Why does this expression involving covariance hold?

Question

Why does the following expression where $W$ is a brownian motion and $\gamma$ is a constant hold?

$Cov[W(t)\int_{0}^{s}e^{-\gamma(s-k)}dW_{k}]=\int_{0}^{min(t,s)}e^{-\gamma(s-k)}dk$

I am assuming it is due to a property similair to: $\mathbb{E}[W(t)W(s)]=min(t,s)$ but now quite sure what property is used in this case.

Answer 1

This is a consequence of Itô isometry :

For two $L^2$ stochastic processes $X$ and $Y$ $$ \operatorname {E} \left[\left(\int _{0}^{T}X_{u}\,\mathrm {d} W_{u}\right)\left(\int _{0}^{T}Y_{u}\,\mathrm {d} W_{u}\right)\right]=\operatorname {E} \left[\int _{0}^{T}X_{u}Y_{u}\,\mathrm {d} u\right]$$

applied to $X_u :=\mathbf1_{\left\{u\le t \wedge s \right\} } $ and $Y_u:=\mathbf1_{\left\{u\le t \wedge s \right\} }e^{-\gamma(s-u)} $