Moments of polynomial functionals of spectrally positive Lévy processes

Abstract

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