Total Duration of Negative Surplus for a Diffusion Surplus
Process with Stochastic Return on Investments
Honglong You, Chuancun Yin
School of Mathematical Sciences, Qufu Normal University, Qufu, China Email: [email protected], [email protected]
Received August 31, 2012; revised September 30, 2012; accepted October 7, 2012
ABSTRACT
In this paper, we consider a Brownian motion risk model with stochastic return on investments. Using the strong Markov property and exploiting the limitation idea, we derive the Laplace-Stieltjes Transform (LST) of the total duration of negative surplus. In addition, two examples are also present.
Keywords: Negative Surplus; Ruin Probability; Laplace-Stieltjes Transform
1. Introduction
Assume that the insurance business is described by the risk process
1
,U t u ct B t t0.
(1.1) Here, is the initial capital; is the fixed rate of premium income; is a standard Brownian motion; and
u c
B x t1 , 00
is a constant, representing the diffusion volatility.
Suppose that the insurer is allowed to invest in an asset or investment portfolio. Following Paulsen and Gjessing [1], we model the stochastic return as a Brownian motion with positive drift. Specifically, the return on the invest- ment generating process is
2
,R t rt B t t0, (1.2)
where r and are positive constants. In (1.2), r is a
fixed interest rate; is another standard Brownian motion independent of , stand- ing for the uncertainty associated with the return on investments at time .
B t t2 , 0t
B x t1 , 0
Let the risk process denote the surplus of the insurer at time under this investments assump- tion. Thus,
X t t, 0t
X t associated with (1.1) and (1.2) is then
the solution of the following linear stochastic integral equation:
0t
d .X t U t
X s R s (1.3)By Paulsen [2] the solution of (1.3) is given by
1
0 d
t
,
X t R t u
R s U s (1.4)where
2 2 2
er t B t.
R t
Note that
X t t
, 0
is a homogeneous strong Markov process, see e.g. Paulsen and Gjessing [1].The risk process (1.4) can be rewritten as
2 0
1
0 0
d
d d
t
t t
X t u X s B s
B s rX s c s
.Because the quadratic variational processes of
2
1
0 d 0 d
t t
X s B s B s
and
2 2 2
0 d
t
X s B
sare the same, where
B t t
, 0
is a standard Brow- nian motion, by Ikeda and Watanabe [3, p. 185] they have the same distribution. Thus, in distribution, we have
2 2 2
0
0
d d .
t t
X t u X s B s
rX s c s
(1.5)with a constant interest. He et al. [7] gived the LST of
the total duration of negative surplus for the classical risk model with debit interest. More recently, Wang and He [8] considered the Brownian motion risk model with interest and derived the LST of total duration of negative surplus. In this paper, we consider a Brownian motion risk model with stochastic return on investments. We will use the limitation idea to obtain the LST of total duration of negative surplus.
The remainder of the paper is organized as follows. In Section 2, we give some preliminary results. In Section 3, by exploiting the limitation idea together with the results obtained in Section 2, we obtain the LST of the total duration of negative surplus. In the last section, we pre- sent two examples.
2. Preliminary Results
Given u
b a,
, where c br
, define
inf 0,
a
T t X t
a
a
b
b
and if the set is empty ,
T
0 inf 0, 0
T t X t
0
and if the set is empty ,
T
inf 0,
b
T t X t
b
and if the set is empty ,
T
u, b
P T
b |X
0 u
P Tu
.
Lemma 2.1 The risk process (1.5) has the strong Markov property: for any finite stopping time T the re- gular conditional expectation of X T
t given FT is
E X T t X T , that is
T
, 0, .E X T t F E X T t X T t a s.
where FT is the information about the process up to
time T, and the equality holds almost surely.
Lemma 2.2 For r, the following ordinary diffe- rential equation
2 2 2
2
x
f x c rx f x f x
(2.1)
has two independent solutions
1
d
x
x x t K t t
(2.2)and
1
d ,
x
x t x K t t
(2.3)where
2 2 2
12 exp 2carctan t ,
K t t
2
2 2
2r 1 8 1
,
2
2 2
1 2
2
1 8 1 .
2
r r
Proof. From Example 2.2 of Paulsen and Gjessing [1],
we get the result.
2
Lemma 2.3 For r 0, c u a r
and
,
c
J a
r
,
a cr
J T T
, define
, , e J
X
a , u I ,
J a
c
u E
r
, , , e J ,
u c
X J
r c
a u E I
r
then
, , ,
c c
u u
c r r
a u
c c
r
a a
r r
,
, c, , a u u a
a u
c c
r
a a
r r
,
where and
resul
are given by (2.2) and (2.3).
Proof. he t can be found in Chapter 16 of
Breiman [9].
r
T
Lemma 2.4 Fo any u
b a,
, then
,u b a
S u S a
P T T
S a
(2.4)
where
b S
0 2 22 2
d
0e d
z rs c s x
s
S x z
is a solution oation
f the equ-
2 2 2
2
x
f x
0.
c rx f x
Proof. By Dynkin’s formula,
,S u
0 d
t Tb Ta
u b a
E S X tT T
LS X s swhere is the generator of diffusion (1.5). It fo that
L llows
X
s
LS X s S X s
2 2 2
2 0.
c rX s S X s
Therefore
.u b a
E S X tT T S u
Since T
bability b a
T
<
u a
P T
is finite, it takes values with pro-
and a
T
Tb
Tb, we ca
with the co limentary
probab n assert ominated
convergence, that
mp , by d ility. Letting t
a
X T T
E S
0
.u b
u
E S
X S u
Expanding the expectation on the left, we have
This, together with 1,
gives the result (2.4).
Lemma 2.5 For the ruin probability for the
risk model (1.5) is given by
<
< u Ta Tb u Tb Ta
E S a I E S b I
.S u
<
<
u a b u b a
P T T P T T
0
u ,
00 d
< .
d
u
u
h z z
(2.5)u
h z z
P T
The probability that the surplus process
X t t
, 0
hit the level b is given by
d, <
d
u .
b
h z z
z
(2.6)where
u b
u b P T
h z
0 2 22 2 d
e .
z rs c s s
h z
Proof. By Lemma 2.4, one can derive (2.5) and (2.6).
3. Total Duration of Negative Surplus
Inpa
this section, we will derive the main result of this per. We assume that the risk process (1.5) does not attain the critical level c
r
. For convenience, we assume that the initial surplus u is positive.
Let the total duration of negative surplus be
0 < <0
= c d .
X t r
I t
For
0 < <c
1 inf t> 0,X t
(1 if the set is empty),
1 inf 1, 0, 0, 1
c
t X t X s s t
r
r
, define two sequences of stopping times of the process (1.5):
(1 if the set is empty),
in general, for k2,3, , recursively define
1
inf > ,
k t k X t
(k if the set is empty),
inf , 0, 0,
k k k
c
t X t X s s t
r
(k if the set is empty).
Let k kk,k1, 2,3, . Given k for
roperty of the surplus pro
some kov p
eriods 1
k , from the strong Mar
cess, we obtain that the p ,i 1, 2,3, ,k
i
are mutually i
on distribution. Let N denot
ndependen ve a
e the num t and ha
ber of
comm i.
1 N
i i
T
. By thSet e monotone convergence
, we ha theorem ve
0 e e .
lim T
u u
E E
First we give the expression for
e TE in e
(3.1)
th 1.
r and
u
following Theorem 3.
Theorem 3.1 Fo u0 r 0, the LST of T is given by
e 1 ,
, 1 0,
u 0, , ,
0, 0, , ,
T
E u
c r
r
,
1 c
u
(3.2)
where
0, c, ,
r
and
u,
are given byLemmas 2.3 and 2.5.
Proof. From Lemma 2.1, we can get
0
0
e e
0, , , k
c r
(3.3) T
u
T T
E E I
c
and
r
0
0
0 e e
0, , , k
c r
T
T T
E E I
c r
(3.4)
om strong Markov property of the surplus process, we get
1 2 1 1
1 1
1 1
1
, , , , , 0
1
1 0
1
e 0 e
1 ,
e
1 ,
1 0,
, e 0, e
, e
1 , 1 0,
i
i
k k k
k
T i
u u u N k
k
k
i
u X t t
k
k u
k
u
E P N E I
u
E I
u
u E E
u E u
1 1 0 .1 0, E e
0 0 0 0 e e lim 1 , lim, 1 0, 0, , ,
lim
1 0, 0, , ,
1
1 0, 0, , ,
lim
1 0, 0, , ,
u T u E E u c u r c r u c r u c r (3.6)
From Lemma 2.5, it follows that
0 0 0 0 0 01 0, 0, , ,
lim
1 0, 0, , ,
d
1 0, , ,
d lim
d
1 0, , ,
d
0, , , d
lim
d 0, , ,
c r c r
h z z c
r
h z z
h z z c
r
h z z
c
h z z
r c
h z z
r
0h z
dz.(3.7) This, together with (3.3) and (3.4), gives (3.2).
Theorem 3.2 For u0 and r 0, the LST of total duration of negative surplus is given by
0e 1
0, , ,0 , 1 0 , ,0
, d
c
h z z
u E u c u r r (3.5)
where 0, c, ,
r
,
u and are givenby Lemmas 2.3 and 2.5.
Proof. It follows from (3.1) and (3.2) that
h z
By ŁHospital’s rule, we get
0 0 0 0 0 0 00, , , d 0, , , d 0, , ,
lim lim
d 0, , 0, , , d
0, , ,0
.
1 0, , ,0 d
c c c
h z z h z z h
r r r
c c
h z z h h h z z
r r
c r c
h z z
r
, z dz
This, together with (3.6) and (3.7), gives (3.5).
4. Examples
In this section we consider two examples.
Example 4.1. Letting
0
in (1.5), we get the risk model
0t
d 0t d
.X t u
crX s s
B s (4.1)From Cai et al. [10], we know that the two independent
solutions of the differential equation
2
2 f x rx c f x f x
are
2
2
2
23
exp 1 , ,
2 2
x
c rx c rx
c rx M
and
x
c rx
2
c rx
2
exp 2 1 , ,1 2 ,
2 2 2
U r
r r
(4.3)
tions can be found in Abramowitz and Stegun [11].
By Lemmas 2.3 and 2.5, we get
where M and U are called the confluent hypergeometric
functions of the first and second kind respectively. More detail on confluent hypergeometric func
0
0<
0, , , e
, 0
c r c r
T T T T
c
E I
r
(4.4)
d, ,
d
uh z z
u
h z z
(4.5)
0
, d
u
u
h z z
(4.6)where
d
h z z
2 2
2
rz cz
e .
h z
According to Theorems 3.1 and 3.2, we get
e 1 ,
, 1 0,
,
0 0,
T u
E u
u
(4.7)
0
0e 1
0 0 d
u
u
E u
h z z
,(4.8)where ,
u,
6).
and (4.5) a .
ark 4.1 The results (4.7) and (4.8) coincide with the main results in Wang and He [7].
Example 4.2. Letting
u are given by (4.2), nd (4
Rem
0
and in (1.5), we get the risk model
0
r
.X t u ct B t (4.9)
It is easy to obtain that the two independent solutions of the ordinary differential equation
2
2 f x cf x f x
are
2 2 2e c c x
x
and
2 2 2e .
x
By Lemmas 2.3 and 2.5, we get
c c x
0 0
2 2
<
2
, , e T
c c
E I
0, T
e ,
22
e ,
cz
h z
22 e
c u
u
and
2 2
, e
c u
u .
According to Theorems 3.1 and 3.2, we have
2
2 2
2 2
2 2
2 2
2 2
2
2
2
e = 1 e
e 1 e e
1 e e
c u T
c c
u c c
c
c c
u E
and
2 2
2 2
2
2 2
e
e 1 e
1 2
2
c u c
u u
E
c c
c
.
5. Conclusion
In in-
corporating stochastic return on investments. We find the LST of the total duration of negative surplus of this pro- cess. However, if the risk model (1.1) is extended to a compound Poisson surplus process perturbed by a diffu- sion, it is difficult to make out. We leave this problem for further research.
REFERENCES
[1] J. Paulsen and H. K. Gjessing, “Optimal Choice of Divi- dend Barriers for a Risk Process with Stochastic Return on Investments,” Insurance: Mathematics and Economics, Vol. 20, No. 3, 1997, pp. 215-223.
doi:10.1016/S0167-6687(97)00011-5
this paper, we have studied the diffusion model
k Theory in a Stochastic Economic Envi- hastic Processes and Their Applications, [2] J. Paulsen, “Ris
ronment,” Stoc
Vol. 46, No. 2, 1993, pp. 327-361. doi:10.1016/0304-4149(93)90010-2
167-6687(93)90996-3
[4] A. D. Egdio dos Reis, “How Long Is the Surplus below Zero?” Insurance: Mathematics and Economics, Vol. 12, No. 1, 1993, pp. 23-38.
doi:10.1016/0
[5] C. S. Zhang and Negative Sur-plus for the Co s That Is Perturbed
p. 245-255. R. Wu, “Total Duration of mpound Poisson Proces
by Diffusion,” Insurance: Mathematics and Economics, Vol. 39, No. 3, 2002, pp. 517-532.
[6] S. N. Chiu and C. C. Yin, “On Occupation Times for a Risk Process with Reserve-Dependent Premium,” Sto- chasticModels, Vol. 18, No. 2, 2001, p
doi:10.1081/STM-120004466
[7] J. M. He, R. Wu and H. Y. Zhang, “Total Duration of Negative Surplus for the Risk Model with Debit Interest,” Statistics and Probability Letters, Vol. 79, No. 10, 2009,
pp. 1320-1326. doi:10.1016/j.spl.2009.02.005
[8] W. Wang and J. M. He, “Total Duration of Negative Sur-
ding,
ommerce,” plus for a Brownian Motion Risk Model with Interest,” Acta Mathematica Sinica, 2012, (Submitted).
[9] L. Breiman, “Probability,” Addison-Wesley, Rea 1968.
[10] J. Cai, H. U. Gerber and H. L. Yang, “Optimal Dividends in an Ornstein-Uhlenbeck Type Model with Credit and Debit Interest,” North American Actuarial Journal, Vol. 10, No. 2, 2006, pp. 94-119.
[11] M. Abramowitz and I. A. Stegun, “Handbook of Mathe- matical Functions: With Formulas, Graphs, and Mathe- matical Tables United States Department of C
