Parisian ruin for the dual risk process in discrete-time

TL;DR

This paper investigates Parisian ruin probabilities in a discrete-time dual risk model, deriving recursive formulas, explicit expressions, and exploring special cases like Binomial, Geometric, and Gambler's ruin models.

Contribution

It introduces new recursive and explicit formulas for Parisian ruin probabilities in discrete-time dual risk models, including special case analyses.

Findings

Derived recursive expression for finite-time Parisian ruin probability.

Obtained explicit formula for infinite-time Parisian ruin probability.

Analyzed special cases such as Binomial, Geometric, and Gambler's ruin models.

Abstract

In this paper we consider the Parisian ruin probabilities for the dual risk model in a discrete-time setting. By exploiting the strong Markov property of the risk process we derive a recursive expression for the fnite-time Parisian ruin probability, in terms of classic discrete-time dual ruin probabilities. Moreover, we obtain an explicit expression for the corresponding infnite-time Parisian ruin probability as a limiting case. In order to obtain more analytic results, we employ a conditioning argument and derive a new expression for the classic infinite-time ruin probability in the dual risk model and hence, an alternative form of the infnite-time Parisian ruin probability. Finally, we explore some interesting special cases, including the Binomial/Geometric model, and obtain a simple expression for the Parisian ruin probability of the Gambler's ruin problem.