Parisian ruin for the dual risk process in discrete-time

Zbigniew Palmowski,Lewis Ramsden,Apostolos D Papaioannou

doi:10.1007/s13385-018-0172-8

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 finite-time Parisian ruin probability, in terms of classic discrete-time dual ruin probabilities. Moreover, we obtain an explicit expression for the corresponding infinite-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 infinite-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.

Highlights

The compound binomial model, first proposed by Gerber [17], is a discrete-time analogue of the classic Cramér–Lundberg risk model which provides a more realistic analysis to the cash flows of an insurance firm
By exploiting the strong Markov property of the risk process we derive a recursive expression for the finite-time Parisian ruin probability, in terms of classic discrete-time dual ruin probabilities
In the compound binomial risk model, it is assumed that income is received via a periodic premium of size one, whilst the initial reserve and the claim amounts are assumed to be integer valued

Summary

Introduction

The compound binomial model, first proposed by Gerber [17], is a discrete-time analogue of the classic Cramér–Lundberg risk model which provides a more realistic analysis to the cash flows of an insurance firm. The aim of this paper is to extend the notion of ruin to the so-called Parisian ruin, which occurs if the process {R∗n}n∈N is strictly negative for a fixed number of periods r ∈ {1, 2, ...} and derive recursive and explicit expressions for the Parisian ruin probability in finite and infinite-time. Lemma 1 In the discrete-time dual risk model, the probability of hitting the zero level from initial capital u ∈ N, in n ∈ N periods, namely Pu( ∗ = n), is given by. = 1 − p0 (1, r + 1), Fig. 2 Equivalence between original and reflected risk processes where (u, t) is the classic finite-time survival probability in the compound binomial risk model, which has been extensively studied in the literature, (see [19] and references therein) and alternatively can be evaluated using Lemma 2. We use the above expressions to derive results for the infinite-time Parisian ruin probabilities, for which, as will be seen, a more analytic expression can be found

Infinite‐time Parisian ruin probability

An alternative approach to the infinite‐time dual ruin probability

Examples

Parisian ruin for the gambler’s ruin problem