Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach

2022/12/21 by Mateusz Kwaśnicki, Kwaśnicki, Mateusz
Decision Sciences · Mathematics · Economics, Econometrics and Finance · #Probability and Risk Models #Random Matrices and Applications #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2212.11390

Abstract

We study spectral-theoretic properties of non-self-adjoint operators arising in the study of one-dimensional Lévy processes with completely monotone jumps with a one-sided barrier. With no further assumptions, we provide an integral expression for the bivariate Laplace transform of the transition density p_t⁺(x, y) of the killed process in (0, ∞), and under a minor regularity condition, a generalised eigenfunction expansion is given for the corresponding transition operator P_t⁺. Assuming additionally appropriate growth of the characteristic exponent, we prove a generalised eigenfunction expansion of the transition density p_t⁺(x, y). Under similar conditions, we additionally show integral formulae for the cumulative distribution functions of the infimum and supremum functionals \underlineX_t and X_t. The class of processes covered by our results include many stable and stable-like Lévy processes, as well as many processes with Brownian components. Our results recover known expressions for the classical risk process, and provide similar integral formulae for some other simple examples of Lévy processes.

Related

• Transition densities of spectrally positive Lévy processes
• Exponential functionals of Lévy processes with jumps
• Short proofs in extrema of spectrally one sided Lévy processes
• Transition densities of one-dimensional Levy processes
• Fluctuation theory for Lévy processes with completely monotone jumps