Expectation of product of stochastic integral and predictable process

Question

I would like to simplify $E[X_t Y_t]$ where $X_t=\int_0^t x_sdWs$ and $Y_t=\int_0^t y_sds$ where $x_s$ and $y_s$ are square integrable (predictable??) processes adapted to the filtration generated by $W_s$. The expectation is taken from time 0.

Ito's lemma didn't get me far. If for instance $y_s$ was deterministic, then the expectation of $E[X_t Y_t]=0$. Is there a similarly simple solution when $y_s$ is not necessarily deterministic?

Thank you

Answer 1

It looks like you have got two process satisfying the SDE $$ \left\{ \begin{matrix} dX_t = x(t, X_t)dW_t \\ dY_t = y(t, W_t)dt \end{matrix}\right.$$

By multivariate Ito's lemma,

$$ dX_tY_t = X_ty(t, W_t) dt + Y_tx(t, X_t)dW_t $$

By integrating and taking expectation, we have

$$ E[X_tY_t] = E\left[\int_0^t X_sy(s,W_s)ds\right] = \int_0^t E[X_sy(s,W_s)]ds$$