Let $X_t = \int^t_0 \sigma_1(s)\,dW(s)$ and $Y_t = \int^t_0 \sigma_2(s)\,dW(s)$ be two stochastic processes. I am interested in simplifying the product $X_tY_t$. In the presence of an expectation we know $E(X_tY_t)= E\int^t_0 \sigma_1(s)\sigma_2(s)\,ds$. But, in its absence can anything be said about $X_t Y_t$?
Using the Ito's product rule we can say that $$X_tY_t = \int^t_0 \left( \int^s_0 \sigma_1(u)\,dW(u) \right) \sigma_2(s) \,dW(s) + \int^t_0 \left( \int^s_0 \sigma_2(u)\,dW(u) \right) \sigma_1(s) \,dW(s)+ \int^t_0 \sigma_1(s) \sigma_2(s)\,ds.$$ Can we have a further simplification not involving the stochastic integral?