On maxima and ladder processes for a dense class of Levy processes

TL;DR

This paper develops a method to approximate the first passage time law of general Levy processes using phase-type jump processes, providing explicit formulas and efficient computation techniques.

Contribution

It introduces a novel approximation approach for Levy processes' passage times using phase-type jumps and derives explicit formulas via Markov additive process embedding.

Findings

Approximation of Levy process passage times by phase-type jump processes

Explicit formulas for passage time laws and ladder processes

Fast convergence of the iterative Wiener-Hopf factorisation solution

Abstract

Consider the problem to explicitly calculate the law of the first passage time T(a) of a general Levy process Z above a positive level a. In this paper it is shown that the law of T(a) can be approximated arbitrarily closely by the laws of T^n(a), the corresponding first passages time for X^n, where (X^n)_n is a sequence of Levy processes whose positive jumps follow a phase-type distribution. Subsequently, explicit expressions are derived for the laws of T^n(a) and the upward ladder process of X^n. The derivation is based on an embedding of X^n into a class of Markov additive processes and on the solution of the fundamental (matrix) Wiener-Hopf factorisation for this class. This Wiener-Hopf factorisation can be computed explicitly by solving iteratively a certain fixed point equation. It is shown that, typically, this iteration converges geometrically fast.