Moments of polynomial functionals of spectrally positive Lévy processes

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)≡W_V(t) ≡ V + J (t) −t, t ⩾ 0 and let τ_V 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 ∫₀^τW (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↦∫₀^τ_xW_x (t) dt defined on x∈ [0,∞).