On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model

TL;DR

This paper simplifies the calculation of the ultimate ruin probability in the renewal risk model by expressing it through roots of the probability generating function, under specific assumptions on the involved discrete random variables.

Contribution

It provides a new explicit formula for survival probability using roots of the generating function, simplifying the classical Pollaczek-Khinchine approach under certain assumptions.

Findings

Expressibility of survival probability via roots of the generating function.

Simplified formula under assumptions of independence and finite support.

Numerical illustrations for specific distributions.

Abstract

In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model. More precisely, we show the expressibility of the distribution function $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-c\theta_i)<u\right),\,u\in\mathbb{N}_0 $$ via the roots of the probability generating function $G_{X-c\theta}(s)=1$, the expectation $\mathbb{E}(X-c\theta)$, and the probability mass function of $X-c\theta$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $c\theta_1,\,c\theta_2,\,\ldots$ are independent copies of $X$ and $c\theta$ respectively, $c>0$, $X$ and $c\theta$ are independent non-negative and integer-valued, and the support of $\theta$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions.