A dual risk model with additive and proportional gains: ruin probability and dividends

TL;DR

This paper analyzes a dual risk model incorporating both additive and proportional gains, deriving ruin probabilities, time to ruin, and dividend distributions, including effects of stochastic perturbations and general interarrival times.

Contribution

It introduces a dual risk model with proportional gains and provides explicit formulas for ruin probabilities and dividend distributions under various stochastic conditions.

Findings

Explicit ruin probability formulas derived.

Distribution of time to ruin characterized.

Dividend payment expectations calculated with stochastic perturbations.

Abstract

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ($i=1,2,\dots$) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature, that is, if the surplus process just before the $i$th arrival is at level $u$, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$. The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.