I am interested in modelling risks associated with events that occur randomly and have longevity to their costs.
The model setup I have is suppose $X_i$ ~ $\operatorname{PoissProcess}(\lambda)$ and each time $X_i$ occurs there is a exponentially decaying cost $e^{-\theta}$. I am interested in the Expected cost between times $t_2$ and $t_1$ but I'm not sure how to find a closed form solution (or if one exists). I can do it if $N(t_2 - t_1) = 1$ but with $N(t_2 - t_1) > 1$ it becomes less clear.
Any help would be much appreciated
Edit I've simplified the problem to a function of finding $\Sigma \mathbb{E}[\mathbb{e}^{\theta s_i}]$ where if given there are K occurrences between $t_1$ and $t_2$ then $s_i$ denotes the time of occurence $i$ with $t_1$ < $s_1$ < $s_2$ < ... < $s_K$ < $t_2$.