Ruin probabilities as recurrence sequences in a discrete-time risk process

TL;DR

This paper uses linear recurrence sequences to derive an exact formula for the ultimate ruin probability in a discrete-time risk process with claims bounded by a fixed integer, providing numerical examples and approximations.

Contribution

It introduces a novel application of recurrence sequence theory to compute ruin probabilities for bounded claim distributions in discrete-time risk models.

Findings

Exact ruin probability formula derived using polynomial zeroes.

Numerical results demonstrate the method's effectiveness.

Approximate ruin probabilities are obtained from the exact formula.

Abstract

We apply the theory of linear recurrence sequences to find an expression for the ultimate ruin probability in a discrete-time risk process. We assume the claims follow an arbitrary distribution with support $\{0,1,\ldots,m\}$, for some integer $m\ge2$. The method requires to find the zeroes of an $m$ degree polynomial and to solve a system of $m$ linear equations. An approximation is derived from the exact ruin formula and several numerical results and plots are provided as examples.