My question is related to Distribution of stochastic integral w.r.t. to centered Poisson process[this].
If I have an inhomogenous poisson process with intensity $\lambda(t)$ what is the distribution of $$X(t) = \int_0^t u(s) dN_s = \sum_{i=i}^{Nt}u(\tau_i)$$ where $u$ is a deterministic function and $\tau_i$ are jump times.
Specifically if I condition on $N_t = n$ and I try to calculate $$\mathbb{E}[X_t|N_t = n] = n * \int_0^tu(s)f_\tau(s)ds$$ for homogenous poisson process $f_\tau$ would be $U([0,t]$ i.e. $1/t$, how would this differ for the inhomogenous case. Thanks for any advice or pointers to any relevant material.