Suppose $W_t$ and $B_t$ are two correlated standard Brownian processes such that $\mathbb{E}[dW_t\cdot dB_t] = \rho dt$. Suppose that $\sigma_1(s)$ and $\sigma_2(s)$ are two deterministic twice differentiable functions.
How can one compute expectation of the following product?
$$ \mathbb{E}\left[\exp\left(\int_0^t \sigma_1(s)dW_s\right)\cdot \int_0^t\sigma_2(s) dB_s \right] $$
and more general case
$$ \mathbb{E}\left[f\left(\int_0^t \sigma_1(s)dW_s\right)\cdot \int_0^t\sigma_2(s) dB_s \right] $$ where $f(x)$ is some nice infinitely differentiable function.