Moments of polynomial functionals of spectrally positive Lévy processes

TL;DR

This work provides a constructive, recursive framework for computing high-order joint moments of functionals A_f and D_g for the area under a spectrally positive Lévy-driven process W_V(t) up to its hitting time of zero. Central to the approach is a closed first-moment formula and a Takács-type recursion that expresses higher-order moments in terms of model primitives (V moments, jump measure ν, and Lévy parameters c, σ), extended to polynomial functionals and generalized to mixed moments via a Gamma-type representation. The paper also develops a detailed analysis of the cost process A_f(x) as a function of the initial level x, including Lipschitz continuity, an auto-covariance structure, and a differential equation for moments in the Id case, and it provides a transform-based treatment for the area under reflected workloads up to an exponential time. Demonstrations show that the framework recovers classical results (Iglehart, Cohen) and yields new joint-moment identities, with broad potential applications in regenerative models, queueing (M/G/1) contexts, and insurance risk. Overall, the results deliver a versatile toolkit for exact and semi-analytic moment computations of area-type functionals in Lévy-driven systems.

Abstract

Let $J(\cdot)$ be a compound Poisson process with rate $λ>0$ 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 $τ_V$ 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 $\int_0^τW(t)\,{\rm d}t$ in terms of the moments of $G(\cdot)$ and $λ$. 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 $x\mapsto \int_0^{τ_x}W_x(t)\,{\rm d}t$ defined on $x\in[0,\infty)$.

Figures (1)

• Figure 1: An illustration of $X_\theta(\cdot)$ when $r=1$ and $J(\cdot)$ is a CPP with nonnegative jumps. Note that conditionally on an undershoot which equals $\delta$, the mirror image of $\left\{X_\theta(t):\tau_\theta\leqslant t\leqslant \zeta_\theta\right\}$ with respect to the horizontal axis is distributed as $\delta+J(t)-t$, revealing the relation to the current work.

Theorems & Definitions (42)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Theorem 2
• Theorem 3: pasta
• proof
• Corollary 1
• proof