Applied Mathematics, 2011, 2, 444-451
doi:10.4236/am.2011.24056 Published Online April 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Moments of Discounted Dividend Payments in the Sparre
Andersen Model with a Constant Dividend Barrier*
Jiyang Tan1, Lin Xiao2, Shaoyue Liu1, Xiangqun Yang2
1Department of Stat i st i c s, Xiangtan University, Xiangtan, China
2Department of Mat hem at ic s, Hunan Normal Un iversity, Changsha, China
E-mail: tanjiyang15@163.com
Received December 23, 2010; revised February 14, 2011; accepted February 26, 2011
Abstract
We consider the Sparre Andersen risk process in the presence of a constant dividend barrier, and propose a
new expected discounted penalty function which is different from that of Gerber and Shiu. We find that ite-
ration mothed can be used to compute the values of expected discounted dividends until ruin and the new
penalty function. Applying the new function and the recursion method proposed in Section 5, we obtain the
arbitrary moments of discounted dividend payments until ruin.
Keywords: Sparre Andersen Model, Expected Discounted Penalty Function, Constant Dividend Barrier,
Recursion, Iteration
1. Introduction
The dividend problem in risk theory was brought out
initially by De Finetti [1] and has been studied exten-
sively in many literatures by now. Much of the literature
on dividend theory is concentrated on the classical risk
model, in which claims occur as a Poisson process. For
the classical risk model with a barrier strategy, Lin et al.
[2] studied the Gerber-Shiu discounted penalty function
at ruin; Dickson and Waters [3] studied arbitrary mo-
ments of the discounted sum of dividend payments until
ruin; Gerber et al. [4] recently developed methods for
estimating the optimal dividend barrier.
The surplus process is not necessarily a compound
Poisson process. Andersen [5] lets claims occur accord-
ing to a more general renewal process. Since then, Sparre
Andersen risk model was studied extensively. For some
recent contributions to Sparre Andersen risk models with
a dividend barrier, see [6-8]. It is worth mentioning that
Albrecher et al. [8] studied a class of Sparre Andersen
risk models with generalized Erlang(n) waiting times in
the presence of a constant dividend barrier b, and gained
some results on the distribution of dividend payments
until ruin. It is natural to ask for developing some me-
thods to get the distribution or moments of discounted
dividend payments in an arbitrary Sparre Andersen mo-
del.
In this paper, we consider the Sparre Andersen model
with arbitrary distributed waiting times in the presence of
a constant dividend barrier b. The analysis is focused on
the evaluation of the new expected discounted penalty
function defined in Section 2, which will permit us to
obtain arbitrary moments of discounted dividend pay-
ments by applying the proposed recursion method.
2. The Model
Consider the Sparre Andersen risk model, which is given
by
 

1
N
i
t
i
Utu ctStu ctX
  
(2.1)
where 0u is the initial surplus, c is a constant pre-
mium rate,
St is the aggregate claim up to time t, N(t)
is the number of claims occurring in (0,t], and i
X
is the
ith claim. Let 12
,,MM denote the inter-claim times,
and assume that
1
.
n
nk
k
LM
We assume that
,1
n
Xn and

,1
n
Mn are in-
dependent sequences of i.i.d. non-negative random va-
riables.
,1
n
Xn have a common distribution
*Supported by the Natural Sciences Foundation of China (grant No.
10871064), and by Scientific Research Funds of Hunan Provincial
Education Department (08C883), and Hunan Provincial Science and
Technology Department (2009FJ3141).
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
445
 
1
Pr
F
xXx with (0) 0F, and

,1
n
Mn
have a common distribution
1
() PrGxM x with
(0) 0G. Assume 1
EM .
The risk process (2.1) is now modified by introducing
a constant dividend barrier b
0b, i.e. whenever the
surplus process reaches the level b, the premium income
is paid out as dividends to shareholders and the modified
surplus process remains at level b until the occurrence of
the next claim. Let


b
Ut denote the modified sur-
plus process, and the random variable D (u,b) denote the
sum of the discounted dividend payments until ruin (with
force of interest 00 ). In the sequel we will be inter-
ested in the kth moment of the sum of discounted divi-
dend payments

 
,| 0,1,2,.
kk
b
WuEDubUu k



Let

 
1
bb
WuWu, which is the expectation. We
will always assume that 0ub.
Let the time of ruin for this modified surplus process

()
b
Ut be


inf0:0 otherwise
bb
TTut Ut 
Obviously, T must be some i
L. Define the stochastic
time
by
() inf:,1.
bii
uLULbi

 
For 0,x let v(x) be a non-negative measurable
function. For 00 we define a new expected dis-
counted penalty function by
 


0,0
buEevUI TUuub





(2.2)
where
I
is the indicator function. The function

bu
is similar to (but different from) Gerber-Shiu
expected discounted penalty function.
Let S denote the space of real-valued measurable func-
tions on [0,b]. Choosing the metric defined by


 
0,
d,sup, ,,
ub
x
yxuyuxyS


,SSdis complete metric space. Obviously,
b
Wu
and

bu
are contained within S because of the mea-
surable property of

vx and the monotone property of

b
Wu
.
3. The Expectation of Discounted Dividends
Until Ruin
Define an operator
:TS S by




 
0
0
00
1
0
dd
dd,
bu cuctt
bt
buc
Tffuctx eFxGt
bxeFxGt D
 


 

 (3.1)
where f = f(u) is an arbitrary function in S, and
1
D



 

0
0
0
00
0
1d, 0
1d, 0
ub t
c
bu c
bu c
cbuc
eG eGt
c
bu
ub GctGt
c











 




(3.2)
Theorem 1. Under the assumption that
1Fb
or
00
, the function
b
Wu is equal to


lim, 0.
n
bn
WuTfub

 (3.3)
As an approximation of

,
b
Wu


n
Tf u satisfies

 




0, 0,
supsup ,
1
n
n
b
ub ub
Tf uWufuTf u

 
(3.4)
where
 
0
0d.
t
F
beGt

Proof. Ruin can not occur in (0, 1
L) and the expecta-
tion of the discounted dividends paid out in this period is
1
D. By the renewal argument we have
01 111
,
L
bb
WuEeW ucLbXD




namely,

 
0
0
()
00
1
()/0
()d d
d,
bu cuctt
bb
bt
b
buc
WuWu ctxeFxGt
Wbxe dFxGtD
 


 

 (3.5)
which is an integral equation for

.
b
Wu Since
,
b
WuS
the Equation (3.5) can be rewritten as


.
bb
WuTW u (3.6)
For arbitrary ,,fg S
we have


0,
d, sup
ub
Tf Tg
 

0
00 d()d
bu cuctt
f
uctx guctxeFxGt
 


0
()/0
|( )()|d()d()
bt
buc
f
bx gbxeFxGt




 

0
()
00
0, 0,
supsupd()d()
bu cuctt
ub ub
f
ugue FxGt





0
()/0 d()d()
bt
bu ceFxGt


 
0
00
d,d d
bt
f
geFxGt



which leads to
0
0
(,) (,)()d().
t
dTfTgd f gFbeGt

(3.7)
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
446
Thus,
T is a contraction on S if F(b) < 1 or 00
(see [9]). (3.3) and (3.4) follow.
Note that when b = 0 we have, for 00 ,
 

0
00
0
01 d,
t
c
WeGt


(3.8)
and for 00 ,

01
0.WcEM
Theorem 1 gives an iteration procedure by which we
can obtain approximations to

b
Wu and error bounds.
In order to gain more information about the sum of dis-
counted dividend payments until ruin, we discuss the
new expected discounted penalty function in Section 4.
4. Expected Discounted Penalty Function
Define an integral operator as follows:




00
/
dd
d,.
bu cuctt
t
bu c
Tffuctx eFxG t
evuctGtf S
 




(4.1)
Theorem 2. The penalty function

bu
is the solu-
tion of the integral equation



.
bb
uTu

(4.2)
And under the assumption that F(b) < 1 or G(b/c) < 1
or 0, the penalty function

bu
is the unique
solution and equal to


lim ,
n
bn
uTfu

(4.3)
where f is an arbitrary real-valued measurable function.
Proof. The discrete time process

,0,1,
bn
UL n
has stationary and independent increments. By the re-
newal argument, we have the integral Equation (4.2).
The uniqueness and the result (4.3) are due to the fact
that
T is a contraction under the conditions F (b) < 1 or
G (b/c) < 1 or 0. In fact, for arbitrary real-valued
measurable functions y(u) and z(u) on [0,b], we have


0, 0,
sup sup
ub ub
Ty Tz


 

00 dd
bu cuctt
yu ctxzu ctxeFxGt
 
 

 

0
[0,][0, ]
sup supd
bu ct
ub ub
yuzueFu ctGt




  
0
0,
supd ,
bc t
ub
yuzu FbeGt


where
 
0d1.
bc t
Fbe Gt

The results are proven.
Remark. 1) Obviously, when u = b we can obtain the
explicit expression

0
()d .
t
bbevbctGt


(4.4)
2) According to Theorem 2, we can obtain the ap-
proximation of
bu
by the iteration method. As an
approximation of
,
bu
n
Tf
satisfy

 




0, 0,
supsup ,
1
n
n
b
ub ub
Tf uufuTfu

 
(4.5)
where



0
0,
sup d
bu ct
ub eFuctGt


.The error
bound (4.5) can be used for estimating the number of
steps necessary to reach a given accuracy.
Now, we give some examples of dividend-related
quantities (such as the probability of the event
bb
uTu
, the kth moment of the discounted divi-
dends paid out in time period

0,
, and the distribution
function of the sum of the dividends paid out in time pe-
riod
0,
, etc.) to illustrate applications of Theorem 2.
Example 4.1. Letting v(x) = 1 and 0, we have

Pr .
bbb
uuTu


The contraction
T is defined by
 
00 dd
1, .
bu cuct
TfufuctxFxGt
bu
GfS
c



 



(4.6)
For any real-valued measurable function f on [0,b], we
have
 

Prlim .
n
bbn
uTuTf

 (4.7)
From (4.4), it is easily seen that

Pr 1
bb
bTb
. (4.8)
Example 4.2. Letting v(x) = 1 and 0, we have


.
bbb
uEeI uTu



Let
Ru
denote this function. Note that

0Pr .
bb
Ruu Tu

Here, the contraction
T is defined by
 


00 dd
d.
bu cuctt
t
buc
Tfu efuctxFxGt
eGt





(4.9)
Choosing f = 0, we have



limd .
nt
buc
n
RuTeGt


(4.10)
The error estimate is
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
447

 
0
0,
supd .
1
n
nt
ub
Tf uRueGt


(4.11)
By (4.4), we have
0
()d ().
t
RbeGt

(4.12)
Example 4.3. The insurer will continuously pay divi-
dends in time period



,Ubc

 at rate c
when T
. Set


0
0
1.
xb
c
c
wx e





Then, the present value of these dividends is

0.ewU
 For arbitrary 0,1,i, letting
 
i
vxw xand 0, yields
 

 

,
i
bbb
uEewUIuTu





which we denote by

,.
i
A
u
The contraction
T is
defined by

 



00 dd
d.
bu cuctt
ti
bu c
Tfu efuctxFxGt
ewuctGt






(4.13)
Choosing

0fu, we have



,limd .
nti
ibuc
n
A
uT ewuctGt






(4.14)
The error estimate is

 
 
0
,
0,
0
0
sup
1d .
1
n
i
ub
in i
t
t
i
Tf uAu
cee Gt





(4.15)
Note that if 0
i , then
 
0
,,iii
A
uAu

is the
ith moment of the discounted dividends paid out in the
period

0,
; if 0,i i.e.

0,vxw x then
 
,0 .
A
uRu

In addition, it should be pointed out
that
 
,0d.
ti
i
A
bewuctGt


(4.16)
Example 4.4. Letting 0, and

vxIx bz
for arbitrary 0z, we have
 
Pr, ,
buU bzT



which is the distribution function of the cumulative divi-
dends in time period



/, .Ubc

 We de-
note this distribution function by

,.uz The contrac-
tion
T is defined by

 

00 dd
.
bu cuct
TfufuctxFxGt
buz bu
GG
cc


 





(4.17)
Choosing
0fu
, we have

,lim.
n
n
buz bu
uzT GG
cc

 

 


(4.18)
The error estimate is

 
0,
sup ,,
1
n
n
ub
bz
Tf uuzGc

 

(4.19)
where



/
0
0,
supd .
bu c
ub
F
uctGt

5. The kth Moment of Discounted Dividend
Payments Until Ruin
In this section, we use
1u
(or 1
) instead of
bu
and use
1
Tu (or 1
T) instead of

.
b
Tu
If ruin doesn’t occur at time

1u
and
12b
Uu
(2
0ub
), we view the process as “starting again”
with “initial surplus” 2
u, and similarly to the definitions
of
b
Tu and
bu
, define the stochastic times
222
TTu and
222
u

respectively for the new
pro cess
2b
Uu. If ruin doesn’t occur at 2
either,
similarly we define
333
TTu and
333
u

. Ap-
plying repeatedly the idea of “starting again”, we can
define two sequences of mutually independent random
variables
,1
i
Ti and
,1
ii
. Suppose that the
claim amount at i
is

1i
X. Then,

ii
ubX
(2i).
Further, suppose that

01
111
,eI T





0,2.
i
iii
i
IXb eITi



Then, i
(2i) are i.i.d. random variables and inde-
pendent of 1
. According to Example 2 in Section 4, we
have
0
1
kk
ERu
and




0
0
222
0
d.
kk
b
k
EEIXbRbX
RbxFx


(5.1)
We denote 2
k
E
by k
R. Assume that

01
1111
,ewUI T






0,2,
i
iiii
i
IXb ewUITi


 
which is the “present value” of the dividends paid in time
period
0, i
in the (i 1)th “starting again” process.
Obviously, i
2i are i.i.d. random variables and
random vectors
11
,
,
22
,
, are mutually in-
dependent. According to Example 3 in Section 4, we have
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
448


0
11 ,,,0,
ij ij j
EAuij



(5.2)



 




0
0
22 2,2
,
0
d,0.
ij ij j
b
ij j
EEIXbAbX
AbxFxij



 

(5.3)
We denote

22
ij
E

by ,ijj
A.
Theorem 3. The kth moment of the sum of discounted
dividend payments until ruin is equal to

  
0,
0
!,1.2,,
!!
k
k
bkiki
i
k
WuA uxk
iki 

(5.4)
where

0,1, 2,
j
xj satisfy the following recursive
formulas:


,
0
1
!
1;1, 0.
!!
j
ji j
jji
i
j
xxAx Rj
iji
 
(5.5)
Proof. It is easily seen that the sum of the discounted
dividend payments until ruin is equal to

1
112 123
11
,,
i
ik
ik
Dub
 

 

(5.6)
where we adopt the convention that 0
11
k
k
. Thus,
we have
 



1 121231234
0
112 23 234
0
!
,!!
!
.
!!
kki
ki
i
kki
iki
i
k
Dub iki
k
iki
  
 


(5.7)
Taking expectation of (5.7) yields

  

0,223234
0
!,
!!
1, 2,.
kki
k
bki
i
k
WuA uE
iki
k


(5.8)
Note that

223234
j
E

 


2 232342345
0
!
!!
j
j
i
i
i
j
Eiji
  



223 34345
0
!
!!
j
j
i
jii
i
j
Eiji
  



223 34 345
0
!
!!
j
j
i
jii
i
jEE
iji
 





,223234
0
!.
!!
j
j
i
ji
i
jAE
iji


Since ,0 ,
j
j
A
R it follows that, for 1,2,,j



223234
223234
,
1
!,
!! 1
j
j
i
j
ji
j
i
E
E
jA
iji R




(5.9)
which leads to

223234 ,0,1,2,.
j
j
xE j


(5.10)
From (5.8) and (5.10), we get (5.4).
6. Numerical Illustration
As an illustration of the results in Sections 3 and 5, con-
sider the case of a Sparre Andersen model with Erlang (2)
interclaim times and Erlang (2) claim amounts, i.e.

11 0.
t
Gt Fttet
 
Let 2, 1.1,c
and 00.03. These accord
with the assumptions in the Example 4.1 in [8].
Let us first consider the expectation of discounted
dividend payments until ruin. Given 0,1, 2,,10b
respectively, according to Theorem 1 we choose the
function
0fu
and determine a number of steps n
necessary to obtain

n
Tf u as an approximation for
b
Wu such that

 
0,
sup 0.0001,
n
b
ub
Tf uWu

see Table 1. Using the iteration procedure, we get some
approximate values of
b
Wu in Table 2. Comparing
with the exact values given by Albrecher et al. [8], it can
be seen that the approximate values in Table 2 are fairly
good. Note that, when b = 0, the numerical value 1.076 is
obtained by (3.8).
For the kth moment of discounted dividend payments
until ruin, we need compute

0,,
ji
A
u
,
j
i
A
(j = 1,2,
, k; 1, 2,,ij
) and
j
R (j = 1,2, , k). In this
example, we only consider three cases: k = 1, 2, 3. In
order to reach an accuracy of 0.00001, the necessary
numbers of steps for iteration are given in Table 3. By
formula (5.4), we obtain the approximate values for
b
Wu again, see Table 4. In Table 5, the approximate
values for the standard deviation

 
22
bb
WuWu
are given. Comparing with the Table 2 in [8], one can
find the approximate values are very excellent too. In
Table 6, approximations for the third moment are dis-
played. In addition, we point out that when b = 0 the
number of steps is not offered in Table 3, and the cor-
responding approximations in Tables 4-6 can be ob-
tained by (4.8) (4.12) and (4.16).
7. Summary
As shown in Section 6, the iteration method and the re-
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
449
Table 1. Numbers of steps for computing the expectation Wb(u) by Theorem 1.
b 1 2 3 4 5 6 7 8 9 10
n 19 76 197 283 307 311 312 312 312 312
Table 2. Approximations for the expectation Wb(u) by Theorem 1.
u
b 0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
1.076\1.076
0.836\0.836
0.856\0.856
0.848\0.848
0.802\0.801
0.730\0.730
0.648\0.648
0.565\0.565
0.487\0.486
0.416\0.416
0.354
1.808\1.808
1.847\1.847
1.829\1.828
1.728\1.728
1.575\1.575
1.397\1.397
1.218\1.218
1.049\1.049
0.897\0.897
0.763
2.846\2.846
2.815\2.815
2.661\2.661
2.424\2.424
2.151\2.151
1.875\1.875
1.615\1.615
1.381\1.381
1.175
3.803\3.803
3.597\3.597
3.277\3.277
2.908\2.908
2.535\2.535
2.184\2.184
1.867\1.867
1.589
4.574\4.574
4.175\4.174
3.705\3.705
3.229\3.229
2.782\2.782
2.379\2.379
2.025
5.143\5.143
4.575\4.575
3.988\3.988
3.436\3.436
2.938\2.9
38
2.500
5.538\5.538
4.840\4.840
4.170\4.170
3.566\3.566
3.035
5.799\5.799
5.010\5.010
4.285\4.285
3.647
5.967\5.967
5.118\5.118
4.357
6.073\6.073
5.185
6.139
a. The exact values given by Albrecher et al. (2005) are in smaller size after \.
Table 3. Numbers of steps for computing δ()
,i
Au (i = 0, 1, 2, 3).
\ n \ u 1 2 3 4 5 6 7 8 9 10
0.03 8 18 39 78 141 220 292 340 366 379
0.06 8 18 36 67 106 143 169 184 191 194
0.09 8 17 34 58 85 107 120 127 130 131
Table 4. Approximations for the expectation Wb(u) by Formula (5.4).
u
b 0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
1.0757
0.8357
0.8564
0.8480
0.8015
0.7302
0.6479
0.5647
0.4865
0.4160
0.3541
1.8082
1.8469
1.8285
1.7283
1.5745
1.3971
1.2177
1.0490
0.8970
0.7635
2.8462
2.8146
2.6605
2.4239
2.1507
1.8746
1.6149
1.3809
1.1753
3.8027
3.5969
3.2774
2.9081
2.5347
2.1836
1.8672
1.5893
4.5740
4.1745
3.7048
3.2292
2.7819
2.3788
2.0246
5.1433
4.5745
3.9881
3.4357
2.9379
2.5005
5.5376
4.8396
4.1703
3.5661
3.0352
5.7989
5.0101
4.2853
3.6474
5.9670
5.1178
4.3570
6.0731
5.1849
6.1393
cursion method proposed in this paper give good appro-
ximations for the arbitrary moments of discounted divi-
dend payments until ruin. The exact mothed presented by
Albrecher et al. [8] can only be used in the model with
generalied Erlang(n)-distributed inter-claim times. The
purpose of this paper is to find an approach which can be
J. Y. TAN ET AL.
Copyright © 2011 SciRes. AM
450
Table 5. Approximations for the standard deviation

 
22
bb
WuWu.
u
b 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
10
0.7440\0.744
1.2397\1.240
1.6667\1.667
1.8637\1.864
1.8841\1.884
1.7972\1.797
1.6564\1.656
1.4958\1.496
1.3343\1.334
1.1815\1.181
1.0415
1.3988\1.399
2.1105\2.110
2.4562\2.456
2.5275\2.528
2.4365\2.436
2.2634\2.263
2.0577\2.058
1.8467\1.847
1.6444\1.644
1.4571
2.1930\2.193
2.6948\2.695
2.8457\2.846
2.7834\2.783
2.6128\2.613
2.3959\2.396
2.1666\2.167
1.9424\1.942
1.7317
2.7416\2.742
2.9887\2.989
2.9813\2.981
2.8365\2.836
2.6288\2.629
2.3987\2.399
2.1675\2.167
1.9458
3.0201\3.020
3.0855\3.085
2.9884\2.988
2.8072\2.807
2.5897\2.590
2.3617\2.362
2.1369
3.1112\3.111
3.0796\3.080
2.9450\2.945
2.7548\2.755
2.5404\2.540
2.3197
3.1040\3.104
3.0348\3.035
2.8918\2.892
2.7048\2.705
2.4974
3.0599\3.060
2.9844\2.984
2.8454\2.845
2.6652
3.0106\3.011
2.9416\2.942
2.8100
2.9690\2.969
2.9095
a. The exact values given by Albrecher et al. (2005) are in smaller size after \.
Table 6. Approximations for the third moment

3
b
Wu.
u
b 0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
3.5755
8.4888
19.421
26.504
26.994
23.379
18.484
13.877
10.113
7.2458
5.1400
20.770
45.111
61.464
62.598
54.214
42.863
32.179
23.453
16.803
11.920
77.832
104.56
106.45
92.189
72.888
54.720
39.881
28.573
20.269
160.74
163.03
141.20
111.63
83.809
61.081
43.762
31.044
239.93
207.79
164.32
123.36
89.909
64.415
45.694
300.33
237.87
178.62
130.19
93.273
66.166
341.11
256.75
187.17
134.11
95.133
366.92
268.21
192.22
136.36
382.71
275.06
195.18
392.21
279.13
397.90
used in an arbitrary Sparre Andersen model. The itera-
tion and the recursion prove helpful in achieving the goal.
I think that more extensive applications about these me-
thods can be found. Obviously, the iteration method can
also be used to compute the Gerber-Shiu penalty func-
tion in Sparre Andersen model, even more complicated
model.
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