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

Abstract

We deal with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy and consider heuristic approximations for the ruin probability. To be more precise, we construct five-and three-moment analogues to the De Vylder approximation. To this end, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. Finally, we analyze the accuracy of the approximations for some typical distributions of premium and claim sizes using statistical estimates obtained by the Monte Carlo methods.

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

“…Theorem 1 below is a special case of Theorem 1 in [36], which is formulated for the Gerber-Shiu function in the model where k 2. The assertion of the theorem is given in [62] for the case k = 1. Theorem 1.…”

“…The rest of the paper is organized as follows. In Section 2, we formulate two theorems, which follow immediately from results obtained in [36,62] and are often referred to in our main results. In Section 3, we get 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.…”

“…Moreover, explicit formulas for the ruin probability as well as for the expected discounted dividend payments are obtained in [36] in the case of exponentially distributed claim and premium sizes. The special case of this model where k = 1 is studied in [62]. In that paper, five-moment and three-moment analogues to the De Vylder approximation for the ruin probability are constructed, the accuracy of those approximations is analyzed and an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes is obtained.…”

“…Thus, in contrast to other papers where non-exponential bounds are usually obtained when exponential bounds do not exist (see, e.g., [48,[55][56][57][58][59][60][61]), our non-exponential bounds are based on the exponential one and proved to be tighter in some cases, especially for relatively small values of the initial surplus. To analyze the accuracy of the bounds, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions as well as apply explicit formulas obtained in [36,62] in the case of exponentially distributed premium and claim sizes. Note that although the exponential case is investigated in detail in [36,62] and explicit formulas are available, it is still worth being considered for the following two reasons: firstly, the bounds seem to be more elegant and easier to apply than the explicit formulas, especially when their accuracy is acceptable or the number of layers is large, and secondly, it is the only case where the explicit formulas are not so complicated and can be used to analyze the accuracy of the bounds if the number of layers is more than one.…”

“…Section 5 is devoted to the construction of explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions. Numerical illustrations are given in Section 6, where, in particular, the explicit formulas obtained in Section 5 and [36,62] are applied in order to investigate how tight the exponential and non-exponential bounds are.…”