The risk model with stochastic premiums and a multi-layer dividend strategy

Abstract

The paper deals with a generalization of the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. First of all, we derive piecewise integro-differential equations for the Gerber-Shiu function and the expected discounted dividend payments until ruin. In addition, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes and find explicit formulas for the ruin probability as well as for the expected discounted dividend payments. Lastly, numerical illustrations for some multi-layer dividend strategies are presented.Keywords Risk model with stochastic premiums, multi-layer dividend strategy, Gerber-Shiu function, expected discounted dividend payments, ruin probability, piecewise integro-differential equation 2010 MSC 91B30, 60G51

Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy

“…The ruin probability of an insurance company is one of the main risk measures considered in risk theory, and the problems of its calculation and approximation have attracted a lot of attention recently (see, e.g., [2,16,22,25,[27][28][29] and references therein). Risk models where shareholders receive dividends from their insurance company have been of great interest to researchers since De Finetti first considered dividend strategies in insurance dealing with a binomial model [12].…”

“…Risk models where shareholders receive dividends from their insurance company have been of great interest to researchers since De Finetti first considered dividend strategies in insurance dealing with a binomial model [12]. The classical risk model and its various modifications with different dividend strategies are investigated in a number of papers (see, e.g., [1,4,8,10,11,15,20,21,26,27,30] and references therein).…”

“…Moreover, we make the additional assumption that the insurance company pays dividends to its shareholders according to a constant dividend strategy, which implies that dividends are paid continuously at a rate d > 0. The strategy can be considered as a multi-layer dividend strategy where the number of layers is equal to one (see, e.g., [27])). Next, we denote a non-negative initial surplus of the insurance company by x, and let X t (x) be its surplus at time t provided that the initial surplus is x.…”

“…Note that the ruin probability is a special case of the expected discounted penalty function, which is introduced in [14] and also called the Gerber-Shiu function. Thus, it is easily seen that the risk model described above is a special case of the model with stochastic premiums and a multi-layer dividend strategy investigated in [27], although in that paper it is assumed that the number of layers is more than one. In [27], piecewise integro-differential equations for the Gerber-Shiu function and the expected discounted dividend payments until ruin are derived.…”

“…Thus, it is easily seen that the risk model described above is a special case of the model with stochastic premiums and a multi-layer dividend strategy investigated in [27], although in that paper it is assumed that the number of layers is more than one. In [27], piecewise integro-differential equations for the Gerber-Shiu function and the expected discounted dividend payments until ruin are derived. In addition, the model is studied in detail in the case of exponentially distributed claim and premium sizes.…”