I'm trying to improve my understanding on inhomogeneous Poisson processes. In particular, I'm focusing on stochastic integrals of the form $$I(T)=\int_0^Tf(t)dW_t$$ with $W_t$ arrival times of a Poisson process with intensity $\lambda_t$. Most of all, I would like to compute expectation values of such integrals, namely $\mathbb{E}[I]$, $\mathbb{E}[I^2]$.
I've collected some results from related problems:
- Covariance with $f(t)=1$
- Attempt to compute $I(T)$ with $\lambda_t=\mu e^{\lambda t},\mu>0$ and $f(t)=e^{\alpha (T-t)}, \alpha>0$
- Calculating $\mathbb{E}[I]$
- Using Rieman-Stieltjes integration for $\mathbb{E}[I]$
And among all computing the moment generating function $\mathbb{E}[e^{\beta I}]$ for an homogeneous Poisson process.
However none of them provide explicitly $\mathbb{E}[I^2]$, or $\mathbb{E}[e^{\beta I}]$ for the inhomogeneous case. I would appreciate if someone would provide examples for specific $f(t), \lambda(t)$, explicit generic expressions, or instructions to compute them myself.
Note: this is related to an attempt in a less precise approach, which apparently leads to contradictory results.