Moments of polynomial functionals of spectrally positive L\'evy 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 $W(t)\equiv W_V(t)\equiv V+J(t)-t$, $t\geqslant 0$ 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 $\int_0^\tau W(t)\,{\rm d}t$ 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} L\'evy 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 $x\mapsto \int_0^{\tau_x}W_x(t)\,{\rm d}t$ defined on $x\in[0,\infty)$.