Moments of polynomial functionals of spectrally positive Lévy processes

Abstract

Let $J\left(\cdot \right)$ be a compound Poisson process with rate $\lambda >0$ and a jumps distribution $G\left(\cdot \right)$ concentrated on $(0,\infty )$. In addition, let $V$ be a random variable which is distributed according to $G\left(\cdot \right)$ and independent from $J\left(\cdot \right)$. Define a new process $W\left(t\right)\equiv W_V\left(t\right)\equiv V+J\left(t\right)-t$, $t⩾0$ and let ${\tau }_V$ be the first time that $W\left(\cdot \right)$ hits the origin. A long-standing open problem due to Iglehart (1971) and Cohen (1979) is to derive the moments of the functional ${\int }_0^{\tau }W\left(t\right)dt$ in terms of the moments of $G\left(\cdot \right)$ and $\lambda$. In the current work, we solve this problem in much greater generality, i.e., first by letting $J\left(\cdot \right)$ belong to a wide class of spectrally positive Lévy processes and secondly, by considering more general class of functionals. We also supply several applications of the existing results, e.g., in studying the process $x\mapsto {\int }_0^{{\tau }_x}{W}_x\left(t\right)dt$ defined on $x\in [0,\infty )$.