Computation of ruin probabilities for general discrete-time Markov models

TL;DR

This paper introduces a novel method to compute ruin probabilities in discrete-time Markov risk models for any initial capital, providing precise approximations with explicit error bounds and applicability to heavy-tailed models.

Contribution

It offers a general technique for calculating ruin probabilities for any initial value using well-posed two-barrier approximations, unlike previous asymptotic methods.

Findings

Provides recursive and fixed-point equations for ruin probabilities.

Introduces a two-barrier approximation with explicit error bounds.

Demonstrates effectiveness through computational examples, including heavy-tailed models.

Abstract

We study the ruin problem over a risk process described by a discrete-time Markov model. In contrast to previous studies that focused on the asymptotic behaviour of ruin probabilities for large values of the initial capital, we provide a new technique to compute the quantity of interest for any initial value, and with any given precision. Rather than focusing on a particular model for risk processes, we give a general characterization of the ruin probability by providing corresponding recursions and fixpoint equations. Since such equations for the ruin probability are ill-posed in the sense that they do not allow for unique solutions, we approximate the ruin probability by a two-barrier ruin probability, for which fixpoint equations are well-posed. We also show how good the introduced approximation is by providing an explicit bound on the error and by characterizing the cases when the error converges to zero. The presented technique and results are supported by two computational examples over models known in the literature, one of which is extremely heavy-tailed.