Consider a stochastic process $X_t$ \begin{equation} dX_t = a(X_t)dt + \sigma dW_t \end{equation} where $W_t$ is a Wiener Process.
I know the expectation value of the product of two stochastic integrals can be written as the expectation value of a time integral.
\begin{equation} \mathbb{E}\Big[\Big( \int^{\tau}_0 F(X_t) dW_t \Big) \cdot \Big(\int^{\tau}_0 G(X_t) dW_t \Big) \Big] = \mathbb{E}\Big[\int^{\tau}_0 F(X_t)\cdot G(X_t) dt \Big] \end{equation}
I am trying to evaluate the following expectation value of the product of a time integral and a stochastic integral:
\begin{equation} C(\tau) = \mathbb{E}\Big[\Big( \int^{\tau}_0 f(X_t) dt \Big) \cdot \Big(\int^{\tau}_0 g(X_t) dW_t \Big) \Big] \end{equation} where $f(X_t)$ and $g(X_t)$ are arbitrary functions of $X_t$, and the stochastic integral is interpreted as an Ito integral.
Is it possible to write $C(\tau)$ as an expectation value of a time integral or a double time integral? \begin{equation} C(\tau) = \mathbb{E}\Big[ \int^{\tau}_0 A(X_t) dt \Big] \quad \textrm{or} \quad \mathbb{E}\Big[\int^{\tau}_0 \int^{\alpha(t)}_0 A(X_t) \cdot B(X_s) ds dt \Big] \end{equation}