﻿ 基于常利率投资和线性阈值分红策略下的绝对破产模型 The Absolute Ruin Risk Model with Constant Interest Investment and a Linear Threshold Dividend Strategy

Statistics and Application
Vol.05 No.01(2016), Article ID:17282,9 pages
10.12677/SA.2016.51005

The Absolute Ruin Risk Model with Constant Interest Investment and a Linear Threshold Dividend Strategy

Ting He, Lijun Wu*

College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang

Received: Mar. 10th, 2016; accepted: Mar. 25th, 2016; published: Mar. 31st, 2016

ABSTRACT

In this paper, the classical absolute ruin risk model and that model with interference are researched based on constant interest investment and a linear threshold barrier dividend strategy. First, renewal equations of moment-generating function and n-th moment with present value of total dividends until absolution ruin are obtained. Second, partial integro-differential equations of Gerber-Shiu function are given.

Keywords:Linear Threshold Dividend, Absolute Ruin Risk Model, Gerber-Shiu Function, Renewal Equation

1. 引言

(1)

(2)

(3)

(4)

Gerber-Shiu期望折现罚金函数为

2.所满足的积分–微分方程

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

1) 当时，考虑一个非常小的时间区间，其中足够小，使得时刻的盈余达不到线性阈值，既有。考虑首次索赔时刻和索赔额，及索赔额是否导致绝对破产，可得

(13)

(14)

2) 采用同样的方法，当

(15)

(16)

3) 当时，令是方程的根，即它表示在之前没有索赔发生时的盈余恢复到0水平的时刻，满足当时有。另一方面当时，表示之前没有索赔发生时的盈余。利用首次索赔时刻和索赔额大小，由全概率公式有

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

3. Gerber-Shiu期望折现罚金函数

(28)

(29)

(30)

(31)

(32)

(33)

(34)

1) 当时,考虑一个非常小的时间区间，其中足够小，使得时刻的盈余达不到线性阈值，既有。考虑首次索赔时刻和索赔额，及索赔额是否导致绝对破产，可得

(35)

2) 采用同样的方法，当

(36)

3) 当时，有

(37)

4. 结论

The Absolute Ruin Risk Model with Constant Interest Investment and a Linear Threshold Dividend Strategy[J]. 统计学与应用, 2016, 05(01): 39-47. http://dx.doi.org/10.12677/SA.2016.51005

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