On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model
Andrius Grigutis
TL;DR
The paper addresses exact computation of the ultimate survival probability in E. Sparre Andersen's renewal risk model under discrete, integer-valued claim sizes and finite-support inter-arrival times. It shows that $\varphi(u)$ can be expressed in closed form using the roots of $G_{X-c\theta}(s)=1$, the mean $\mathbb{E}(X-c\theta)$, and the pmf of $X-c\theta$, and it develops a finite-time recurrence $\varphi(u,T)$ with a root-based linear system to obtain the initial values. A generating function $\Xi(s)=\sum_{i\ge0}\varphi(i+1)s^i$ is derived, relating to $G_{\mathcal M}(s)$ and the distribution of an auxiliary variable $\mathcal M$, and explicit formulas for the initial probabilities $\pi_i$ are provided when the roots are simple. The work includes numerical examples demonstrating exact survival probabilities for various discrete distributions and discusses the impact of truncating inter-arrival times, offering a practical approach for precise ruin probabilities in discrete renewal risk models.
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 $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-cθ_i)<u\right),\,u\in\mathbb{N}_0 $$ via the roots of the probability generating function $G_{X-cθ}(s)=1$, the expectation $\mathbb{E}(X-cθ)$, and the probability mass function of $X-cθ$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $cθ_1,\,cθ_2,\,\ldots$ are independent copies of $X$ and $cθ$ respectively, $c>0$, $X$ and $cθ$ are independent non-negative and integer-valued, and the support of $θ$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions.