The compound binomial model with randomly paying dividends to shareholders and policyholders

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Abstract

Considering surplus of a joint stock insurance company based on compound binomial model, set up thresholds a1, a2 for shareholders and policyholders respectively. When surplus is no less than the thresholds, the company randomly pays dividends to shareholders and policyholders with probabilities q1, q2 respectively. For this model, we have derived the recursive formulas of both the expected discount penalty function and ruin probability, and the distribution function of the deficit at ruin.

Introduction

The compound binomial model which is the classic risk model has been numerously studied, for examples Bara et al. (2008), Cai and Dickson (2002), Cheng et al. (2000), DeVylder (1996), Gong and Yang (2001), Isckson (1994), Shiu (1989). Based on the compound binomial risk, Tan and Yang (2006) considered that insurance company randomly paid dividends to policyholder: given a threshold, when the surplus is no less than the threshold, the company randomly decides whether to pay dividends to policyholders or not. Tan and Yang (2006) had studied the ruin problem for this model.

In this paper, we extend the model in Tan and Yang (2006), and consider a joint stock insurance company which randomly pays dividends to shareholders and policyholders on the basis of the compound binomial model: given two thresholds, the company randomly does decide to pay dividends to shareholders and policyholders according to the thresholds. We have built this sort of model, and derived the recursive formulas of both the expected discount penalty function and the ruin probability, and the distribution function of the deficit at ruin.

Section snippets

The model and preliminaries

Let the following essential factors be given:

1. Three non-negative integers u,a1,a2; without loss of generality, we assume that a2a1 and moreover that a2>a1.

2. Four stochastic processes ξ,η(1),η(2),X in some probability space (Ω,F,P).

(a) ξ={ξt,t=1,2,} is independent and identically distributed; the common distribution is the binomial distribution B(p), p(0,1),q=1p.

(b) η(i)={ηt(i),t=1,2,} is independent and identically distributed; the common distribution is the binomial distribution B(qi),

Recursive formulas

Let P(0)=0,P(n)=k=1nf(k),n=1,2,, P¯(n)=1P(n),n=0,1,2,, μ=EX=n=0+nf(n)=n=0+P¯(n)<+. In this article, we always assume E(ξ1X1+η1(1)+η1(2))=pμ+q1+q2<1, which leads to a positive security loading. We denote the security loading by θ: θ=1pμq1q2pμ>0.

Remark (iii). The following fact will be used in the proof of Theorem 1: p1p2+q1p2+p1q2+q1q2=(q1+p1)(q2+p2)=1. For the compound binomial model with randomly paying dividends to shareholders and policyholders. We have the following result.

Theorem 1

LetH(x

Application of formulas

Example 1

Letting ω(x,y)=1, the ϕ(u)=E[I(T<+)|U(0)=u]=ψ(u), when a1>0, by Theorem 1, ψ(0),ψ(1),,ψ(a2) can be obtained by the following linear equations qψ(0)q2ψ(a21)q1ψ(a11)=δ,qψ(u+1)+(pf(1)1)ψ(u)+pk=0u1ψ(k)f(u+1k)=Δ1(u+1),0ua11,qp1ψ(u+1)+[qq1+pp1f(1)1]ψ(u)+pk=0u1ψ(k)T(u+1k)=Δ2(u+1),a1ua21, where Δ1(u+1)=pP¯(u+1),Δ2(u+1)=pL(u+1), δ=pk=a2+G(k+1)+pk=a1a22L(k+1)+pk=0a12P¯(k+1)+pp2L(a2)+pp1P¯(a1). When a1=0, ψ(0),ψ(1),,ψ(a2) can be obtained by the following linear equations qp1ψ

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Supported by National Natural Science Foundation of China (NSF10871064) and Key Laboratory of Comput. and Stoch. Math. and its Appl., Universities of Hunan Province, Hunan Normal University.

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