Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where:

$X_s$ is a wide sense stationary process with known probability density function, statistical moments, and (higher order) spectral density function(s).

$f(t)$ is a known real function.

$T>0$ is a constant and $0<t<T$.

$Y$ is an unknown random variable which satisfies $u(T)=0$.