Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

Olena Ragulina,Jonas Šiaulys

doi:10.3390/math8111885

Abstract

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples.

Highlights

One of the central objects investigated in risk theory is the ruin probability, which is the probability that the surplus of an insurance company becomes negative in some time interval and implies that the company is no longer able to reimburse claims
This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy
We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes

Summary

Introduction

One of the central objects investigated in risk theory is the ruin probability, which is the probability that the surplus of an insurance company becomes negative in some time interval and implies that the company is no longer able to reimburse claims. To calculate the ruin probability in different risk models, a great number of approaches has been proposed and studied recently (see the monographs [1,2,3,4,5,6] and references therein). Since Gerber and Shiu [7] introduced the expected discounted penalty function for the classical risk model, the ruin probability has often been investigated together with the surplus prior to ruin and the deficit at ruin in various risk models (see, e.g., [8,9,10,11,12,13,14] and references therein). Risk models where the insurance company pays dividends to its shareholders have attracted great interest since De Finetti [15] considered dividend strategies for a binomial model. We mention only a few papers [16,17,18,19,20,21,22,23,24] devoted to the investigation of risk models with different dividend strategies

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