
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).