# Ruin Probability in Compound Poisson Process with Investment

Authors: Yong Wu; Xiang Hu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

In classical risk models, the surplus process is defined asU(t)=u+ct-∑i=1N(t)Xi, where U(0)=u≥0 is the initial surplus, c>0 is the premium rate, {N(t),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 thenth risky asset follows
(1.1)dRn(t)=μnRn(t)dt+σnRn(t)dBn(t),R(0)>0,
where R(0) 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(t)dBn(t)=ρmndt,n,m=1,2,…. Furthermore, we assume that the insurance company invests a fixed proportion ωn in nth risky asset. It is natural that ∑n=1Nωn=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
(1.2)S(t)=u+ct-∑i=1N(t)Xi+∑n=1N∫0tωnS(s)Rn(s)dRn(s),S(0)=u.
Letting T denote the time of ruin, we have T=inf{t∣S(t)<0} and T=∞ for all t≥0. The probability of ruin from initial surplus u is defined as
(1.3)Ψ(u)=P(T<∞∣S(0)=u).
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∣S(t)<0,S(h)>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∣S(t)=0,S(h)>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<∞∣S(0)=u) and Ψd(u)=P(Td<∞∣S(0)=u). Clearly, the ruin probability Ψ(u) can be decomposed as
(1.4)Ψ(u)=Ψs(u)+Ψd(u).
In addition, it follows from the oscillating nature of the sample path of {S(t)} that
(1.5)Ψd(0)=Ψ(0)=1,Ψs(0)=0.
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 Ψ(u),Ψs(u),Ψd(u). Secondly, we consider the special case when the claim sizes are exponentially distributed. Finally, we obtain a lower bound of Ψ(u).


## 2. Integro-Differential Equation for Ruin Probability

In this section, we will derive integro-differential equations forΨ(u), Ψs(u), Ψd(u). These equations will be used in Section 3.Theorem 2.1.
Assume thatΨ(u) is twice continuously differentiable. Then, for any u>0,Ψ(u) satisfies the following integro-differential equation
(2.1)λΨ(u)=(c+∑n=1Nωnμnu)Ψ′(u)+12(∑n=1Nωn2σn2∑m=1Nρmn)u2Ψ′′(u)+λ∫0uΨ(u-x)dF(x)+λF¯(u)
and the boundary condition is as follows:
(2.2)Ψ(+∞)=0,Ψ(0)=1.Proof.
Denote(2.3)Ut=u+ct+∑n=1N∫0tωnS(s)Rn(s)dRn(s),Wt=∑i=1N(t)Xi.
Consider the surplus process S(t) in an infinitesimal time interval (0,t]. Note that N(t) is a homogeneous Poisson process, and there are three possible cases in (0,t] as follows.
Case one. The probability is1-λt if here is no claim in (0,t].
Case two. The probability isλt if there is only one claim in (0,t].
Case three. The probability iso(t) if there is more than one claim in (0,t].
Therefore,(2.4)Ψ(u)=(1-λt)E[Ψ(Ut)]+λtE[∫0∞Ψ(Ut-x)dF(x)]+o(t)=(1-λt)E[Ψ(Ut)]+λtE[∫0UtΨ(Ut-x)dF(x)]+λtE[F¯(Ut)]+o(t).
Let Δ=ct+∑n=1N∫0t(ωnS(s)/Rn(s))dRn(s). It follows from Taylor’s formula that
(2.5)E[Ψ(Ut)]=E[Ψ(u+Δ)]=Ψ(u)+Ψ′(u)E(Δ)+12Ψ′′(u)E(Δ2)+E(o(Δ2)).
By Ito’s formula, we have
(2.6)lim⁡t→0E(Δ)t=lim⁡t→0ct+E[∑n=1N∫0t(ωnS(s)/Rn(s))dRn(s)]t=c+∑n=1Nωnμnu,lim⁡t→0E(Δ2)t=lim⁡t→0E[∑n=1N∫0t(ωnS(s)/Rn(s))dRn(s)]2+o(t)t=∑n=1Nωn2σn2μnu∑m=1Nρmnu2.
Similarly,
(2.7)lim⁡t→0E[o(Δ2)]t=0.
Hence,
(2.8)lim⁡t→0E[Ψ(Ut)]-Ψ(u)t=(c+∑n=1Nωnμnu)Ψ′(u)+12(∑n=1Nωn2σn2∑m=1Nρmn)u2Ψ′′(u).
According to (2.4) and (2.8), we have
(2.9)λΨ(u)=(c+∑n=1Nωnμnu)Ψ′(u)+12(∑n=1Nωn2σn2∑m=1Nρmn)u2Ψ′′(u)+λ∫0uΨ(u-x)dF(x)+λF¯(u).
From the definition of Ψ(u), we easily obtain the boundary condition.Theorem 2.2.
Assume thatΨs(u) is twice continuously differentiable. Then, for any u>0,Ψs(u) satisfies the following integro-differential equation:
(2.10)λΨs(u)=(c+∑n=1Nωnμnu)Ψs′(u)+12(∑n=1Nωn2σn2∑m=1Nρmn)u2Ψs′′(u)+λ∫0uΨs(u-x)dF(x)+λF¯(u),
and the boundary condition is as follows:
(2.11)Ψs(+∞)=0,Ψs(0)=0.Proof.
Since the proof of this theorem is similar to that of Theorem2.1, we omit it.Theorem 2.3.
Assume thatΨd(u) is twice continuously differentiable. Then, for any u>0,Ψd(u) satisfies the following integro-differential equation:
(2.12)λΨd(u)=(c+∑n=1Nωnμnu)Ψd′(u)+12(∑n=1Nωn2σn2∑m=1Nρmn)u2Ψd′′(u)+λ∫0uΨd(u-x)dF(x),
and the boundary condition is as follows:
(2.13)Ψd(+∞)=0,Ψd(0)=1.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 forΨ(u), 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
(3.1)h1(u)=c+au,h2(u)=12bu2,
where
(3.2)a=∑n=1Nωnμn,b=∑n=1Nωn2σn2∑m=1Nρmn.Corollary 3.1.
Under the assumptions of Theorems2.1–2.3, if F(x)=1-e-βx,x>0,β>0, then, for any u>0,Ψ(u),Ψs(u), and Ψd(u) satisfy the following integro-differential equation:
(3.3)h2(u)ξ′′′(u)+[h1(u)+h2′(u)+βh2(u)]ξ′′(u)+[βh1(u)+h1′(u)-λ]ξ′(u)=0,
where ξ(u) is any of Ψ(u),Ψs(u), and Ψd(u). The boundary conditions of Ψ(u),Ψs(u) and Ψd(u) are as follows:
(3.4)cΨ′(0+)=0,cΨs′(0+)=-λ,cΨd′(0+)=λ.Proof.
WhenF is a exponential distribution, (2.1) can be expressed as Theorems 2.1–2.3, and we obtain the following equation:
(3.5)λΨ(u)-h1(u)Ψ′(u)-h2(u)Ψ′′(u)=λe-βu(β∫0uΨ(x)eβxdx+1).
Taking derivative about u on both sides of (3.5), we obtain
(3.6)λβΨ(u)-(λ-h1′(u))Ψ′(u)+[h1(u)+h2′(u)]Ψ′′(u)+h2(u)Ψ′′′(u)=λβe-βu(β∫0uΨ(x)eβxdx+1).
According to the above expression and (2.1), (3.3) holds. The boundary condition of Ψ(u) 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).Ifλ=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,
(3.7)Ψ(u)≥Ψ0(u),u≥0.
Equation (2.1) implies that Ψ0(u) satisfies the following differential equation:
(3.8)(c+au)Ψ′(u)+12bu2Ψ′′(u)=0.
The solution of (3.8) is given by Ψ0(u)=1-g(u)/g(+∞), where
(3.9)g(u)=∫0u(bx2)-(a/b)e(2c/bx)dx.
Thus, we obtain a lower bound for Ψ(u).



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*Source: 286792-2012-05-28.xml*