Asymptotics for the ruin time of a piecewise exponential Markov process with jumps

TL;DR

This paper analyzes the asymptotic behavior of the ruin time for a class of Ornstein-Uhlenbeck processes with jumps driven by compound Poisson processes, deriving explicit formulas and limits for ruin probabilities and undershoot distributions.

Contribution

It introduces a method to compute the joint Laplace transform of passage time and undershoot for processes with mixed exponential jumps, extending previous models to include two-sided jumps and asymptotic analysis.

Findings

Explicit Laplace transform for passage time and undershoot.

Asymptotic ruin probability for negative drift as initial state grows.

Limit distribution of undershoot in the negative drift case.

Abstract

In this paper a class of Ornstein--Uhlenbeck processes driven by compound Poisson processes is considered. The jumps arrive with exponential waiting times and are allowed to be two-sided. The jumps are assumed to form an iid sequence with distribution a mixture (not necessarily convex) of exponential distributions, independent of everything else. The fact that downward jumps are allowed makes passage of a given lower level possible both by continuity and by a jump. The time of this passage and the possible undershoot (in the jump case) is considered. By finding partial eigenfunctions for the infinitesimal generator of the process, an expression for the joint Laplace transform of the passage time and the undershoot can be found. From the Laplace transform the ruin probability of ever crossing the level can be derived. When the drift is negative this probability is less than one and its asymptotic behaviour when the initial state of the process tends to infinity is determined explicitly. The situation where the level to cross decreases to minus infinity is more involved: The level to cross plays a much more fundamental role in the expression for the joint Laplace transform than the initial state of the process. The limit of the ruin probability in the positive drift case and the limit of the distribution of the undershoot in the negative drift case is derived.