Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 286792, 7 pages
doi:10.1155/2012/286792
Research Article
Ruin Probability in Compound Poisson Process
with Investment
Yong Wu and Xiang Hu
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
Correspondence should be addressed to Xiang Hu, huxiangaq@126.com
Received 1 February 2012; Accepted 29 March 2012
Academic Editor: Shiping Lu
Copyright q 2012 Y. Wu and X. Hu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider that the surplus of an insurer follows compound Poisson process and the insurer
would invest its surplus in risky assets, whose prices satisfy the Black-Scholes model. In the risk
process, we decompose the ruin probability into the sum of two ruin probabilities which are caused
by the claim and the oscillation, respectively. We derive the integro-differential equations for these
ruin probabilities these ruin probabilities. When the claim sizes are exponentially distributed,
third-order differential equations of the ruin probabilities are derived from the integro-differential
equations and a lower bound is obtained.
1. Introduction
Nt
In classical risk models, the surplus process is defined as Ut u ct − i1 Xi , where
U0 u ≥ 0 is the initial surplus, c > 0 is the premium rate, {Nt, t ∈ R } is a homogeneous
Poisson process with rate λ > 0, and {Xi , i ∈ N } is a sequence of independent and identically
distributed i.i.d. nonnegative random variables with distribution F, denoting claim sizes.
In this paper, we assume that the surplus can be invested in risky assets. The price of
the nth risky asset follows
dRn t μn Rn tdt σn Rn tdBn t,
R0 > 0,
1.1
where R0 denotes the initial value of the risky asset, μn and σn are fixed constants, and
{Bn t, t > 0} are standard Brownian motions. Moreover, Bn t and Bm t are correlated with
dBm tdBn t ρmn dt, n, m 1, 2, . . .. Furthermore, we assume that the insurance company
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Journal of Applied Mathematics
invests a fixed proportion ωn in nth risky asset. It is natural that N
n1 ωn 1, but here we
think that ωn < 0 is reasonable, which means that the investor can borrow money from risky
market. Thus, the surplus of the insurance company can be expressed as
St u ct −
Nt
i1
Xi N t
ωn Ss
dRn s,
Rn s
n1 0
1.2
S0 u.
Letting T denote the time of ruin, we have T inf{t | St < 0} and T ∞ for all t ≥ 0. The
probability of ruin from initial surplus u is defined as
Ψu P T < ∞ | S0 u.
1.3
In this paper, we decompose the ruin probability into the sum of two ruin probabilities which
are caused by the claim and the oscillation, respectively see 1–3. We denote by Ts inf{t |
St < 0, Sh > 0, 0 < h < t} and Ts ∞ for all t ≥ 0; namely, Ts is the ruin time at which ruin
is caused by a claim. We also denote by Td inf{t | St 0, Sh > 0, 0 < h < t} and Td ∞
for all t ≥ 0; namely, Td is the ruin time at which ruin is caused by oscillation. Clearly, T min{Ts , Td }. Moreover, we denote ruin probabilities in the two situations, respectively, by
Ψs u P Ts < ∞ | S0 u and Ψd u P Td < ∞ | S0 u. Clearly, the ruin probability
Ψu can be decomposed as
Ψu Ψs u Ψd u.
1.4
In addition, it follows from the oscillating nature of the sample path of {St} that
Ψd 0 Ψ0 1,
Ψs 0 0.
1.5
The ruin in the compound Poisson process by geometric Brownian motion has been studied
extensively in the literature see 4–7 and references therein. In 8–11, ruin in the compound Poisson risk process under interest force has been studied. In this paper, we investigate
the ruin probabilities in the risk process 1.2. We first derive integro-differential equation for
Ψu, Ψs u, Ψd u. Secondly, we consider the special case when the claim sizes are exponentially distributed. Finally, we obtain a lower bound of Ψu.
2. Integro-Differential Equation for Ruin Probability
In this section, we will derive integro-differential equations for Ψu, Ψs u, Ψd u. These
equations will be used in Section 3.
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3
Theorem 2.1. Assume that Ψu is twice continuously differentiable. Then, for any u > 0, Ψu satisfies the following integro-differential equation
u
N
N
N
1 2 2
c ωn μn u Ψ u ωn σn ρmn u2 Ψ u λ Ψu − xdFx λFu
2 n1
0
n1
m1
2.1
λΨu and the boundary condition is as follows:
Ψ∞ 0,
2.2
Ψ0 1.
Proof. Denote
Ut u ct N t
ωn Ss
dRn s,
Rn s
n1 0
Wt Nt
2.3
Xi .
i1
Consider the surplus process St in an infinitesimal time interval 0, t. Note that Nt is a
homogeneous Poisson process, and there are three possible cases in 0, t as follows.
Case one. The probability is 1 − λt if here is no claim in 0, t.
Case two. The probability is λt if there is only one claim in 0, t.
Case three. The probability is ot if there is more than one claim in 0, t.
Therefore,
Ψu 1 − λtEΨUt λtE
∞
ΨUt − xdFx ot
0
1 − λtEΨUt λtE
U
t
ΨUt − xdFx λtE FUt ot.
2.4
0
Let Δ ct N t
n1 0 ωn Ss/Rn sdRn s.
It follows from Taylor’s formula that
1
EΨUt EΨu Δ Ψu Ψ uEΔ Ψ uE Δ2 E o Δ2 .
2
2.5
By Ito’s formula, we have
ct E
EΔ
lim
lim
t→0
t→0
t
E Δ
t→0
t
lim
2
lim
t→0
E
t
N
n1 0 ωn Ss/Rn sdRn s
c
t
t
N
n1
ωn Ss/Rn sdRn s
0
t
N
ωn μn u,
n1
2
ot
N
n1
ωn2 σn2 μn u
2.6
N
ρmn u2 .
m1
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Journal of Applied Mathematics
Similarly,
E o Δ2
0.
lim
t→0
t
2.7
Hence,
EΨUt − Ψu
lim
t→0
t
N
N
N
1 2 2
c ωn μn u Ψ u ωn σn ρmn u2 Ψ u.
2
n1
n1
m1
2.8
According to 2.4 and 2.8, we have
N
N
1 2 2
ωn σn ρmn u2 Ψ u
λΨu c ωn μn u Ψ u 2
n1
n1
m1
u
λ Ψu − xdFx λFu.
N
2.9
0
From the definition of Ψu, we easily obtain the boundary condition.
Theorem 2.2. Assume that Ψs u is twice continuously differentiable. Then, for any u > 0, Ψs u
satisfies the following integro-differential equation:
N
N
N
1 2 2
λΨs u c ωn μn u Ψs u ωn σn ρmn u2 Ψs u
2
n1
n1
m1
u
λ Ψs u − xdFx λFu,
2.10
0
and the boundary condition is as follows:
Ψs ∞ 0,
Ψs 0 0.
2.11
Proof. Since the proof of this theorem is similar to that of Theorem 2.1, we omit it.
Theorem 2.3. Assume that Ψd u is twice continuously differentiable. Then, for any u > 0, Ψd u
satisfies the following integro-differential equation:
N
N
N
1 2 2
λΨd u c ωn μn u Ψd u ω σ
ρmn u2 Ψd u
2 n1 n n m1
n1
u
λ Ψd u − xdFx,
0
2.12
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5
and the boundary condition is as follows:
Ψd ∞ 0,
Ψd 0 1.
2.13
Proof. It follows from 1.4 that Ψ u Ψs u Ψd u and Ψ u Ψs u Ψd u. Adding
2.1 to 2.10, we obtain 2.12. From the definition of Ψd u, we easily obtain the boundary
condition.
3. Ruin with Exponential Claim Sizes
In this section, we will derive the differential equations for Ψu, Ψs u, and Ψd u with
exponential claim sizes, respectively. When λ 0, the surplus process 1.2 reduces to a pure
diffusion risk model, in which the surplus follows Brownian motions with drift, and we can
get lower bounds of the ruin probabilities. For simplicity, we denote
h1 u c au,
h2 u 1 2
bu ,
2
3.1
where
a
N
ωn μn ,
n1
N
N
ωn2 σn2 ρmn .
b
n1
3.2
m1
Corollary 3.1. Under the assumptions of Theorems 2.1–2.3, if Fx 1 − e−βx , x > 0, β > 0, then,
for any u > 0, Ψu, Ψs u, and Ψd u satisfy the following integro-differential equation:
h2 uξ u h1 u h2 u βh2 u ξ u βh1 u h1 u − λ ξ u 0,
3.3
where ξu is any of Ψu, Ψs u, and Ψd u. The boundary conditions of Ψu, Ψs u and Ψd u
are as follows:
cΨ 0 0,
cΨs 0 −λ,
cΨd 0 λ.
3.4
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Journal of Applied Mathematics
Proof. When F is a exponential distribution, 2.1 can be expressed as Theorems 2.1–2.3, and
we obtain the following equation:
u
λΨu − h1 uΨ u − h2 uΨ u λe−βu β Ψxeβx dx 1 .
3.5
0
Taking derivative about u on both sides of 3.5, we obtain
λβΨu − λ − h1 u Ψ u h1 u h2 u Ψ u h2 uΨ u
u
λβe−βu β Ψxeβx dx 1 .
3.6
0
According to the above expression and 2.1, 3.3 holds. The boundary condition of Ψu in
3.4 can be derived if we let u → 0 . Similarly, we can derive the third-order differential
equations for Ψs u and Ψd u.
If λ 0, then the surplus process 1.2 reduces to a pure diffusion risk model, in which
the surplus follows Brownian motions with drift. Denote the ruin probability in this pure
diffusion risk process by Ψ0 u. Clearly,
Ψu ≥ Ψ0 u,
u ≥ 0.
3.7
Equation 2.1 implies that Ψ0 u satisfies the following differential equation:
1
c auΨ u bu2 Ψ u 0.
2
3.8
The solution of 3.8 is given by Ψ0 u 1 − gu/g∞, where
gu u bx2
−a/b
e2c/bx dx.
3.9
0
Thus, we obtain a lower bound for Ψu.
Acknowledgments
This research was supported by the Preliminary Doctoral Program for Mathematics of
Chongqing University of Technology. The authors would like to express their thanks to the
referees for helpful suggestions.
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