Following pp. 36-39 I have stucked with calculating covariance structure of the process $$ Z(z) = Z_N(z) - \int _{(0, z)} Z_Y(y) d\Lambda (y), $$ where $Z_N$ and $Z_Y$ are zero mean Gaussian processes and $\Lambda$ is some measure (p. 36) without "jumps" for simplicity (that is $\Delta \Lambda = 0$).
By Fubini's theorem: $\mathbb E Z(z) = \mathbb EZ_N(z) - \mathbb E\int _{(0, z)} Z_Y d\Lambda = 0 - \int _{(0,z)}\mathbb EZ_Y d\Lambda = 0$. Hence, $$ \mathrm{cov}[Z(s), Z(t)] = \mathbb E\left[ Z_N(s) - \int _{(0, s)} Z_Y d\Lambda \right]\left[Z_N(t) - \int _{(0, t)} Z_Y d\Lambda \right] $$ By direct calculation: $\mathbb E[Z_N(s)Z_N(t)] = Z_N(s\wedge t) - Z_N(s) Z_N(t)$.
Please, give me an advise how can I deal with: $$ \mathbb E \left[\int _{(0, s)} Z_Y d\Lambda \int _{(0, t)} Z_Y d\Lambda \right] $$ and $$ \mathbb E\left[ Z_N(s) \int _{(0, t)} Z_Y d\Lambda \right] $$ in order to obtain the final result?