The risk model with stochastic premiums, dependence and a threshold dividend strategy

TL;DR

This paper extends classical risk models by incorporating dependence via copulas and a threshold dividend strategy, deriving equations for ruin probabilities and dividends, with explicit formulas in special cases and numerical results.

Contribution

It introduces a generalized risk model with dependence structures and dividend strategies, providing explicit formulas and numerical illustrations for key quantities.

Findings

Explicit ruin probability formulas without dependence.

Expected discounted dividends in the absence of dependence.

Numerical illustrations demonstrating model behavior.

Abstract

The paper deals with a generalization of the risk model with stochastic premiums where dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by Farlie--Gumbel--Morgenstern copulas. In addition, dividends are paid to its shareholders according to a threshold dividend strategy. We derive integral and integro-differential equations for the Gerber--Shiu function and the expected discounted dividend payments until ruin. Next, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes. In particular, we find explicit formulas for the ruin probability in the model without either dividend payments or dependence as well as for the expected discounted dividend payments in the model without dependence. Finally, numerical illustrations are presented.