Applied Mathematics, 2011, 2, 1046-1050
doi:10.4236/am.2011.28145 Published Online August 2011 (http://www.SciRP.org/journal/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
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


,,CrT to the set of real numbers. In
other words, a fuzzy process

,
X
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
X
instead of longer notation


,
X
t

.
Definition 2. (Liu [7]) A fuzzy process t
X
is said to
have independent increments if
1021
,,,
kk
ttt tt t
1
X
XX XX X


 (1)
are independent fuzzy variables for any times
01 k
tt t



. A fuzzy process t
X
is said to have
stationary increments if, for any given, the
0s
s
ts
X
X


are identically distributed fuzzy variables for
all .
0t
*Supported by the Innovation Project of Guangxi Graduate Education
(Grant No. 2011105960202M31).

Q. Y. HUANG ET AL.
1047
2.2. Liu Process
Definition 3. (Liu [7]) A fuzzy process is said to be
a Liu process if t
C
1)
0
o
C
2) has stationary and independent increments
t
3) every increment
C
s
ts
uted fuzzy variable with expected value and vari-
ance
CC
 is a normally distrib-
et
22
t

, whose membership function is

1
21 exp,
6
xut
x
xR
t


 
 





 (2)
The parameters and e


0 
0
are called the drift
and diffusion coefficients, respectively. The Liu process
is said to be standard if and e1

. The Liu proc-
ess plays the role of the counterpart of Brownian motion.
Definition 4. (Liu [7]) Let be a standard Liu
process. Then the fuzzy process t
C

exp
tt
Get  C
is called a geometric Liu process, or sometimes expo-
nential Liu process. Li [8] has deduced that Gt is of a
log nor mal me mber shi p fun c tion

1
ln
21 exp,0
6
xet
xx
t


 
 





 (3)
and the expected value is


6csc 6 ,6
e
Ee   had been proved by
Li and Qin [8].
3. Fuzzy Model of Discount Factor
In this section we review a fuzzy model of discount fac-
tor which presented by Gao and Zhao, the model is as-
sumed that interests rate is a trapezoidal fuzzy variables.
Accumulation function for force of interest is


y
t,
as insurer, T is present value of payment at the end of
year. As insured, the present value for Survival annu-
ity paid 1 per year is .
V
m
m
Y


1
,0
,
0,
yKm
T
ytttn
emK
VKn








n
where

is a trapezoidal fuzzy variable ,
.

,,,abc d
ab dc
4. An Actuarial Model for Fuzzy Process
4.1. The Model
The paper hypothesizes that interest rate has two com-
ponents: risk-free interest and risk interest.

is used to
represent force of interest,

0t
y
ttC

 is the ac-
cumulation function for force of interest, where 0

is a
part of interests with no risk, t is a standard Liu proc-
ess, used to describe the interests on risk.
C

is

-coefficient.
We have the discount factor:
 
exp
t
Vyt
The actuarial model for Liu process is:


0t
y
ttC


 
exp
t
Vyt
4.2. Expected Value for Vt
t
Cis a normally distributio n fuzz y variable


22
,
t
CNett 
1
. Because is a standard Liu process
with 0,e
t
C



,


t
CN 2 0,t. If 6t
 it
is easy to prove that the expected va lue is
, then





 

0
0
0
0
0
0
0
0
0
0
1
0
0
6
06
exp
dd
1ln d
1ln
11exp d
6
1
1d
1
1
1
t
tt
tC
t
t
t
t
t
t
t
EVEt C
eEe
eCrexxCrex
eCr xx
x
ex
t
ex
x
e
x

 





























 
































x

0
d
6csc6
t
x
et t
 




t
EV . Otherwise, we have
. Some Actuarial Formula of Pure Life
he pure premium of the n-year continuous life insur-
5Premium of the N-Year Continuous
Insurance
T

1048 Q. Y. HUANG ET AL.
5.2. Pure Premium of the N-Year Continuous
Life Insurance under the Assumption of
Gompertz
Under the assumption of ,
ance with payment


bt is





0
1
AEV bt
:0
0
d
6csc6 d
where 6
n
xnTttxx t
nttxxt
EVp t
btett pt
t











.1. Pure Premium of the N-Year Continuous
nder the assumption of
5Life Insurance under the Assumption of de
Moivre
Ude
M
oivre, we have
1
xt
x
t


,  
1
txxt
p
x


Then





0
0
1
Ab
:0
0
6csc6 d
6csc6d ,6
nt
xntxx t
nt
tet tpt
tb tettt
x











where

1:
x
n
A
respected to pure premium of the n-year
ous a constant, that is to say continu life insurance.
If the compensation is

t b, so we have b



06c
nt
be



0
1:0
0
sc6 d
6csc6d ,6
xn txx t
nt
Attpt
btettt
x








If the payment amount is increasing linearly,


tb t, then b




0
0
1:0
0
6csc6 d
6csc 6d
6
t
xn txx t
nt
n
I
Abtet tpt
btte tt
x
t














If the payment amount is increasing geometrically,


k
t t, b


0
0
1:0
1
0
6csc6 d
6csc 6d
6
nt
k
xntxx t
nt
k
A
tett pt
te tt
x
t















If the payment amount is increasing exponentially,

t
t eb

,



0
0
1:0
0
6csc6 d
6csc 6d
6
nt
t
xntxx t
nt
A
eett pt
tet t
x
t















Gompertz ,
x
t
xt BC



0,B 1C,and

1
ln xt
BCC
xt C
txxt
pBCe




If the compensation is a constant, that is to say


bt b

, so we have






0
0
0
1:0
1
ln
0
6c sc6 d
n
bt etBCet 
1
ln
0
6csc6 d
6csc6d
6
xt
xt
nt
xntxx t
CC
ntxt C
B
tCC
xt C
Abte ttpt
bBtCet t
t



 












If the payment amount is increasing linearly,
B



btb t

, then








0
0
0
1:0
1
ln
0
6c sc6 d
n
bt tetBC et
Bb




1
ln
0
6csc6 d
6csc6d
6
xt
xt
nt
xntxx t
BCC
ntxt C
B
tCC
xt C
IAb tettpt
ttCe tt
t
















If the payment amount is increasing geometrically,


k
bt t

,






1:0
1
1
1ln
0
6csc6 d
6csc6d
6
xt
xt
nt
xntxx t
BCC
n
B
tCC
nkxtC
Abtet tpt
BtCett
t








 






If the payment amount is increasing exponentially,
ln
0
6c
sc6 d
kt xt
C
tt
etBC et






t
bt e

,







0
0
0
1:0
1
ln
0
6c sc6 d
6c
B
et etBCet 
1
ln
0
6csc6 d
sc6d
6
xt
xt
nt
xntxx t
BCC
nt
txt
C
tCC
nxt C
Abte ttpt
BCe tt
t




 







 





Q. Y. HUANG ET AL.
1049
5.3. Pure Premium of the N-Year Continuous
Life Insurance under the Assumption of
Makeham
Under the assumption of
M
akeham ,
,0,1,
xt
xt
A
BCBCA B

 



1
ln
xt C
txxt
pABCe



the compensation is a constant, that is to s
b, so we h
xt
B
AtC C 
If ay
ave

bt









0
1nt
0
:0
0
1
ln
1
6csc6 d
6csc6
d6
csc 6d
6
xt
xt
xn txx t
ntxt
B
AtC Cn
0
0
ln
x
t
C
C C
B
At C
A
bte



tt pt
bte tABC
etbtABC
tt
t


 














If the payment amount is increasing linearly,
, then
e




btb t












0
0
0
0
ln xt
1:0
0
1
ln
1
6csc6 d
6csc6
d6
csc 6d
6
xt
nt
xntxx t
ntxt
B
AtC Cn
x
t
C
C C
C
Abte ttpt
btte tABC
ettbtABC
ett
t



 
 

 













If the payment amount is increasing geometrically,
B
At 

k
bt t










0
0
0
1:0
0
11
ln
0
d6
1
ln
6csc6 d
6csc6
csc 6d
6
xt
xt
nt
xntxx t
nt
kxt
B
AtC Cnkxt
C
B
AtC C
C
A
btett pt
ttetA BC
ett
t


 
 












If the payment amount is increasing exponentially,
et tABC




t
bt e










0
0
0
1:0
0
1
ln
0
1
ln
6csc6 d
6csc6
d6
csc 6d
6
xt
xt
nt
xntxx t
nt
tx
B
AtC Cnxt
C
B
AtC C
C
t
A
btett pt
etetA BC
ettABC
et
t


 
 









 






t
5. Life Insurance under the Assumption of
Weibull
Under the assumption of
0
4. Pure Premium of the N-Year Continuous
Weibull ,

,0,
n
xt kx tkn 

 


1
1
1n
n
knxx t
n
txxt
pkxte 







If the compensation is a constant, that is to say


bt b

, so we have




 



0
1
0
1:0
1
6csc6 d
d6
csc 6d
6
n
nt
xntxx t
ntknx
A
0
1
11
0
11
6c sc6
n
nn
nn
t
kn xxtknx
0
nt
btett pt
tbke
t t
t





 










If the payment amount is increasing linearly,
bt
e tkxt
e




 





tx t e





btb t

, then


0
1
06csc6 d
nt
bte ttpt







 



0
1
11
1
0
:
0
11
1
0
6c sc6
d6
csc 6
6
n
nn
n
xn txx t
nn
t
kn xxtkn x
nntknxt
A
tb tetkxt
etke
tb txtet t
t






 


 

 











If the payment amount is increasing geometrically,
d


a
bt t



0
1
06csc6 d
nt


 



0
1
11
1
0
:
0
11
1 1
0
6c sc6
d6
csc 6d
6
n
nn
n
xn txx t
nn
t a
kn xxtkn x
nntkn xt
a
A
btett pt




tt etkxt
etke
txte tt
t






 


  














Q. Y. HUANG ET AL.
1050
If the payment amount is increasing exponentially,

t
bt e





 

 

0
0
1
11
1
0
1:0
0
11
1
0
6csc6 d
6csc6
d6
csc 6d
6
n
nn
n
nt
xntxx t
nn
t
t
kn xxtkn x
nntkn xt
A
btett pt
teet kxt
etke
tx tet t
t



























6. Conclusions
The main contribution of this paper is to suggest a new
actuarial for fuzzy markets by means of Liu process.
Some actuarial formulas of pure premium of the n
continuous life insurance on the proposed fuzzy m
are investigated.
7.
[1] L. A. Zadeh, “Fuzzy Sets,” Information and Control
8, No. 3, 1965, pp. 338-353.
doi:10.1016/S0019-9958(65)90241-X

-year
odel
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