A decomposition of the ruin probability for the
risk process perturbed by diffusion
q
Guojing Wang
Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China Received 1 November 1999; received in revised form 1 July 2000; accepted 19 September 2000
Abstract
In this paper, we consider the ruin probabilities (caused by oscillation or by a claim) of the classical risk process perturbed by diffusion and the risk process with return on investments. We will prove their twice continuous differentiability and derive the integro-differential equations satisfied by them. We will present the explicit expressions for them when the claims are exponentially distributed. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Risk process; Ruin probability; Integro-differential equation
1. Introduction
Let(Ω, F, P )be a complete probability space containing all the random variables we meet in the following. The classical risk process perturbed by diffusion is
Rt0=u+ct+σ Wt0−
Nt X
k=1
Zk. (1.1)
In (1.1),u, candσ are all positive constants,udenotes the initial capital of an insurance company andcthe rate of premium income;{Nt}is a Poisson process with parameterλ > 0, it counts the total numbers of the claims in the interval(0, t];{Zk},k≥ 1 is a non-negative sequence of i.i.d. random variables,Zk the amount of thekth claim,{Wt0}the standard Brownian motion, it stands for the uncertainty associated with the income of the insurance company at timet,R0t the surplus of the insurance company at timet,{Nt},{Zk}and{Wt0}are mutually independent. The process (1.1) is a homogenous strong Markov process.
We now follow the steps in Paulsen and Gjessing (1997) to introduce the risk process with deterministic return on investments. Our risk process is, in fact, a special case of (2.4) in Paulsen and Gjessing (1997).
The basic process is the surplus generating process (1.1). The deterministic return on investments generating process is
It =rt, (1.2)
q
Supported by NNSF (Grant No. 19971047) in China and Doctoral Foundation of Suzhou University. E-mail address: rgwang@suda.edu.cn (G. Wang).
wherer is a constant, it stands for a constant rate of return on investments. The risk process with deterministic return on investments is then the solution of the linear stochastic differential equation
Rt =R0t +
Z t
0
Rs−dIs. (1.3)
By Paulsen and Gjessing (1997), the solution of (1.3) is
Rt =exp{rt}
and (1.4) is also a homogenous strong Markov process. Relation (1.4) can be rewritten as
Rt =exp{rt}
Dufresne and Gerber (1991) decompose the ruin probability of the risk process (1.1) into two parts: the ruin probability caused by oscillation and the ruin probability caused by a claim. They obtain explicit expressions for these two different kinds of ruin probability in the form of series, making the assumption that the ruin probabilities are twice differentiable. Motivated by this idea in Dufresne and Gerber (1991), we will decompose the ruin probability of the process in (1.4) correspondingly into two parts. We will prove the twice continuous differentiability of these two different kinds of ruin probability and present their explicit expressions when the claims are exponentially distributed. Note that the twice continuous differentiability ensures that the solutions of the integro-differential equations satisfied by the ruin probabilities respectively exist.
2. The classical risk process perturbed by diffusion
Leta >0, defineτa =inf{s:|Ws0| =a}. Forx ∈[−a, a], put
Denote byΨd0(u)the ruin probability caused by oscillation andΨs0(u)the ruin probability caused by a claim. We have (see (1.5) in Dufresne and Gerber (1991))
IfF (z)has continuous density function on [0,+∞), thenP (∪+∞k=1{R0T
k =0})=0, and therefore
Ψd0(u)=P (Tu0<+∞, RT00
u =0), (2.4)
Ψs0(u)=P (Tu0<+∞, RT00
u <0). (2.5)
From (2.4) and (2.5) we see that
Ψd0(u)=
0, u <0,
1, u=0, (2.6)
Ψs0(u)=
1, u <0,
0, u=0. (2.7)
Theorem 2.1. Letu >0, supposeF (z)has continuous density function on [0,+∞), then the probabilityΨd0(u)
satisfies the following integral equation:
Ψd0(u)=1 2
Z +∞
0
(Ψd0(ct)+Ψd0(2u+ct))exp{−λt}hu σ, t
dt
+
Z +∞
0
λexp{−λs}ds
Z u/σ
−u/σ
Hu σ, s, x
dx
Z u+cs+σ x 0
Ψd0(u+cs+σ x−z)dF (z). (2.8)
Proof. Denote byAdthe event of ruin due to oscillation. LetFt =σ{Rs0, s≤t}. DefineMtbyMt =E[I (Ad)|Ft]. Note that{Mt, t ≥ 0}is aFt-martingale. PutT =τu/σ∧T1, we haveP (T < +∞)≤ P (T1 <+∞)=1. By optional stopping theorem together with the homogeneous strong Markov process ofRt0, we get
Ψd0(u)=EM0=E[MT]=E[E[I (Ad)|FT]]=E[Ψd0(R0T)]. (2.9) Therefore
Ψd0(u)=E[Ψd0(RT0)]=E[Ψd0(u+cτu/σ+σ Wτ0u/σ)I (τu/σ < T1)]
+E[Ψd0(u+cT1+σ WT01 −Z1)I (τu/σ ≥T1)]=I1+I2. (2.10)
Exactly similar to the calculation ofI10andI20in Theorem 3.1 in Wang and Wu (2000) we get
I1= 1 2
Z +∞
0
(Ψd0(ct)+Ψd0(2u+ct))exp{−λt}hu σ, t
dt, (2.11)
I2=
Z +∞
0
λexp{−λs}ds
Z u/σ
−u/σ
Hu σ, s, x
dx
Z u+cs+σ x 0
Ψd0(u+cs+σ x−z)dF (z), (2.12) respectively. Formula (2.8) now follows from (2.9)–(2.12). The proof is completed. h
Suppose that the first jump of the sample function of the process Rt0 occurs at timet and has magnitude z. Since no ruin occurs up to timeT, conditioned on{T1=t < τu/σ, Wt0=x}, we see thatAudoccurs if and only if
z≤u+ct+σ xandAud+ct+σ x−zoccurs. The probability of the eventAdu+ct+σ x−zisΨd0(u+cs+σ x−z). Summing all over(t, x, z)∈(0,+∞)[−u/σ, u/σ](0, u+ct+σ x−z] we get the equality (2.12). Therefore, Theorem 2.1 holds.
Theorem 2.2. SupposeF (z)has continuous density function on [0,+∞), thenΨd0(u)is twice continuously differ-entiable in the interval(0,+∞).
Proof. For any ε0 > 0 it is sufficient to show that Ψd0(u) is twice continuously differentiable in the interval
(ε0,+∞). Replacingτu/σ byτε0/σ in the proof of Theorem 2.1, then for anyu∈(ε0,+∞)we have
Ψd0(u)=1 2
Z +∞
0
(Ψd0(u−ε0+ct)+Ψd0(u+ε0+ct))exp{−λt}h
ε0
σ, t
dt
+
Z +∞
0
λexp{−λs}ds
Z ε0/σ −ε0/σ
Hε0 σ, s, x
dx
Z u+cs+σ x
0
Ψd0(u+cs+σ x−z)dF (z). (2.13) By changing the variable of integration, we can moveuin theΨd0(u)in the integrands on the right-hand side of (2.13) intoHorh. Note that the density functionh(a, t )is twice continuously differentiable intand, forx 6=0,H (a, t, x) is twice continuously differentiable inxand satisfies limt↓0Hx′(a, t, x)=limt↓0Hx′′(a, t, x)=0. Therefore, by the dominated convergence theorem and equality (2.13) we can verify thatΨd0(u)is twice continuously differentiable
in the interval(ε0,+∞). h
Theorem 2.3. Letu > 0, Suppose F (z)has continuous density function on [0,+∞), thenΨd0(u) satisfies the following integro-differential equation:
1 2σ
2Ψ0′′
d (u)+cΨ 0′
d (u)=λΨ 0 d(u)−λ
Z u
0
Ψd0(u−z)dF (z). (2.14)
Proof. The proof is exactly similar to that of Theorem 3.3 in Wang and Wu (2000). h
Similar to Theorems 2.2 and 2.3, we have the following theorem.
Theorem 2.4. SupposeF (z)has continuous density function on [0,+∞), thenΨs0(u)is twice continuously differ-entiable in the interval(0,+∞).
Theorem 2.5. Letu > 0, supposeF (z)has continuous density function on [0,+∞), thenΨs0(u) satisfies the integro-differential equation
1 2σ
2Ψ0′′
s (u)+cΨ0 ′
s (u)=λΨs0(u)−λ
Z u
0
Ψs0(u−z)dF (z)−λ(1−F (u)). (2.15) Proposition 2.1. SupposeF (z)has continuous density function on [0,+∞), thenΨd0(u)is continuous on [0,+∞). Proof. By Theorem 2.2, it is sufficient to show thatΨd0(0+)=Ψd0(0). SinceP (T0
0+ =T00=0)=1 (see Lemma 4.1 in Wang and Wu (2000)) andRt0is right continuous, by the dominated convergence theorem we get
lim u↓0Ψ
0
d(u)=lim u↓0E[I (T
0
u <+∞, R0T0
u =0)]=E[limu↓0I (T
0
u <+∞, RT00
u =0)]
=E[I (T00<+∞, R00=0)]=1=Ψd0(0).
Similar to Proposition 2.1, we have the following proposition.
Proposition 2.2. SupposeF (z)has continuous density function on [0,+∞), thenΨs0(u)is continuous on [0,+∞).
3. The case with deterministic return on investments
LetBt =σR0texp{−rs}dWs0,Btis Itˇo’s stochastic integral, its variance process ishBit =(σ2/2r)(1−exp{−2rt}). Letv(s)=inf{t:hBit > s}, then
v(s)= 1 2rln
σ2
σ2−2rs
. (3.1)
Fors < σ2/2r, setWs =Bv(s), it is well known thatWs is a local Brownian motion starting from the origin and running for an amount of timeσ2/2r.
LetTu = inf{t ≥ 0 : Rt < 0}and definingTu = +∞ifRt ≥ 0 for allt ≥ 0. Denote byΨ (u) the ruin probability of the risk process (1.4), thenΨ (u) =P (Tu < +∞). Foru > 0 we assumeE[Rt] = exp{rt}[u+
(1/r)(1−exp{−rt})(c−λµ)]>0 (see Lemma 2.1 in Wang and Wu (1998)), i.e., we assumec−λµ >0 so that we haveΨ (u) <1.
For the risk process (1.4) we denote byΨd(u)andΨs(u)the ruin probability caused by oscillation and the ruin probability caused by a claim, respectively. We still have
Ψ (u)=Ψd(u)+Ψs(u). (3.2)
SupposeF (z)has continuous density function on [0,+∞), similar to (2.4) and (2.5) we have
Ψd(u)=P (Tu<+∞, RTu =0), (3.3)
Ψs(u)=P (Tu<+∞, RTu<0). (3.4)
From (3.3) and (3.4) we get that
Ψd(u)=
0, u <0,
1, u=0, (3.5)
Ψs(u)=
1, u <0,
0, u=0. (3.6)
Theorem 3.1. Letu > 0, supposeF (z)has continuous density function on [0,+∞), then the probabilityΨd(u)
satisfies the following integral equation:
Ψd(u)=
Z +∞
0
λexp{−λs}ds
Z u
−u
H (u, v−1(s), x)dx
×
Z exp{rs}[u+x+(c/r)(1−exp{−rs})]
0
Ψd
exp{rs}hu+x+c
r(1−exp{−rs})−zexp{−rs}
i
dF (z)
+1
2
Z σ2/2r 0
h
Ψd
exp{rv(s)}h2u+c
r(1−exp{−rv(s)})
i
+Ψd
c
r(exp{rv(s)} −1)
i
Proof. Letτu0=inf{v(s):|Bv(s)| =u},τu =inf{s:|Bv(s)| =u}, thenτu0=v(τu). PutT =T1∧τu0. Similar to
We now present the expressions forI1andI2.
I11=E
Then exactly similar to the calculations ofI1andI2in Theorem 2.1 in Wang and Wu (1998), we can verify that
I11andI21are equal to the first term and the second term on the right-hand side of (3.7), respectively. The proof is
completed. h
Similar to Theorem 2.2, we have the following theorem.
Theorem 3.2. SupposeF (z)has continuous density function on [0,+∞), thenΨd(u)is twice continuously
differ-entiable in(0,+∞).
Theorem 3.3. Suppose F (z) has continuous density function on [0,+∞), then Ψd(u) satisfies the following
Therefore, by a standard use of Itˇo’s formula, we get
Similar to Theorems 3.2 and 3.3 and Propositions 2.1 and 2.2, we have the following theorem.
Theorem 3.4. SupposeF (z)has continuous density function on [0,+∞), thenΨs(u)is twice continuously
differ-entiable in(0,+∞).
Theorem 3.5. Suppose F (z) has continuous density function on [0,+∞), then Ψs(u) satisfies the following
Proposition 3.2. Suppose F (z) has continuous density function on [0,+∞), then Ψs(u) is continuous on [0,+∞).
4. Examples
This section will present the explicit expression for the ruin probability caused by oscillation and the ruin probability caused by a claim when the claims are exponentially distributed.
SupposeF (z)has continuous density function on [0,+∞), from Sections 2 and 3, we can derive thatΨd0(u), Ψs0(u), Ψd(u) and Ψs(u) are, respectively, the bounded continuous solutions of the following boundary value problems:
1 2σ
2Ψ0′′
d (u)+cΨ0 ′
d (u)=λΨd0(u)−λ
Z u
0
Ψd0(u−z)dF (z), u >0, (4.1)
Ψd0(0)=1, Ψd0(+∞)=0, (4.2)
1 2σ
2Ψ0′′
s (u)+cΨ 0′
s (u)=λΨ 0 s(u)−λ
Z u
0
Ψs0(u−z)dF (z)−λ(1−F (u)), u >0, (4.3)
Ψs0(0)=0, Ψs0(+∞)=0, (4.4)
1 2σ
2Ψ′′
d(u)+(ru+c)Ψd′(u)=λΨd(u)−λ
Z u
0
Ψd(u−z)dF (z), u >0, (4.5)
Ψd(0)=1, Ψd(+∞)=0, (4.6)
1 2σ
2Ψ′′
s(u)+(ru+c)Ψs′(u)=λΨs(u)−λ
Z u
0
Ψs(u−z)dF (z)−λ(1−F (u)), u >0, (4.7)
Ψs(0)=0, Ψs(+∞)=0. (4.8)
LetF (z) = 1−exp{−αz},(α > 0), then the above boundary value problems can be, in turn, reduced to the following problems:
1 2σ
2Ψ0′′′
d (u)+(12ασ 2
+c)Ψd0′′(u)+(αc−λ)Ψd0′(u)=0, u >0, (4.9) Ψd0(0)=1, Ψd0(+∞)=0, 12σ2Ψd0′′(0+)+cΨd0′(0+)=λ, (4.10)
1 2σ
2Ψ0′′′
s (u)+(12ασ 2
+c)Ψs0′′(u)+(αc−λ)Ψs0′(u)=0, u >0, (4.11) Ψs0(0)=0, Ψs0(+∞)=0, 12σ2Ψs0′′(0+)+cΨs0′(0+)= −λ, (4.12)
1 2σ
2Ψ′′′
d (u)+(12ασ 2
+ru+c)Ψd′′(u)+(αru+αc+r−λ)Ψd′(u)=0, u >0, (4.13)
Ψd(0)=1, Ψd(+∞)=0, 12σ2Ψd′′(0+)+cΨd′(0+)=λ, (4.14) 1
2σ 2Ψ′′′
s (u)+(12ασ 2
+ru+c)Ψs′′(u)+(αru+αc+r−λ)Ψs′(u)=0, u >0, (4.15)
It is easy to solve the problem (4.9) and (4.10) and the problem (4.11) and (4.12). Note that
(12ασ2+c)2> (21ασ2+c)2−2σ2(αc−λ)=(12ασ2−c)2+2λσ2>0. (4.17) The solution of (4.9) and (4.10) is that
Ψd0(u)=1+c1exp{−λ1u} +c2exp{−λ2u}, (4.18)
where
λ1=
(12ασ2+c)−
q
(12ασ2+c)2−2σ2(αc−λ)
σ2 , (4.19)
λ2=
(12ασ2+c)+
q
(12ασ2+c)2−2σ2(αc−λ)
σ2 , (4.20)
c1=
σ2λ22−2cλ2−2λ
σ2(λ2
2−λ21)−2c(λ2−λ1)
, (4.21)
c2= −
σ2λ21+2cλ1+2λ
σ2(λ2
2−λ21)−2c(λ2−λ1)
. (4.22)
The solution of (4.11) and (4.12) is
Ψs0(u)=c11exp{−λ1u} +c21exp{−λ2u}, (4.23)
where
c11= 2λ
σ2(λ2 2−λ
2
1)−2c(λ2−λ1)
, (4.24)
c12= −2λ σ2(λ2
2−λ21)−2c(λ2−λ1)
. (4.25)
We now consider the boundary value problem (4.13) and (4.14). Let
β =c r −
ασ2
2r , (4.26)
U (a, b, x)= 1 Γ (a)
Z +∞
0
exp{−xt}ta−1(1+t )b−a−1dt, a >0, x >0, (4.27)
F (a, b, x)= Γ (b) Γ (b−a)Γ (a)
Z 1
0
exp{xt}ta−1(1−t )b−a−1dt, b > a >0. (4.28) Put
¯
Q′1(x)=expn−αx+ r
σ2(x+β)
2oUλ 2r,
1 2,
r
σ2(x+β)
2, (4.29)
¯
Q′2(x)=(x+β)expn−αx+ r
σ2(x+β)
2oF1 2 +
λ 2r,
3 2,
r
σ2(x+β)
2. (4.30)
and Gjessing (1997) use them to express the non-ruin probability of (1.4). Wang and Wu (1999) use them to express the distribution of the surplus of (1.4) at the time of ruin. Here we will use them to express ourΨd(u)andΨs(u).
Following the steps to solve the differential equation (2.16) in Paulsen and Gjessing (1997) we can get the general solution of (4.13) as follows:
Ψd(u)=c02+c21Q1(u)+c22Q2(u), (4.31)
To find the solution of (4.13) and (4.14) we should determine the arbitrary constantsc20, c21andc22according to boundary value conditions (4.14). Lettingδ =(r/σ2)β2, some calculations give that
c20=1, (4.34)
Therefore, we get the explicit expression
Ψd(u)=1−
k2+λQ2(+∞)
C Q1(u)+
k1+λQ1(+∞)
C Q2(u). (4.40)
Similar to solving the boundary value problem (4.13) and (4.14) we can get the solutionΨs(u)of boundary value problem (4.15) and (4.16) as follows:
Ψs(u)=
References
Dufresne, F., Gerber, H.U., 1991. Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance: Mathematics and Economics 10, 51–59.
Paulsen, J., Gjessing, H.K., 1997. Ruin theory with stochastic return on investments. Advances in Applied Probability 29, 965–985. Revuz, D., Yor, M., 1991. Continuous Martingales and Brownian Motion. Springer, Berlin.
Slater, L.J., 1960. Confluent Hypergeometric Functions. Cambridge University Press, London.
Wang, G., Wu, R., 1998. Ruin theory for risk process with return on investments. Insurance: Mathematics and Economics, submitted for publication.
Wang, G., Wu, R., 1999. Notes on risk process with return on investments. Insurance: Mathematics and Economics, submitted for publication. Wang, G., Wu, R., 2000. Some distributions for classical risk process that is perturbed by diffusion. Insurance: Mathematics and Economics
