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Moments of polynomial functionals of spectrally positive Lévy processes

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  • Glynn, Peter
  • Jacobovic, Royi
  • Mandjes, Michel

Abstract

Let J(⋅) be a compound Poisson process with rate λ>0 and a jumps distribution G(⋅) concentrated on (0,∞). In addition, let V be a random variable which is distributed according to G(⋅) and independent from J(⋅). Define a new process W(t)≡WV(t)≡V+J(t)−t, t⩾0 and let τV be the first time that W(⋅) hits the origin. A long-standing open problem due to Iglehart (1971) and Cohen (1979) is to derive the moments of the functional ∫0τW(t)dt in terms of the moments of G(⋅) and λ. In the current work, we solve this problem in much greater generality, i.e., first by letting J(⋅) 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↦∫0τxWx(t)dt defined on x∈[0,∞).

Suggested Citation

  • Glynn, Peter & Jacobovic, Royi & Mandjes, Michel, 2025. "Moments of polynomial functionals of spectrally positive Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s030441492500167x
    DOI: 10.1016/j.spa.2025.104726
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