Applied Mathematics, 2011, 2, 1046-1050

doi:10.4236/am.2011.28145 Published Online August 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

Some Actuarial Formula of Life Insurance for

Fuzzy Markets*

Qiaoyu Huang, Liang Lin, Tao Sun

College of Science , Guilin University of Technology, Guilin, China

E-mail: Lzcst135@163.com

Received May 13, 2011; revised June 16, 2011; accepted June 25, 2011

Abstract

This paper presents an actuarial model of life insurance for fuzzy markets based on Liu process. At first,

some researches about an actuarial model of life insurance for stochastic market and concepts about fuzzy

process have been reviewed. Then, an actuarial model of life insurance for fuzzy process is formulated.

Keywords: Fuzzy Actuarial Model, Fuzzy Process, Liu Process, Geometric Liu Process, Actuarial Formula

1. Introduction

The concept of fuzzy set was initiated by Zadeh [1] via

membership function in 1965. In order to measure a

fuzzy event, Liu and Liu [2] introduced the concept of

credibility measure in 2002. Li and Liu [3] gave a suffi-

cient and necessary condition for credibility measure in

2006 (cf. Liu [4]). Credibility theory was founded by Liu

[5] in 2004 and refined by Liu [6] in 2007 as a branch of

mathematics for studying the behavior of fuzzy phe-

nomena. Credibility theory is deduced from the normal-

ity, monotonicity, self-duality, and maximality axioms.

Liu [7] recently introduced the concepts of fuzzy process,

Liu process and the geometric Liu process which will be

commonly used model in finance for the value of an as-

set in a fuzzy environment. The two types of fundamen-

tal and important fuzzy processes, Liu process and the

geometric Liu process, are the counterparts of Brownian

motion and the geometric Brownian motion, respec-

tively.

Li [8] presented an alternative assumption that stock

price follows geometric Liu process. It is an application

of fuzzy process for stock markets. Peng [9] presented A

General Stock Model for Fuzzy Markets based on Liu’s

research in 2008. In 2010, Gao and Zhao [10] first pre-

sented fuzzy interests rate for insurance market.

The remainder of this paper is structured as follows.

Section 2 is intended to introduce some useful concepts

of fuzzy process as they are needed. Section 3 reviews

Gao and Zhao’s actuarial model for fuzzy market. An

actuarial model of life insurance with payouts increased

for fuzzy process is formulated in Section 4. Some actu-

arial formula of pure premium of the n-year continuous

life insurance model is discussed in Section 5, as dis-

count factor is a fuzzy process. Finally, some remarks are

made in the concluding section.

2. Preliminaries

In this section, we will introduce some useful definitions

and properties about fuzzy process.

2.1. Fuzzy Process

Definition 1. (Liu [7]) Given an index set T and a

credibility space

,,Cr , a fuzzy process is a func-

-tion from

,,CrT to the set of real numbers. In

other words, a fuzzy process

,

t

is a function of

two variables such that the fun ction

*,

Xt

is a fuzzy

variable for each . For simplicity, sometimes we sim-

ply use the symbol

*

t

t

instead of longer notation

,

t

.

Definition 2. (Liu [7]) A fuzzy process t

is said to

have independent increments if

1021

,,,

kk

ttt tt t

1

XX XX X

(1)

are independent fuzzy variables for any times

01 k

tt t

. A fuzzy process t

is said to have

stationary increments if, for any given, the

0s

ts

X

are identically distributed fuzzy variables for

all .

0t

*Supported by the Innovation Project of Guangxi Graduate Education

(Grant No. 2011105960202M31).