \right]ds = \int_0^T \lambda g(s)E\left[ \eta(s) \right]ds = 0$