劉麗芳

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具馬氏調(diào)制費率的風(fēng)險模型罰金函數(shù)的期望

劉麗芳

(1. 中南大學(xué) 數(shù)學(xué)與統(tǒng)計學(xué)院,湖南 長沙, 410000; 2. 湖南文理學(xué)院 數(shù)學(xué)與計算科學(xué)學(xué)院,湖南 常德, 415000)

主要考慮馬氏風(fēng)險模型的罰金函數(shù)的數(shù)學(xué)期望. 在具有馬氏調(diào)制費率, 索賠額服從指數(shù)分布情形下, 得出罰金函數(shù)的數(shù)學(xué)期望所滿足的積分方程.

馬氏調(diào)制;風(fēng)險模型;罰金函數(shù)

1 模型介紹

近年來, 許多學(xué)者對馬氏風(fēng)險模型中的破產(chǎn)概率以及破產(chǎn)時間的數(shù)學(xué)期望進行了較為深入的研究[1-3], 本文主要是在文獻[2]的基礎(chǔ)上, 對馬氏風(fēng)險模型罰金函數(shù)的期望進一步探討, 具體模型如下:

2 主要結(jié)果

對式(2)求導(dǎo)并整理得:

證畢.

,

將(9)式代入(10)式中即得(8)式, 證畢.

[1] Wagner Christian. Time in the red in a two state Markov model[J]. Insurance: Mathematics and Economics, 2002, 31: 365-372.

[2] Wang Han-Xing, Fang Da-Fan. Ruin Probability under a Markovian Risk Model[J]. Acta Mathematical Applications(English Series), 2003, 19(4): 621-630.

[3] Cai J, Dickson D C M. Ruin probabilities with a Markov chain interest model[J]. Insurance: Mathematics and Economics, 2004, 35: 513-525.

[4] Asmussen S. Ruin probabilities[M]. Singapore: World Scientific, 2000: 35-76.

[5] Jan Grandell. Aspects of Risk Theory[M]. NewYork: Springer-Verlag, 1991: 26-48.

The expectation of penalty function in risk model with Markov-Modulated premium rates

LIU Li-fang

(School of Mathematics and Computing Sciences, Hunan University of Arts and Science,Changde 415000, China)

The penalty function expectation was considered in Markov risk model. In the case, where the premium rate was Markov-modulated and the claim distribution was exponential, a integro-differential equation was gotten about the penalty function expectation.

Markov-modulated; risk model; penalty function

O 211.9

1672-6146(2012)04-0014-03

10.3969/j.issn.1672-6146.2012.04.003

2012-10-15

湖南省自然科學(xué)基金(09JJ6016); 湖南省教育廳青年項目(10B073)

劉麗芳(1974-), 女,博士研究生, 研究方向為概率統(tǒng)計. E-mail: lifangliu0318@163.com

(責任編校: 劉曉霞)