Directory UMM :Data Elmu:jurnal:I:Insurance Mathematics And Economics:Vol26.Issue1.2000:

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Some distributions for classical risk process that is

perturbed by diffusion

Guojing Wang

a,b,∗

, Rong Wu

a_{Department of Physics, Hebei University, Baoding 071002, China} b_{Department of Mathematics, Nankai University, Tianjin 300071, China}

Received December 1998; received in revised form July 1999

Abstract

In this paper we discuss the classical risk process that is perturbed by diffusion. We prove some properties of the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin. We present the simple and explicit expression for these distributions when the claims are exponentially distributed. ©2000 Published by Elsevier Science B.V. All rights reserved.

Keywords: Risk process; Ruin probability; Supremum distribution; Surplus distribution at the time of ruin; Integro-differential equation

1. Introduction

Let(,J, P )be a complete probability space containing all objects defined in the following. We consider the classical risk process that is perturbed by diffusion

Rt =u+ct+σ Wt −

k=1

Zk, (1.1)

whereudenotes the initial capital,cthe premium income,_{Wt :t≥0}is a standard Brownian motion,{Nt :t ≥0}

is a Poisson process with parameterλ >0, it counts the number of the claims in the interval(0, t],_{Zk, k ≥1}is

a non-negative sequence of i.i.d. random variables,Zk denotes the amount of thekth claim.Rt is the surplus of an

insurance company at time t. We assume throughout the paper that_{Nt},{Wt}and{Zk}are independent. Denote

the distribution function of the claims byF and their mean value byµ. LetT1, T2, . . . be the occurrence times of

the claims and setT0=0.

The model (1.1) is first introduced by Gerber (1970) and further studied by many authors during the last few years, such as Dufresne and Gerber (1991), Veraverbeke (1993), Furrer and Schmidli (1994), Schmidli (1995), Zhang

Supported by NNSF (grant no. 16971047) and DSF in China.

∗_{Corresponding author. Address: Department of Mathematics, Nankai University, Tianjin 300071, China.} 0167-6687/00/$ – see front matter ©2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 ( 9 9 ) 0 0 0 3 5 - 9

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(1997) and so on. Many results on ruin probability and other ruin problems have been obtained by above-mentioned works. The purpose of this paper will be to discuss the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the same time of ruin. We first prove some properties of these two distributions and then give some special examples for those distributions we consider. The ideas and methods we used in this paper are motivated by Grandell (1991), Paulsen (1993) and Paulsen and Gjessing (1997), can be used to the risk model with return on investment, see Wang and Wu (1998).

In Section 2, we introduce the supremum distribution of the risk process before ruin when ruin occurs. We first derive the integral equation satisfied by it. Then by using the integral equation we prove its differentiability. Finally we give the integro-differential equation satisfied by it. In the last part of this section we define the supremum distribution of the risk process before ruin.

In Section 3, we discuss the surplus distribution at the time of ruin. We also derive the integral equation and the integro-differential equation satisfied by it. The contents in this section are parallel those in Section 2.

In Section 4, we get the simple and explicit expression for the ruin probability, the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin by solving boundary value problems with same differential equation of third order but with different boundary values with each other when the claims are exponentially distributed. For general expressions of these distributions, we can refer to Dufresne and Gerber (1991 ) for the series expression of ruin probability and to Zhang (1997) for the series expression of the surplus distribution at the time of ruin.

2. Supremum distribution before ruin

LetTu = inf{t ≥ 0 : Rt < 0}andTu = +∞ if Rt ≥ 0 for all t ≥ 0. ThenTu is the time of ruin. The

ruin probability9(u)is defined by9(u)₌P (inft≥0Rt <0)and the non-ruin probability8(u)=1−9(u)=

P (inft≥0Rt ≥ 0). We have9(u)= P (Tu <+∞). Foru > 0, we assume throughout this paper thatE[Rt] =

u₊(c₋λµ)t >0, i.e., we assumec₋λµ >0, so that we have9(u) <1. Letx > u >0, set

G(u, x)₌P ( sup

0≤t≤Tu

Rt ≥x, Tu<+∞) (2.1)

G(u, x)stands for the probability that the supremum value of the risk process before ruin reaches or surpasses the levelx when ruin occurs. We call it the supremum distribution of the risk process before ruin when ruin occurs. It is obvious that

G(u, x)₌

0, u_≤0, x >0,

9(u), u_≥x >0. (2.2)

Letτa=inf{s:|Ws| =a}. Put

H (a, t, x)₌(2π t )−1/2 +∞

k=−∞

exp

−₂1_t(x₊4ka)2

−exp

−₂1_t(x₋2a₊4ka)2

, (2.3)

h(a, t )₌ 1

2√2πat −3/2

+∞

k=−∞

(4k₊1)exp

−a

2t(4k+1)

+(4k₋3)exp

−a

2t(4k−3)

−2(4k₋1)exp

−a

2t(4k−1)

(2.4)

It follows from Revuz and Yor (1991) (pp. 105–106) thatP (Ws ∈dx, τa> s)=H (a, s, x)dxandP (τa ∈ds)=

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Fort _≥0 letθt be the shift operators fromto itself defined byRs(θtω)=Rs+t(ω). For finite stopping time T,

we define the mapθT fromto itself byθT(ω)=θt(ω)ifT (ω)=t(see Revuz and Yor (1991), p. 34 and p. 97).

Clearly, we haveRt◦θT =Rt+T.

Theorem 2.1. Letx > u >0,thenG(u, x)satisfies the following integral equation:

G(u, x)₌exp_{−λt0}

Z a

−a

G(u₊ct0+σ y, x)H (a, t0, y)dy +

Z t0

λexp_{−λs_}ds

Z a

−a

H (a, s, y)dy

Z u+cs+σ y 0

G(u₊cs₊σ y₋z, x)dF (z)

Z t0

(G(u₊ct₊σ a, x)₊G(u₊ct₋σ a, x))exp_{−λt_}h(a, t )dt, (2.5)

wheret0≤(1/2c)(x−u),0< a≤(1/2σ )((x−u)∧u).

Proof. Letτa =inf{t:|Wt| =a}, setT =t0∧τa∧T1. Fort ∈(0, T ), we have 0< Rt < x. ThusP (T ≤Tu)=1.

SinceT _≤t0and on{Tu<+∞}we haveTu =T+Tu◦θT, therefore, by homogeneous strong Markovian property

ofRt, we get

G(u, x)₌E

I sup

0≤t≤Tu

Rt ≥x, Tu<+∞ !#

I sup

T≤t≤Tu

Rt ≥x, T ≤Tu<+∞ !#

I sup

0≤t≤Tu◦θr

Rt◦θT ≥x,0≤Tu◦θT <+∞ !#

I sup

0≤t≤Tu

Rt ≥x, Tu<+∞ !

◦θT #

=E[G(RT, x)] (2.6)

We now compute the expectation on the right-hand side of equality (2.6).

E[G(RT, x)]=EG Rt0, x

, t0< τa∧T1+EG Rτa, x

, τa≤T1, τa≤t0 +E

G RT1, x

, T1≤τa, T1≤t0=I1+I2+I3. (2.7)

By assumption of independence we have

I1=EG u+ct0+σ Wt0, x

I (t0< τa) I (t0< T1) =E[I (T1> t0)]EG u+ct0+σ Wt0

I (τa> t0) =exp_{−λt0}

Z a

−a

G(u₊ct0+σ y, x)H (a, t0, y)dy. (2.8)

By Proposition 2.8.3 in Port and Stone (1978) we get

P Wτa =a, τa∈dt

=P Wτa = −a, τa∈dt

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By equality (2.9) we get

I2=EG u+cτa+σ Wτa, x

I (τa≤T1) I (τa≤t0) =E

G (u₊cτa+σ a, x) I (τa≤T1) I (τa ≤t0) I Wτa =a

G(u₊cτa−σ a, x)I (τa ≤T1)I (τa≤t0)I Wτa = −a

= Z t0

G (u₊ct₊σ a, x)exp_{−λt_}P Wτa =a, τa∈dt

+ Z t0

G (u₊ct₋σ a, x)exp_{−λt_}P Wτa = −a, τa∈dt

=1₂ Z t0

(G (u₊ct₊σ a, x)₊G (u₊ct₋σ a, x))exp_{−λt_}h (a, t )dt. (2.10)

Now,

I3=EG u+cT1+σ WT1−Z1, x

I (T1≤τa) I (T1≤t0) =

Z t0

λexp_{−λs_}ds

Z _+∞

dF (z)E[G (u₊cs₊σ Ws−z, x) I (τa≥s)]

= Z t0

λexp_{−λs_}ds

Z _+∞

dF (z)

Z a

−a

G (u₊cs₊σ y₋z, x) H (a, s, y)dy

= Z t0

λexp_{−λs_}ds

Z a

−a

H (a, s, y)dy

Z u+cs+σ y 0

G(u₊cs₊σ y₋z, x)dF (z). (2.11)

Formula (2.5) now follows from equalities (2.6),(2.7) and (2.8) and (2.10) and (2.11).

Theorem 2.2. SupposeF (z) has continuous density function on [0,_+∞). ThenG(u, x) is twice continuously differentiable inuin interval(0, x).

Proof. For arbitraryε0 > 0, it is sufficient to show thatG(u, x)is twice continuously differentiable in u in

in-terval (ε0, x−ε0). If we set t0 < (1/2c)ε0 and a < (1/2σ )ε0 (note that t0 and a do not depend on u), we

still have formula (2.5). By changing the variable of integration, we can moveuinG’s in the integrand on the right-hand side of equality (2.5) into H or h. Then by the fine properties H and h we can verify that the theorem

holds.

Theorem 2.3. SupposeF (z)has continuous density function on [0,_+∞), letx > u >0. ThenG(u, x)satisfies the following integro-differential equation:

1 2σ

2_G′′

u(u, x)+cG′u(u, x)=λG(u, x)−λ

Z u

G(u₋z, x)dF (z). (2.12)

Proof. Letε, t >0 such thatε < u < x₋εandT_tε ₌inf_{s >0 :u₊cs₊σ Ws ∈/ (ε, x−ε)} ∧t. Note that

G′_u(u, x)andG′′_u(u, x)are bounded on [ε, x₋ε] and thereforeRs∧Ttε

0 σ G′u(u+cv+σ Wv, x)dWvis a martingale.

PutT ₌T_tε_∧T1. Similar to (2.6) we have

G(u, x)₌EG RTtε∧T1, x

. (2.13)

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Therefore, by Itô formula, we get

G(u, x)₌exp_{−λt_}E

G u₊cT_tε₊σ WTε t , x

Z t 0

λexp_{−λs_}

G u₊cT_sε₊σ WTε s, x

;T_sε < s +

Z +∞

EG(u₊cs₊σ Ws −z, x);Tsε=s

dF (z)

=exp_{−λt_}

(

G(u, x)₊E

" Z Ttε

(cG′_u(u₊cs₊σ Ws, x)+

σ2

2 G

′′

u(u+cs+σ Ws, x))ds #)

+ Z t

λexp_{−λs_}

G u₊cT_sε₊σ WTε s, x

;T_sε< s +

Z _+∞

G(u₊cs₊σ Ws −z, x);Tsε=s

dF (z)

ds. (2.14)

Dividing by t gives 1₋exp_{−λt_}

t G(u, x)=exp{−λt}

(

Z T_tε 0

(cG′_u(u₊cs₊σ Ws, x)+

σ2

2 G

′′

u(u+cs+σ Ws, x))ds #)

Z t 0

λexp_{−λs_}

G(u₊cT_sε₊σ WTε s, x);T

ε s < s

Z +∞

EG(u₊cs₊σ Ws−z, x);Tsε=s

dF (z)

ds. (2.15)

Lettingt _→0 gives formula (2.12).

We now consider the distributionŴ(u, x)defined byŴ(u, x)₌P (sup₀_≤_{t <T}

uRt ≥x).Ŵ(u, x)can be decomposed

as follows:

Ŵ(u, x)₌P sup

0≤t <Tu

Rt ≥x, Tu<+∞ !

+P sup

0≤t <Tu

Rt ≥x, Tu= +∞

=G(u, x)₊8(u). (2.16)

Indeed, since Rt is a process with stationary and independent increments and E[Rt] = (c−λu)t > 0, thus

limt↑+∞Rt = +∞ P-a.e. Therefore,

P sup

0<t <Tu

Rt ≥x, Tu= +∞

=P (Tu= +∞)=8(u). (2.17)

Ŵ(u, x)is the probability that the supremum value of the risk process before ruin reaches or surpasses the level x. We call it the supremum distribution of the risk process before ruin.

3. Surplus distribution at the time of ruin

We consider the surplus distribution at the time of ruinD(u, y)defined by

D(u, y)₌P Tu<+∞, RTu≥ −y

, y >0. (3.1)

D(u, y)indicates the probability that the surplus of an insurance company at the time of ruin is in the interval [₋y,0). Evidently, we have

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D(u, y)₌I[0,y](−u), u≤0,

D(_+∞, y)₌0. (3.2)

Theorem 3.1. Letu >0, thenD(u, y)satisfies the following integral equation

D(u, y)₌1

Z +∞

(D(ct, y)₊D(2u₊ct, y))exp_{−λt_}hu σ, t

Z +∞

λexp_{−λs_}ds

Z +u/σ

−u/σ

σ, s, x

Z u+cs+σ x 0

D(u₊cs₊σ x₋z, y)dF (z)

+λ

Z u+cs+σ x+y u+cs+σ x

dF (z)

. (3.3)

Proof. LetT ₌ τu/σ ∧T1. Fort ∈ (0, T ), we haveRt > 0, thusP (T ≤ Tu) = 1. Therefore, we haveTu =

T ₊Tu◦θT on{Tu<+∞}. By homogeneous strong Markov property forRtwe get

D(u, y)₌P Tu<+∞, RTu ≥ −y

=EI Tu<+∞, RTu ≥ −y

=EI T _≤Tu<+∞, RTu ≥ −y

=EI Tu◦θT <+∞, RT+Tu◦θT ≥ −y

=E[I (Tu<+∞, RTu≥ −y)◦θT]=E[D(RT, y)] (3.4)

By equality (3.4), we get

D(u, y)₌EhDRτu/σ∧T₁, y

=ED u₊cτu/σ+σ Wτu/σ, y

, τu/σ < T1

+ED u₊cT1+σ WT1−Z1, y

, τu/σ ≥T1=I10+I20. (3.5)

By equality (2.9) we get

I₁0₌E[D(u₊cτu/σ+σ Wτu/σ, y)I (τu/σ < T1)]=

2E[(D(2u+cτu/σ, y)+D(cτu/σ, y))I (τu/σ < T1)]

Z _+∞

(D(2u₊ct, y)₊D(ct, y))exp_{−λt_}hu σ, t

dt. (3.6)

Similar to (2.11) we have

I₂0₌E

D u₊cT1+σ WT1−Z1, y

I (τu/σ ≥T ) =

Z _+∞

λexp_{−λs_}ds

Z _+∞

dF (z)

Z ₊u/σ

−u/σ

D(u₊cs₊σ x₋z, y)Hu

σ, s, x

Z _+∞

λexp_{−λs_}ds

Z ₊u/σ

−u/σ

σ, s, x

Z u+cs+σ x 0

D(u₊cs₊σ x₋z, y)dF (z)

+λ

Z u+cs+σ x+y u+cs+σ x

dF (z)

(3.7)

Formula (3.3) now follows from equalities (3.4), (3.5), (3.6) and (3.7). Corresponding to Theorem 2.2, we have the following Theorem 3.2.

Theorem 3.2. SupposeF (z) has continuous density function on [0,_+∞). Then D(u, x)is twice continuously differentiable in u in interval(0,_+∞).

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Theorem 3.3. Letu > 0, supposeF (z)has continuous density function on [0,_+∞). ThenD(u, x)satisfies the following integro-differential equation

1 2σ

2_D′′

u(u, y)+cDu′(u, y)=λD(u, y)−λ

Z u

D(u₋z, y)dF (z)₋λ

Z u₊y u

dF (z). (3.8)

Proof. Letε, t, M >0 such thatε < u < M andT_tε,M ₌inf_{s >0 :u₊cs₊σ Ws ∈/ (ε, M)} ∧t. Note that

D_u′(u, y)andD′′_u(u, y)are bounded on [ε, M] and therefore,Rs∧Ttε,M

0 σ D′u(u+cv+σ Wv, y)dWvis a martingale.

PutT ₌T_tε,M _∧T1. Similar to (2.13) we have

D(u, x)₌EhDR

Ttε,M∧T1, y i

. (3.9)

Similar to (2.15) we have 1₋exp_{−λt_}

t D(u, y)=exp{−λt}

(

Z T_tε,M 0

(cD′_u(u₊cs₊σ Ws, y)+

σ2

2 D

′′

u(u+cs+σ Ws, y))ds #)

+1_t Z t

λexp_{−λs_}nEhDu₊cT_sε,M₊σ W_Tε,M s , y

;T_sε,M < si

Z +∞

EhD(u₊cs₊σ Ws−z, x);Tsε,M =s i

dF (z)

ds (3.10)

Lettingt _→0 gives formula (3.8).

4. Examples

This section will give the explicit expressions for the ruin probability, the supremum distribution of the risk process before ruin and the surplus distribution at the time of ruin when the claims are exponentially distributed. Corresponding to Lemma 4.2 and Propositions 2.2 and 2.3 in Wang and Wu (1998) we have the following Lemma 4.1 and Proposition 4.1 and 4.2, respectively:

Lemma 4.1. P (T₀+=T0=0)=1.

Proposition 4.1. SupposeF (z)has continuous density function on [0,_+∞), then8(u)is continuous on [0,_+∞).

Proposition 4.2. SupposeF (z)has continuous density function on [0,_+∞), thenG(u, x)is continuous on [0, x].

Proposition 4.3. Lety >0. SupposeF (z)has continuous density function on [0,_+∞), thenD(u, y)is continuous inuon [0,_+∞).

Proof. By Theorem 3.2, it is sufficient to show thatD(0+_{, y)}₌_D(₀_{, y)}_{. By Lemma 4.1, we have lim}_n

→+∞I (T1/n< +∞)₌1 P — a.e. By Lemma 4.1 and the law of iterated logarithm for Brownian motion we get limn→+∞RT1/n≥

−y P-a.e., i.e., limn→+∞I (RT1/n ≥ −y)=1 P-a.e.

Thus by Fatou Lemma we get

1_≥ lim

n_→+∞D

₁

n, y

≥ lim

n_→+∞D

₁

n, y

≥E

lim

n_→+∞I (RT1/n≥ −y)I (T1/n<+∞)

Therefore,D(0+, y)₌limn→+∞D(1/n, y)=1=D(0, y). This completes the proof.

Suppose now F (z) has continuous density function on [0,_+∞). Following the ideas to prove Theorem 2.2 or Theorem 3.2, we can prove that8(u)is twice continuously differentiable in(0,_+∞).

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Following the proof of (2.12) we can prove that formula (2.1) in Dufresne and Gerber (1991 ) holds. Therefore, it follows from Proposition 4.1 that8(u)is the bounded continuous solution of the following boundary value problem:

(1/2)σ28′′_u(u)₊c8′_u(u)₌λ8(u)₋λR₀u8(u₋z)dF (z), u >0,

8(0)₌0 8(_+∞)₌limu↑+∞8(u)=1.

(I)

From Theorem 2.3, boundary conditions (2.2) and Proposition 4.2, it follows thatG(u, x)is the bounded continuous solution of the following boundary value problem

(1/2)σ2G′′_u(u, x)₊cG′_u(u, x)₌λG(u, x)₋λRu

0 G(u−z, x)dF (z), 0< u < x,

G(0, x)₌0, x >0 G(x, x)₌9(x), (II)

and from Theorem 3.3, boundary conditions (3.2) and Proposition 4.3 it follows that D(u, y) is the bounded continuous solution of the following boundary value problem

(1/2)σ2D_u′′(u, y)₊cD_u′(u, y)₌λD(u, y)₋λR₀uD(u₋z, y)dF (z)₋λR_uu+y dF (z), u >0,

D(0, y)₌1, D(_+∞, y)₌limu↑+∞D(u, y)=0.

(III)

Remark 4.1. If the bounded continuous solution of (II) is unique, then by comparing the boundary value problems

(I) and (II) we get

G(u, x)₌ 8(u)

8(x)9(x). (4.1)

In fact, we have the following proposition.

Proposition 4.4. IfF (z)has continuous density function on [0,_+∞), then (4.1) holds.

Proof. Note that8(u)andG(u, x)solves (I) and (II), respectively. LetTux,ε=inf{t :Rt ∈/(ε, x−ε)}. Assumef (u)

solves (I), thenf (R_Tx,ε

u ,∧t)is a martingale. This impliesf (u)=E[f (RTux,ε∧t)]. Lettingt → +∞showsf (u)= E[f (R_Tx,ε

u )]. Lettingε → 0 givesf (u) = E[f (RTux,0∧t)] = f (x)P (RTux,0 = x). This givesP (RTux,0 = x) =

Ŵ(u, x) ₌ f (u)/f (x). Lettingx _{→ +∞}showsf (u) ₌ 8(u). Letf (u, x)solve (II), thenf (R_Tx,ε

u ∧t, x) is a

martingale. We havef (u, x) ₌ E[f (R_Tx,ε

u ∧t, x)]. Lettingt → +∞showsf (u, x) = E[f (RTux,ε, x)]. Letting ε_→0 givesf (u, x)₌E[f (R

Tux,0, x)]=9(x)P (RTux,0 =x)=9(x)8(u)/8(x). This ends the proof. Remark 4.2. In the proof of Proposition 4.4 we get the formulaŴ(u, x)₌8(u)/8(x). It can also been proved by

a different “martingale approach” (see (Zhang, 1997)).

WhenF (z)₌1₋exp_{−αt_}α >0, the boundary value problems (I),(II) and (III) can be reduced to the boundary value problems (Ia),(IIa) and (IIIa) as follows respectively:

(1/2)σ28′′′_(u)₊ ₍₁_/₂_)ασ2 +c

8′′_(u)₊_(αc₋_λ)8′_(u)₌₀_, _{u >}₀_,

8(0)₌0, 8(_+∞)₌limu↑+∞8(u)=1, (1/2)σ28′′(0+)+c8′(0)=0,

. (Ia)

(1/2)σ2G′′_u′(u, x)₊ (1/2)ασ2₊cG′′_u(u, x)₊(αc₋λ)G′_u(u, x)₌0,0< u < x,

G(0)₌0, G(x, x)₌9(x), (1/2)σ2G′′_u(0+, x)₊cG′_u(0, x)₌0, (IIa)

and

(1/2)σ2D_u′′′(u, y)₊ (1/2)ασ2₊cD′′_u(u, y)₊(αc₋λ)D_u′(u, y)₌0, u >0, D(0, y)₌1, D(_+∞, y)₌limu↑+∞D(u, y)=0, (1/2)σ2Du′′(0+, y)+cDu′(0+, y) =λexp_{−αy_}

(IIIa)

and the solutions of (Ia),(IIa) and (IIIa) are all unique. Note that

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1. The solution of boundary value problem (Ia) is

8(u)₌1₋c1exp{−λ1u} −c2exp{−λ2u}, (4.3)

whereλ1, λ2>0,

λ1=

σ2 



1 2ασ

2 +c₋

₁

2ασ

2₋_c

+2λσ2 

, (4.4)

λ2=

σ2 



1 2ασ

2 +c₊

₁

2ασ

2₋_c

+2λσ2 

, (4.5)

c1=

σ2λ2₂₋2λ2c

σ2_(λ2

2−λ21)−2c(λ2−λ1)

, (4.6)

and

c2= −

σ2λ2₁₋2λ1c

σ2_(λ2 2−λ

1)−2c(λ2−λ1)

. (4.7)

2. The solution of the boundary value problem (IIIa) is

D(u, y)₌c3exp{−λ1u} +c4exp{−λ2u} u≥0 (4.8)

where

c3=

σ2λ2₂₋2λ2c−2λexp{−αy}

σ2_(λ2 2−λ

1)−2c(λ2−λ1)

, (4.9)

and

c4=

2λexp_{−σ y_{} −}σ2λ2₁₊2λ1c

σ2_(λ2 2−λ

1)−2c(λ2−λ1)

. (4.10)

3. From equality (4.1) we get that the solution of (IIa) is

G(u, x)₌8(u)

8(x)9(x)=

1₋c1exp{−λ1u} −c2exp{−λ2u}

1₋c1exp{−λ1x} −c2exp{−λ2x}

(c1exp{−λ1x} +c2exp{−λ2x}). (4.11)

4. From Remark 4.2 and formula (4.3) we get

Ŵ(u, x)₌ 1−c1exp{−λ1u} −c2exp{−λ2u}

1₋c1exp{−λ1x} −c2exp{−λ2x}

. (4.12)

Acknowledgements

The authors would express their hearty thanks to the referee for presenting them the new proofs of Theorems 2.3 and 3.3 and the proof of Proposition 4.4.

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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.

Furrer, H.J., Schmidli, H., 1994. Exponential inequalities for ruin probabilities of risk process perturbed by diffusion. Insurance: Mathematics and Economics 15, 23–36.

Gerber, H.U., 1970. An extension of the renewal equation and its application in the collective theory of risk. Scandinavian Actuarial Journal, 205–210.

Grandell, J., 1991. Aspects of Risk Theory. Springer, New York.

Paulsen, J., 1993. Risk theory in a stochastic economic environment. Stochastic Processes and their Application 46, 327–361. Paulsen, J., Gjessing, H.K., 1997. Ruin theory with stochastic return on investment. Advances in Applied Probability 29, 965–985. Port, S., Stone, C., 1978. Brownian Motion and Classical Potential Theory. Academic Press, New York.

Revuz, D., Yor, M., 1991. Continuous Martingales and Brownian Motion. Springer, Berlin.

Schmidli, H., 1995. Cramer–Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insurance: Mathematics and Economics 16, 135–149.

Veraverbeke, N., 1993. Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance: Mathematics and Economics 13, 57–62.

Wang, G., Wu, R., 1998. Ruin theory for risk process with return on investments. Insurance: Mathematics and Economics, submitted for publication.

Zhang, C., 1997. Some problems on the classical risk model perturbed by diffusion and a class of insurance risk models. Ph.D. Thesis, Nankai University.

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Therefore, by Itô formula, we get

G(u, x)₌exp_{−λt_}E

G u₊cT_tε₊σ WTε t , x

Z t

λexp_{−λs_}

G u₊cT_sε₊σ WTε s, x

;T_sε < s +

Z +∞ 0

EG(u₊cs₊σ Ws −z, x);Tsε=s

dF (z)

=exp_{−λt_} (

G(u, x)₊E "

Z Ttε

(cG′_u(u₊cs₊σ Ws, x)+

σ2

2 G

′′

u(u+cs+σ Ws, x))ds

+ Z t

λexp_{−λs_}

G u₊cT_sε₊σ WTε s, x

;T_sε< s +

Z _+∞ 0

G(u₊cs₊σ Ws −z, x);Tsε=s

dF (z)

ds. (2.14)

Dividing by t gives 1₋exp_{−λt_}

t G(u, x)=exp{−λt} (

E "

t Z T_tε

(cG′_u(u₊cs₊σ Ws, x)+

σ2

2 G

′′

u(u+cs+σ Ws, x))ds

+1 t

Z t

λexp_{−λs_}

G(u₊cT_sε₊σ WTε s, x);T

ε s < s

+ Z +∞

EG(u₊cs₊σ Ws−z, x);Tsε=s

dF (z)

ds. (2.15)

Lettingt _→0 gives formula (2.12).

We now consider the distributionŴ(u, x)defined byŴ(u, x)₌P (sup₀_≤_{t <Tu}Rt ≥x).Ŵ(u, x)can be decomposed as follows:

Ŵ(u, x)₌P sup

0≤t <Tu

Rt ≥x, Tu<+∞

+P sup

0≤t <Tu

Rt ≥x, Tu= +∞

=G(u, x)₊8(u). (2.16) Indeed, since Rt is a process with stationary and independent increments and E[Rt] = (c−λu)t > 0, thus limt↑+∞Rt = +∞ P-a.e. Therefore,

P sup

0<t <Tu

Rt ≥x, Tu= +∞

=P (Tu= +∞)=8(u). (2.17)

Ŵ(u, x)is the probability that the supremum value of the risk process before ruin reaches or surpasses the level x. We call it the supremum distribution of the risk process before ruin.

3. Surplus distribution at the time of ruin

We consider the surplus distribution at the time of ruinD(u, y)defined by

D(u, y)₌P Tu<+∞, RTu≥ −y

, y >0. (3.1)

D(u, y)indicates the probability that the surplus of an insurance company at the time of ruin is in the interval [₋y,0). Evidently, we have

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D(u, y)₌I[0,y](−u), u≤0,

D(_+∞, y)₌0. (3.2)

Theorem 3.1. Letu >0, thenD(u, y)satisfies the following integral equation

D(u, y)₌1

Z +∞ 0

(D(ct, y)₊D(2u₊ct, y))exp_{−λt_}hu σ, t

+ Z +∞

λexp_{−λs_}ds Z +u/σ

−u/σ

Hu σ, s, x

Z u+cs+σ x

D(u₊cs₊σ x₋z, y)dF (z)

+λ

Z u+cs+σ x+y u+cs+σ x

dF (z)

. (3.3)

Proof. LetT ₌ τu/σ ∧T1. Fort ∈ (0, T ), we haveRt > 0, thusP (T ≤ Tu) = 1. Therefore, we haveTu =

T ₊Tu◦θT on{Tu<+∞}. By homogeneous strong Markov property forRtwe get

D(u, y)₌P Tu<+∞, RTu ≥ −y

=EI Tu<+∞, RTu ≥ −y

=EI T _≤Tu<+∞, RTu ≥ −y

=EI Tu◦θT <+∞, RT+Tu◦θT ≥ −y

=E[I (Tu<+∞, RTu≥ −y)◦θT]=E[D(RT, y)] (3.4)

By equality (3.4), we get

D(u, y)₌EhDRτu/σ_∧T₁, y

=ED u₊cτu/σ+σ Wτu/σ, y

, τu/σ < T1

+ED u₊cT1₊σ WT1−Z1, y

, τu/σ ≥T1=I10+I20. (3.5)

By equality (2.9) we get

I₁0₌E[D(u₊cτu/σ+σ Wτu/σ, y)I (τu/σ < T1)]=

2E[(D(2u+cτu/σ, y)+D(cτu/σ, y))I (τu/σ < T1)]

Z _+∞ 0

(D(2u₊ct, y)₊D(ct, y))exp_{−λt_}hu σ, t

dt. (3.6)

Similar to (2.11) we have

I₂0₌E

D u₊cT1+σ WT1−Z1, y

I (τu/σ ≥T )

= Z _+∞

λexp_{−λs_}ds Z _+∞

dF (z) Z ₊u/σ

−u/σ

D(u₊cs₊σ x₋z, y)Hu σ, s, x

= Z _+∞

λexp_{−λs_}ds Z ₊u/σ

−u/σ

Hu σ, s, x

Z u+cs+σ x

D(u₊cs₊σ x₋z, y)dF (z)

+λ

Z u+cs+σ x+y

u+cs+σ x

dF (z)

(3.7) Formula (3.3) now follows from equalities (3.4), (3.5), (3.6) and (3.7).

Corresponding to Theorem 2.2, we have the following Theorem 3.2.

Theorem 3.2. SupposeF (z) has continuous density function on [0,_+∞). Then D(u, x)is twice continuously differentiable in u in interval(0,_+∞).

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Theorem 3.3. Letu > 0, supposeF (z)has continuous density function on [0,_+∞). ThenD(u, x)satisfies the following integro-differential equation

1 2σ

2_D′′

u(u, y)+cDu′(u, y)=λD(u, y)−λ

Z u

D(u₋z, y)dF (z)₋λ Z u₊y

dF (z). (3.8)

Proof. Letε, t, M >0 such thatε < u < M andT_tε,M ₌inf_{s >0 :u₊cs₊σ Ws ∈/ (ε, M)} ∧t. Note that

D_u′(u, y)andD′′_u(u, y)are bounded on [ε, M] and therefore,Rs∧Ttε,M

0 σ D′u(u+cv+σ Wv, y)dWvis a martingale. PutT ₌T_tε,M _∧T1. Similar to (2.13) we have

D(u, x)₌EhDR

Ttε,M∧T1, y i

. (3.9)

Similar to (2.15) we have 1₋exp_{−λt_}

t D(u, y)=exp{−λt} (

E "

t Z T_tε,M

(cD′_u(u₊cs₊σ Ws, y)+

σ2

2 D ′′

u(u+cs+σ Ws, y))ds

+1_t Z t

λexp_{−λs_}nEhDu₊cT_sε,M₊σ W_Tε,M s , y

;T_sε,M < si +

Z +∞ 0

EhD(u₊cs₊σ Ws−z, x);Tsε,M =s

dF (z)

ds (3.10)

Lettingt _→0 gives formula (3.8).

4. Examples

Lemma 4.1. P (T₀+=T0=0)=1.

Proposition 4.1. SupposeF (z)has continuous density function on [0,_+∞), then8(u)is continuous on [0,_+∞).

Proposition 4.2. SupposeF (z)has continuous density function on [0,_+∞), thenG(u, x)is continuous on [0, x].

Proposition 4.3. Lety >0. SupposeF (z)has continuous density function on [0,_+∞), thenD(u, y)is continuous inuon [0,_+∞).

Proof. By Theorem 3.2, it is sufficient to show thatD(0+_{, y)}₌_D(₀_{, y)}_{. By Lemma 4.1, we have lim}_n

→+∞I (T1/n<

+∞)₌1 P — a.e. By Lemma 4.1 and the law of iterated logarithm for Brownian motion we get limn→+∞RT1/n≥

−y P-a.e., i.e., limn→+∞I (RT1/n ≥ −y)=1 P-a.e. Thus by Fatou Lemma we get

1_≥ lim

n_→+∞D

₁ n, y

≥ lim

n_→+∞D

₁ n, y

≥E

lim

n_→+∞I (RT1/n≥ −y)I (T1/n<+∞)

Therefore,D(0+, y)₌limn_→+∞D(1/n, y)₌1₌D(0, y). This completes the proof.

Suppose now F (z) has continuous density function on [0,_+∞). Following the ideas to prove Theorem 2.2 or Theorem 3.2, we can prove that8(u)is twice continuously differentiable in(0,_+∞).

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(1/2)σ28′′_u(u)₊c8′_u(u)₌λ8(u)₋λR₀u8(u₋z)dF (z), u >0, 8(0)₌0 8(_+∞)₌limu↑+∞8(u)=1.

(I) From Theorem 2.3, boundary conditions (2.2) and Proposition 4.2, it follows thatG(u, x)is the bounded continuous solution of the following boundary value problem

(1/2)σ2G′′_u(u, x)₊cG′_u(u, x)₌λG(u, x)₋λRu

0 G(u−z, x)dF (z), 0< u < x,

G(0, x)₌0, x >0 G(x, x)₌9(x), (II)

and from Theorem 3.3, boundary conditions (3.2) and Proposition 4.3 it follows that D(u, y) is the bounded continuous solution of the following boundary value problem

(1/2)σ2D_u′′(u, y)₊cD_u′(u, y)₌λD(u, y)₋λR₀uD(u₋z, y)dF (z)₋λR_uu+y dF (z), u >0, D(0, y)₌1, D(_+∞, y)₌limu↑+∞D(u, y)=0.

(III)

Remark 4.1. If the bounded continuous solution of (II) is unique, then by comparing the boundary value problems

(I) and (II) we get

G(u, x)₌ 8(u)

8(x)9(x). (4.1)

In fact, we have the following proposition.

Proposition 4.4. IfF (z)has continuous density function on [0,_+∞), then (4.1) holds.

Proof. Note that8(u)andG(u, x)solves (I) and (II), respectively. LetTux,ε=inf{t :Rt ∈/(ε, x−ε)}. Assumef (u) solves (I), thenf (R_Tx,ε

u ,∧t)is a martingale. This impliesf (u)=E[f (RTux,ε∧t)]. Lettingt → +∞showsf (u)=

E[f (R_Tx,ε

u )]. Lettingε → 0 givesf (u) = E[f (RTux,0∧t)] = f (x)P (RTux,0 = x). This givesP (RTux,0 = x) =

Ŵ(u, x) ₌ f (u)/f (x). Lettingx _{→ +∞}showsf (u) ₌ 8(u). Letf (u, x)solve (II), thenf (R_Tx,ε

u ∧t, x) is a martingale. We havef (u, x) ₌ E[f (R_Tx,ε

u ∧t, x)]. Lettingt → +∞showsf (u, x) = E[f (RTux,ε, x)]. Letting

ε_→0 givesf (u, x)₌E[f (R

Tux,0, x)]=9(x)P (RTux,0 =x)=9(x)8(u)/8(x). This ends the proof.

Remark 4.2. In the proof of Proposition 4.4 we get the formulaŴ(u, x)₌8(u)/8(x). It can also been proved by

a different “martingale approach” (see (Zhang, 1997)).

WhenF (z)₌1₋exp_{−αt_}α >0, the boundary value problems (I),(II) and (III) can be reduced to the boundary value problems (Ia),(IIa) and (IIIa) as follows respectively:

(1/2)σ28′′′_(u)₊ ₍₁_/₂_)ασ2 +c

8′′_(u)₊_(αc₋_λ)8′_(u)₌₀_, _{u >}₀_,

8(0)₌0, 8(_+∞)₌limu↑+∞8(u)=1, (1/2)σ28′′(0+)+c8′(0)=0,

. (Ia)

(1/2)σ2G′′_u′(u, x)₊ (1/2)ασ2₊cG′′_u(u, x)₊(αc₋λ)G′_u(u, x)₌0,0< u < x,

G(0)₌0, G(x, x)₌9(x), (1/2)σ2G′′_u(0+, x)₊cG′_u(0, x)₌0, (IIa)

and

(1/2)σ2D_u′′′(u, y)₊ (1/2)ασ2₊cD′′_u(u, y)₊(αc₋λ)D_u′(u, y)₌0, u >0, D(0, y)₌1, D(_+∞, y)₌limu_↑+∞D(u, y)₌0, (1/2)σ2D_u′′(0+, y)₊cD_u′(0+, y) =λexp_{−αy_}

(IIIa)

and the solutions of (Ia),(IIa) and (IIIa) are all unique. Note that

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1. The solution of boundary value problem (Ia) is

8(u)₌1₋c1exp_{−λ1u_{} −}c2exp_{−λ2u_}, (4.3)

whereλ1, λ2>0,

λ1=

σ2 



1 2ασ

2 +c₋

s ₁

2ασ

2₋_c 2

+2λσ2 

, (4.4)

λ2₌ 1 σ2





1 2ασ

2 +c₊

s ₁

2ασ

2₋_c 2

+2λσ2 

, (4.5)

c1₌ σ

2_λ2 2−2λ2c σ2_(λ2

2−λ21)−2c(λ2−λ1)

, (4.6)

and

c2_{= −} σ

2_λ2 1−2λ1c σ2_(λ2

2−λ 2

1)−2c(λ2−λ1)

. (4.7)

2. The solution of the boundary value problem (IIIa) is

D(u, y)₌c3exp{−λ1u} +c4exp{−λ2u} u≥0 (4.8)

where

c3=

σ2λ2₂₋2λ2c−2λexp{−αy} σ2_(λ2

2−λ 2

1)−2c(λ2−λ1)

, (4.9)

and

c4=

2λexp_{−σ y_{} −}σ2λ2₁₊2λ1c σ2_(λ2

2−λ 2

1)−2c(λ2−λ1)

. (4.10)

3. From equality (4.1) we get that the solution of (IIa) is

G(u, x)₌8(u) 8(x)9(x)=

1₋c1exp{−λ1u} −c2exp{−λ2u}

1₋c1exp{−λ1x} −c2exp{−λ2x}

(c1exp_{−λ1x_{} +}c2exp_{−λ2x_}). (4.11) 4. From Remark 4.2 and formula (4.3) we get