Exact expression of ultimate time survival probability in homogeneous discrete-time risk model

Abstract

In this work, we set up the generating function of the ultimate time survival probability $ \varphi(u+1) $, where $ \varphi(u) = \mathbb{P}\left(\sup\limits_{n\geqslant 1}\sum\limits_{i = 1}^{n}\left(X_i- \kappa\right)<u\right), $ $ u\in\mathbb{N}_0, \, \kappa\in\mathbb{N} $ and the random walk $ \left\{\sum_{i = 1}^{n}X_i, \, n\in\mathbb{N}\right\} $ consists of independent and identically distributed random variables $ X_i $, which are non-negative and integer-valued. We also give expressions of $ \varphi(u) $ via the roots of certain polynomials. The probability $ \varphi(u) $ means that the stochastic process $ u+ \kappa n-\sum\limits_{i = 1}^{n}X_i $ is positive for all $ n\in\mathbb{N} $, where a certain growth is illustrated by the deterministic part $ u+ \kappa n $ and decrease is given by the subtracted random part $ \sum_{i = 1}^{n}X_i $. Based on the proven theoretical statements, we give several examples of $ \varphi(u) $ and its generating function expressions, when random variables $ X_i $ admit Bernoulli, geometric and some other distributions.