Right now, I want to figure out the covariance of a stochastic integral and a riemann integral in the following form:
$$E\left(\int_{0}^{t}\exp[B(t)-B(s)]ds \cdot \int_{0}^{t}\exp[B(t)-B(s)]dW(s)\right).$$ where $B(\cdot)$ and $W(\cdot)$ are independent Brownian motions, or $d\langle B, W \rangle_t = \gamma dt$.
My guess to this is 0 and my calculation is
$$E\left(\int_{0}^{t}\exp[B(t)-B(s)]ds \cdot \int_{0}^{t}\exp[B(t)-B(s)]dW(s)\right) = E\left(\int_{0}^{t}\exp[B(t)-B(s)]\int_{0}^{t}\exp[B(t)-B(r)]dr dW(s)\right) = 0 $$ by virtue of $E(\int \cdot dW) = 0$. Am I correct?