# A Note on a Generalized Discounted Penalty Function in a Sparre Andersen Risk Model Perturbed by Diffusion

Authors: Chaolin Liu; Zhimin Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider a Sparre Andersen risk model perturbed by diffusion where the interclaim times are generalized Erlang(n) distribution. Generalized discounted penalty functions incorporating the maximum surplus before ruin are studied. We derive the integrodifferential equations and give the solutions for the generalized discounted penalty functions.


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## Body

## 1. Introduction

Consider the following Sparre Andersen risk model perturbed by a Brownian motion:(1)U(t)=u+ct-∑i=1N(t)‍Xi+σB(t),
where u≥0 is the initial surplus and c>0 is the premium rate collected per unit time. {X1,X2,…}, representing the individual claim amounts, is a sequence of independent and identically distributed (i.i.d.) random variables distributed like a strictly positive variable X with cumulative distribution function (c.d.f.) F(x)=ℙ(X≤x), probability density function (p.d.f.) f(x), and Laplace transform f^(s)=∫0∞‍e-sxf(x)dx. {N(t),t≥0}, a renewal process counting the number of claims up to time t, is defined by N(t)=sup{n:V1+⋯+Vn≤t}, where {Vi}i=1∞ denoting the interclaim time random variables are i.i.d. like a generic variable V with c.d.f. K, p.d.f. k, and Laplace transform k^. Finally, {B(t),t≥0} is a standard Brownian motion starting from 0, and σ>0 is the dispersion parameter. We assume that {Xi}, {Vi}, and {B(t)} are mutually independent and that c𝔼V>𝔼X is a net profit condition.LetT=inf{t≥0:U(t)<0} be the ruin time with the understanding that T=∞ if U(t)≥0 for all t≥0; that is, ruin does not occur. Two ruin-related quantities of interest in ruin theory are the surplus immediately before ruin U(T-) and the deficit at ruin |U(T)|. A unified tool to study these ruin quantities is the Gerber-Shiu discounted penalty function. Recently, some researchers are interested in generalizing the Gerber-Shiu function by incorporating other quantities. One generalization is to consider the maximum surplus prior to ruin, namely, U¯(T)=sup0≤t<TU(t), and this results in the following generalized discounted penalty function:
(2)ϕ123(u)=𝔼[e-δTw123(U(T-),|U(T)|,U¯(T))×I(T<∞)∣U(0)=ue-δTw123(U(T-),|U(T)|,U¯(T))],
where δ≥0 is the interest force, w123(x1,x2,x3):ℝ+3→ℝ+ is a measurable function satisfying some integrability conditions, and I(A) is the indicator function of event A.In this paper, we are interested in the specific penalty function(3)w123(x1,x2,x3)=w(x1,x2)I(x3<b),
where b>0 and w(x1,x2):ℝ+2→ℝ+ is a measurable function. Thus, ϕ123(u) reduces to the following generalized discounted penalty function for 0≤u<b:
(4)ϕ(u;b)=𝔼[e-δTw(U(T-),|U(T)|)×I(U¯(T)<b,T<∞)∣U(0)=u].
Note that ruin can be caused by either a claim or oscillation of the Brownian motion. Set w(0,0)=1 without loss of generality. We can decompose ϕ(u;b) as
(5)ϕ(u;b)=ϕw(u;b)+ϕd(u;b),
where
(6)ϕw(u;b)=𝔼[e-δTw(U(T-),|U(T)|)×I(U¯(T)<b,U(T)<0,T<∞)∣U(0)=u],ϕd(u;b)=𝔼[e-δTI(U¯(T)<b,U(T)=0,T<∞)∣U(0)=u]
are, respectively, the discounted penalty functions caused by a claim and oscillation of the Brownian motion.Letb→∞; then ϕ(u;b), ϕw(u;b), and ϕd(u;b) reduce to the original discounted penalty functions, denoted by ϕ(u;∞), ϕw(u;∞), and ϕd(u;∞), which have been well studied by Li and Garrido [1].The marginal distribution ofU¯(T) has been studied by Li and Dickson [2] in a Sparre Andersen risk model and Li and Lu [3] in a Markov-modulated risk model. Recently, Cheung and Landriault [4] have studied the generalized discounted penalty function ϕ123(u) in the MAP risk model. In this paper, we focus on the evaluation of the generalized discounted penalty functions ϕw(u;b) and ϕd(u;b). In Section 2, we show that ϕw(u;b) and ϕd(u;b) satisfy some integrodifferential equations with boundary conditions. The solutions of the integrodifferential equations will be studied in Section 3. We show that ϕw(u;b) and ϕd(u;b) can be expressed via ϕw(u;∞) and ϕd(u;∞) and the solutions of a homogeneous integrodifferential equation.


## 2. Integrodifferential Equations

In this section, we show thatϕw(u;b) and ϕd(u;b) satisfy some integrodifferential equations with boundary conditions. Before presenting our main results, we need some preliminaries.


### 2.1. Preliminaries

Consider a spectrally negative Lévy process{X(t),t≥0} defined on {Ω,ℱ,𝔽={ℱt}t≥0,ℙ} which satisfies the usual conditions. Let ℙx and 𝔼x be shorthand for ℙ(·∣X(0)=x) and 𝔼(·∣X(0)=x) with the understanding that ℙ0=ℙ and 𝔼0=𝔼 for the special case x=0.The Laplace exponent ofX(t) is defined as
(7)ψ(β)=1tln𝔼[eβX(t)],
which is finite at least on the positive half axis since X(t) does not have positive jumps. Furthermore, ψ(β) is convex and limβ→∞ψ(β)=∞. Define the right inverse
(8)Φ(q)=sup{β≥0:ψ(β)=q},
for each q≥0.Forq≥0, the scale function W(q)(x):ℝ→[0,∞) is defined as the continuous function on [0,∞) such that
(9)∫0∞‍e-sxW(q)(x)dx=1ψ(β)-q,β>Φ(q),
and W(q)(x)=0 for x<0. For q≥0, define the function Z(q)(x) by
(10)Z(q)(x)=1+q∫0x‍W(q)(y)dy,
with Z(q)(x)=1 for x≤0.Forx∈ℝ, define
(11)τx+=inf{t>0:X(t)>x},τx-=inf{t>0:X(t)<x}.
The scale functions play an important role in studying the one-sided and two-sided exit problems for spectrally negative Lévy process (see, e.g., Section 8 of [5]). By formula (8.9) of [5], we have for x≤a and q≥0(12)Q(q)(x;a):=𝔼x[e-qτ0-I(τ0-<τa+)]=Z(q)(x)-Z(q)(a)W(q)(x)W(q)(a).
The q-potential measure killed on exiting [0,a] is defined as
(13)R(q)(x,dy;a)=∫0∞‍e-qtℙx(X(t)∈dy,t<τ0-∧τa+)dt,
for q≥0, where τ0-∧τa+=min{τ0-,τa+}. By Theorem 8.7 of [5], we know that R(q)(x,dy;a) has a density r(q)(x,y;a) given by
(14)r(q)(x,y;a)=W(q)(x)W(q)(a-y)W(q)(a)-W(q)(x-y),
for x,y∈[0,a].We will reproduce formulae (12) and (14) when X(t) is a Brownian motion with drift; that is, X(t)=ct+σB(t). In this case, we have
(15)ψ(β)=cβ+12σ2β2,Φ(q)=-cσ2+c2σ4+2qσ2.
Inverting the Laplace transform (9) gives
(16)W(q)(x)=e-q1x-e-q2x(σ2/2)(q2-q1),
where q1=c/σ2-c2/σ4+2q/σ2, q2=c/σ2+c2/σ4+2q/σ2. Then, by (10), we have
(17)Z(q)(x)=q2e-q1x-q1e-q2xq2-q1.
By (16) and (17), we can reproduce (12) and (14) as follows:
(18)Q(q)(x;a)=e-q2x-q1a-e-q1x-q2ae-q1a-e-q2a,x≤a,r(q)(x,y;a)=(e-q1x-q2a-e-q2x-q1a)(eq1y-eq2y)(σ2/2)(q2-q1)(e-q1a-e-q2a),0≤y<x≤a,r(q)(x,y;a)=(e-q1x-e-q2x)(e-q1(a-y)-e-q2(a-y))(σ2/2)(q2-q1)(e-q1a-e-q2a),0≤x≤y≤a.


### 2.2. Main Results

Forq>0, denote by eq an exponential r.v. with mean 1/q. Then, the interclaim generic variable V can be expressed as V=eλ1+⋯+eλn, where eλ1,…,eλn are mutually independent. Let ℐ be the identity operator.Now, we first consider the generalized discounted penalty functionϕd(u;b).Theorem 1.
For0<u<b, ϕd(u;b) is differentiable at least 2n times and satisfies the following integrodifferential equation:
(19)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)=∫0u‍ϕd(u-x;b)f(x)dx,
with boundary conditions
(20)∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=0=1,∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=b=0,
for j=0,1,…,n-1.Proof.
Prior to the first claim, the surplus process behaves like the Brownian motionX(t)=u+ct+σB(t) starting from u. By considering whether or not ruin occurs prior to the first claim, we have
(21)ϕd(u;b)=𝔼u[e-δτ0-I(τ0-<τb+,τ0-<V)]+∫t∈(0,∞)‍∫y∈(0,b)‍∫x∈(0,y)‍e-δtk(t)×ℙu(X(t)∈dy,t<τ0-∧τb+)×ϕd(y-x;b)f(x)dxdt=𝔼u[e-δτ0-I(τ0-<τb+,τ0-<V)]+∫0b‍σd(y;b)𝔼u[e-δVI(X(V)∈dy,V<τ0-∧τb+)],
where σd(y;b)=∫0y‍ϕd(y-x;b)f(x)dx.
Now forj=1,…,n, let ej=∑i=jn‍eλi,
(22)Aj(u;b)=𝔼u[e-δτ0-I(τ0-<τb+∧ej)],Bj(u,dy;b)=𝔼u[e-δejI(X(ej)∈dy,ej<τ0-∧τb+)],
and define
(23)ϕd,j(u;b)=Aj(u;b)+∫0b‍σd(y;b)Bj(u,dy;b).
Then, we have ϕd(u;b)=ϕd,1(u;b).
Forj=1,…,n-1, we have by the Markov property,
(24)Aj(u;b)=𝔼u[e-δτ0-I(τ0-<τb+∧eλj)]+𝔼u[e-δτ0-I(τ0-<τb+,eλj<τ0-<ej+1)]=𝔼u[e-(λj+δ)τ0-I(τ0-<τb+)]+∫0∞‍∫0b‍λje-(λj+δ)tℙu(X(t)∈dx,t<τ0-∧τb+)×𝔼x[e-δτ0-I(τ0-<τb+∧ej+1)]dt=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)Aj+1(x;b).
Similarly, we have
(25)Bj(u,dy;b)=λj∫0b‍R(λj+δ)(u,dx;b)Bj+1(x,dy;b).
Thus, (23) becomes
(26)ϕd,j(u;b)=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)Aj+1(x;b)+λj∬0bσd(y;b)R(λj+δ)(u,dx;b)Bj+1(x,dy;b)=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)ϕd,j+1(x;b),
while for j=n, (23) reads
(27)ϕd,n(u;b)=Q(λn+δ)(u;b)+λn∫0b‍R(λn+δ)(u,dx;b)ϕd,n+1(x;b),
with the understanding that ϕd,n+1(x;b):=σd(x;b).
Letqj,1=c/σ2-c2/σ4+2(λj+δ)/σ2, qj,2=c/σ2+c2/σ4+2(λj+δ)/σ2. Then, by formulae (18), we have for j=1,…,n,
(28)ϕd,j(u;b)=e-qj,2u-qj,1b-e-qj,1u-qj,2be-qj,1b-e-qj,2b+λj(e-qj,1u-qj,2b-e-qj,2u-qj,1b)(σ2/2)(qj,2-qj,1)(e-qj,1b-e-qj,2b)×∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx+λj(e-qj,1u-e-qj,2u)(σ2/2)(qj,2-qj,1)(e-qj,1b-e-qj,2b)×∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx.
From (28), we obtain the boundary conditions
(29)ϕd,j(0;b)=1,ϕd,j(b;b)=0.
Furthermore, it is readily seen from (28) that ϕd,j(u) is differentiable with respect to u in (0,b). From this fact, we can check that ϕd,j(u) is twice differentiable with respect to u in (0,b).
Note that(30)(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,1u-qj,2b∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx]=(∂∂u+qj,2ℐ)[(e-qj,2b-e-(qj,1-qj,2)u-qj,2b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,2u-qj,1b∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx]=(∂∂u+qj,1ℐ)[(e-(qj,2-qj,1)u-qj,1b-e-qj,1b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,1u∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx]=(∂∂u+qj,2ℐ)[(e-(qj,1-qj,2)u-qj,2b-e-qj,1b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,2u∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx]=(∂∂u+qj,1ℐ)[(e-qj,2b-e-(qj,2-qj,1)u-qj,1b)ϕd,j+1(u)].
Then, applying the operator (∂/∂u+qj,1ℐ)(∂/∂u+qj,2ℐ) to both sides of (28) and using previous identities, we can obtain
(31)ϕd,j+1(u;b)=(λj+δλjℐ-cλj∂∂u-σ22λj∂2∂u2)ϕd,j(u;b).
Recursively, we obtain
(32)ϕd,j+1(u)=∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b).
In particular, by (32), the twice differentiability of ϕd,n(u;b) implies that ϕd(u;b) is 2n times differentiable. Setting j=n in (32) gives the integrodifferential equation (19). Finally, by (29) and (32), we obtain the boundary conditions (20).Now we derive integrodifferential equation forϕw(u). Similar to Theorem 1, we have the following.Theorem 2.
Letω(u)=∫u∞‍w(u,x-u)f(x)dx. If ω(u) is differentiable, then for 0<u<b, ϕw(u;b) is differentiable at least 2n times and satisfies the following integrodifferential equation:
(33)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕw(u;b)=∫0u‍ϕw(u-x;b)f(x)dx+ω(u),
with boundary conditions
(34)∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=0=0,∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=b=0,
for j=0,1,…,n-1.Proof.
Similar to (21), we have
(35)ϕw(u)=∫0b‍σw(y;b)𝔼u[e-δVI(X(V)∈dy,V<τ0-∧τb+)],
where σw(y)=∫0y‍ϕw(y-x;b)f(x)dx+ω(y). The rest of the proof is exactly the same as Theorem 1.Remark 3.
Different from Li and Garrido [1], we analyze the differentiability and derive the integrodifferential equation for the generalized discounted penalty function at the same time. Instead of using Taylor’s expansion, the techniques used in the proof of Theorems 1 and 2 are based on the one-sided and two-sided exit results in Lévy process. We remark that such techniques have also been successfully used in analyzing the dependent risk model perturbed by diffusion (see, e.g., Zhang and Yang [6]).Remark 4.
We have significantly relaxed the condition on the2n times differentiability of the Gerber-Shiu functions presented in Propositions 2 and 4 of Li and Garrido [1], where the twice differentiability of ω(u) and f(x) has been assumed.


## 2.1. Preliminaries

Consider a spectrally negative Lévy process{X(t),t≥0} defined on {Ω,ℱ,𝔽={ℱt}t≥0,ℙ} which satisfies the usual conditions. Let ℙx and 𝔼x be shorthand for ℙ(·∣X(0)=x) and 𝔼(·∣X(0)=x) with the understanding that ℙ0=ℙ and 𝔼0=𝔼 for the special case x=0.The Laplace exponent ofX(t) is defined as
(7)ψ(β)=1tln𝔼[eβX(t)],
which is finite at least on the positive half axis since X(t) does not have positive jumps. Furthermore, ψ(β) is convex and limβ→∞ψ(β)=∞. Define the right inverse
(8)Φ(q)=sup{β≥0:ψ(β)=q},
for each q≥0.Forq≥0, the scale function W(q)(x):ℝ→[0,∞) is defined as the continuous function on [0,∞) such that
(9)∫0∞‍e-sxW(q)(x)dx=1ψ(β)-q,β>Φ(q),
and W(q)(x)=0 for x<0. For q≥0, define the function Z(q)(x) by
(10)Z(q)(x)=1+q∫0x‍W(q)(y)dy,
with Z(q)(x)=1 for x≤0.Forx∈ℝ, define
(11)τx+=inf{t>0:X(t)>x},τx-=inf{t>0:X(t)<x}.
The scale functions play an important role in studying the one-sided and two-sided exit problems for spectrally negative Lévy process (see, e.g., Section 8 of [5]). By formula (8.9) of [5], we have for x≤a and q≥0(12)Q(q)(x;a):=𝔼x[e-qτ0-I(τ0-<τa+)]=Z(q)(x)-Z(q)(a)W(q)(x)W(q)(a).
The q-potential measure killed on exiting [0,a] is defined as
(13)R(q)(x,dy;a)=∫0∞‍e-qtℙx(X(t)∈dy,t<τ0-∧τa+)dt,
for q≥0, where τ0-∧τa+=min{τ0-,τa+}. By Theorem 8.7 of [5], we know that R(q)(x,dy;a) has a density r(q)(x,y;a) given by
(14)r(q)(x,y;a)=W(q)(x)W(q)(a-y)W(q)(a)-W(q)(x-y),
for x,y∈[0,a].We will reproduce formulae (12) and (14) when X(t) is a Brownian motion with drift; that is, X(t)=ct+σB(t). In this case, we have
(15)ψ(β)=cβ+12σ2β2,Φ(q)=-cσ2+c2σ4+2qσ2.
Inverting the Laplace transform (9) gives
(16)W(q)(x)=e-q1x-e-q2x(σ2/2)(q2-q1),
where q1=c/σ2-c2/σ4+2q/σ2, q2=c/σ2+c2/σ4+2q/σ2. Then, by (10), we have
(17)Z(q)(x)=q2e-q1x-q1e-q2xq2-q1.
By (16) and (17), we can reproduce (12) and (14) as follows:
(18)Q(q)(x;a)=e-q2x-q1a-e-q1x-q2ae-q1a-e-q2a,x≤a,r(q)(x,y;a)=(e-q1x-q2a-e-q2x-q1a)(eq1y-eq2y)(σ2/2)(q2-q1)(e-q1a-e-q2a),0≤y<x≤a,r(q)(x,y;a)=(e-q1x-e-q2x)(e-q1(a-y)-e-q2(a-y))(σ2/2)(q2-q1)(e-q1a-e-q2a),0≤x≤y≤a.


## 2.2. Main Results

Forq>0, denote by eq an exponential r.v. with mean 1/q. Then, the interclaim generic variable V can be expressed as V=eλ1+⋯+eλn, where eλ1,…,eλn are mutually independent. Let ℐ be the identity operator.Now, we first consider the generalized discounted penalty functionϕd(u;b).Theorem 1.
For0<u<b, ϕd(u;b) is differentiable at least 2n times and satisfies the following integrodifferential equation:
(19)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)=∫0u‍ϕd(u-x;b)f(x)dx,
with boundary conditions
(20)∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=0=1,∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=b=0,
for j=0,1,…,n-1.Proof.
Prior to the first claim, the surplus process behaves like the Brownian motionX(t)=u+ct+σB(t) starting from u. By considering whether or not ruin occurs prior to the first claim, we have
(21)ϕd(u;b)=𝔼u[e-δτ0-I(τ0-<τb+,τ0-<V)]+∫t∈(0,∞)‍∫y∈(0,b)‍∫x∈(0,y)‍e-δtk(t)×ℙu(X(t)∈dy,t<τ0-∧τb+)×ϕd(y-x;b)f(x)dxdt=𝔼u[e-δτ0-I(τ0-<τb+,τ0-<V)]+∫0b‍σd(y;b)𝔼u[e-δVI(X(V)∈dy,V<τ0-∧τb+)],
where σd(y;b)=∫0y‍ϕd(y-x;b)f(x)dx.
Now forj=1,…,n, let ej=∑i=jn‍eλi,
(22)Aj(u;b)=𝔼u[e-δτ0-I(τ0-<τb+∧ej)],Bj(u,dy;b)=𝔼u[e-δejI(X(ej)∈dy,ej<τ0-∧τb+)],
and define
(23)ϕd,j(u;b)=Aj(u;b)+∫0b‍σd(y;b)Bj(u,dy;b).
Then, we have ϕd(u;b)=ϕd,1(u;b).
Forj=1,…,n-1, we have by the Markov property,
(24)Aj(u;b)=𝔼u[e-δτ0-I(τ0-<τb+∧eλj)]+𝔼u[e-δτ0-I(τ0-<τb+,eλj<τ0-<ej+1)]=𝔼u[e-(λj+δ)τ0-I(τ0-<τb+)]+∫0∞‍∫0b‍λje-(λj+δ)tℙu(X(t)∈dx,t<τ0-∧τb+)×𝔼x[e-δτ0-I(τ0-<τb+∧ej+1)]dt=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)Aj+1(x;b).
Similarly, we have
(25)Bj(u,dy;b)=λj∫0b‍R(λj+δ)(u,dx;b)Bj+1(x,dy;b).
Thus, (23) becomes
(26)ϕd,j(u;b)=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)Aj+1(x;b)+λj∬0bσd(y;b)R(λj+δ)(u,dx;b)Bj+1(x,dy;b)=Q(λj+δ)(u;b)+λj∫0b‍R(λj+δ)(u,dx;b)ϕd,j+1(x;b),
while for j=n, (23) reads
(27)ϕd,n(u;b)=Q(λn+δ)(u;b)+λn∫0b‍R(λn+δ)(u,dx;b)ϕd,n+1(x;b),
with the understanding that ϕd,n+1(x;b):=σd(x;b).
Letqj,1=c/σ2-c2/σ4+2(λj+δ)/σ2, qj,2=c/σ2+c2/σ4+2(λj+δ)/σ2. Then, by formulae (18), we have for j=1,…,n,
(28)ϕd,j(u;b)=e-qj,2u-qj,1b-e-qj,1u-qj,2be-qj,1b-e-qj,2b+λj(e-qj,1u-qj,2b-e-qj,2u-qj,1b)(σ2/2)(qj,2-qj,1)(e-qj,1b-e-qj,2b)×∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx+λj(e-qj,1u-e-qj,2u)(σ2/2)(qj,2-qj,1)(e-qj,1b-e-qj,2b)×∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx.
From (28), we obtain the boundary conditions
(29)ϕd,j(0;b)=1,ϕd,j(b;b)=0.
Furthermore, it is readily seen from (28) that ϕd,j(u) is differentiable with respect to u in (0,b). From this fact, we can check that ϕd,j(u) is twice differentiable with respect to u in (0,b).
Note that(30)(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,1u-qj,2b∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx]=(∂∂u+qj,2ℐ)[(e-qj,2b-e-(qj,1-qj,2)u-qj,2b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,2u-qj,1b∫0u‍(eqj,1x-eqj,2x)ϕd,j+1(x;b)dx]=(∂∂u+qj,1ℐ)[(e-(qj,2-qj,1)u-qj,1b-e-qj,1b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,1u∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx]=(∂∂u+qj,2ℐ)[(e-(qj,1-qj,2)u-qj,2b-e-qj,1b)ϕd,j+1(u)],(∂∂u+qj,1ℐ)(∂∂u+qj,2ℐ)×[e-qj,2u∫ub‍(e-qj,1(b-x)-e-qj,2(b-x))ϕd,j+1(x;b)dx]=(∂∂u+qj,1ℐ)[(e-qj,2b-e-(qj,2-qj,1)u-qj,1b)ϕd,j+1(u)].
Then, applying the operator (∂/∂u+qj,1ℐ)(∂/∂u+qj,2ℐ) to both sides of (28) and using previous identities, we can obtain
(31)ϕd,j+1(u;b)=(λj+δλjℐ-cλj∂∂u-σ22λj∂2∂u2)ϕd,j(u;b).
Recursively, we obtain
(32)ϕd,j+1(u)=∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b).
In particular, by (32), the twice differentiability of ϕd,n(u;b) implies that ϕd(u;b) is 2n times differentiable. Setting j=n in (32) gives the integrodifferential equation (19). Finally, by (29) and (32), we obtain the boundary conditions (20).Now we derive integrodifferential equation forϕw(u). Similar to Theorem 1, we have the following.Theorem 2.
Letω(u)=∫u∞‍w(u,x-u)f(x)dx. If ω(u) is differentiable, then for 0<u<b, ϕw(u;b) is differentiable at least 2n times and satisfies the following integrodifferential equation:
(33)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕw(u;b)=∫0u‍ϕw(u-x;b)f(x)dx+ω(u),
with boundary conditions
(34)∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=0=0,∏i=1j‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;b)|u=b=0,
for j=0,1,…,n-1.Proof.
Similar to (21), we have
(35)ϕw(u)=∫0b‍σw(y;b)𝔼u[e-δVI(X(V)∈dy,V<τ0-∧τb+)],
where σw(y)=∫0y‍ϕw(y-x;b)f(x)dx+ω(y). The rest of the proof is exactly the same as Theorem 1.Remark 3.
Different from Li and Garrido [1], we analyze the differentiability and derive the integrodifferential equation for the generalized discounted penalty function at the same time. Instead of using Taylor’s expansion, the techniques used in the proof of Theorems 1 and 2 are based on the one-sided and two-sided exit results in Lévy process. We remark that such techniques have also been successfully used in analyzing the dependent risk model perturbed by diffusion (see, e.g., Zhang and Yang [6]).Remark 4.
We have significantly relaxed the condition on the2n times differentiability of the Gerber-Shiu functions presented in Propositions 2 and 4 of Li and Garrido [1], where the twice differentiability of ω(u) and f(x) has been assumed.


## 3. The Solutions

In this section, we derive the solutions of the integrodifferential equations (19) and (33).We relax the restriction0<u<b to u>0 in equations (19) and (33) and note by Theorem 1 of Li and Garrido [1] that
(36)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕd(u;∞)=∫0u‍ϕd(u-x;∞)f(x)dx,∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)ϕw(u;∞)=∫0u‍ϕw(u-x;∞)f(x)dx+ω(u).
Thus, by the general theory of differential equations, we have
(37)ϕd(u;b)=ϕd(u;∞)+∑i=12n‍kd,ivi(u),ϕw(u;b)=ϕw(u;∞)+∑i=12n‍kw,ivi(u),
where kd,i’s and kw,i’s are constants determined by the boundary conditions (20), (34), and v1(u),…,v2n(u) are linearly independent solutions of the following homogeneous integrodifferential equation:
(38)∏i=1n‍(λi+δλiℐ-cλi∂∂u-σ22λi∂2∂u2)v(u)=∫0u‍v(u-x)f(x)dx,u≥0.We remark thatϕd(u;∞) and ϕw(u;∞) have been well investigated by Li and Garrido [1]. If the p.d.f. f has a rational Laplace transform (a ratio of two polynomials), the solutions vi(u)’s to the homogeneous integrodifferential equation (38) can be obtained by Laplace transforms as follows.Assume that the claim sizeX is rationally distributed with
(39)f^(s)=qm-1(s)qm(s),
where qm(s) is a polynomial of degree m without zeros in the right half complex plane and qm-1(s) is a polynomial of degree m-1 satisfying qm-1(0)=qm(0). Assume without loss of generality that the leading coefficient of qm(s) is 1.Let(40)B(s)=∏i=1n‍(λi+δλi-cλis-σ22λis2)=∑i=02n‍Bisi.
Taking Laplace transforms on both sides of (38) gives(41)B(s)v^(s)-C(s)=v^(s)f^(s),
where C(s)=∑i=12n‍∑j=0i-1‍Bisjv(i-1-j)(0) is a polynomial of degree 2n-1 or less. Then (41) gives
(42)v^(s)=C(s)qm(s)B(s)qm(s)-qm-1(s).
By Theorem 2 of [1], the denominator of (42) can be factorized as follows:
(43)B(s)qm(s)-qm-1(s)=D∏i=1n‍(s-ρi)∏j=1n+m‍(s+Rj),
where D=∏i=1n‍(-σ2/2λi)n, ρi’s, and -Rj’s are zeros of the denominator of (42) lying in the right and left half complex plane, respectively. If ρi’s and Rj’s are distinct, we have by partial fraction
(44)v^(s)=∑i=1n‍ais-ρi+∑j=1n+m‍bjs+Rj,
where
(45)ai=C(ρi)qm(ρi)D∏r=1,r≠in‍(ρi-ρr)∏j=1n+m‍(ρi+Rj),bj=C(-Rj)qm(-Rj)D∏i=1n‍(-Rj-ρi)∏r=1,r≠jn+m‍(Rr-Rj).
Upon inversion, (44) gives
(46)v(u)=∑i=1n‍aieρiu+∑j=1n+m‍bje-Rju,u≥0.
Finally, the 2n linearly independent solutions v1(u),…,v2n(u) can be obtained by specifying the initial conditions
(47)vi(j)(0)=I(j=i-1),i=1,…,2n,j=0,1,…,2n-1,
in C(s).In the rest, we pay attention to the classical compound Poisson risk model perturbed by diffusion; that is,n=1. From Li [7], we know that the two linearly independent solutions v1(u) and v2(u) to (38) (n=1) can be chosen to be
(48)v1(u)=ϕd(u;∞),v2(u)=∫0u‍ϕd(u-x;∞)eρ1xdx,
where ρ1 is the solution of equation λ1+δ-cs-(σ2/2)s2-λ1f^(s)=0 in the right half complex plane.By the boundary conditions(49)ϕd(0;b)=1,ϕd(b;b)=0,ϕw(0;b)=0,ϕw(b;0)=0,
we obtain
(50)kd,1=0,kd,2=-ϕd(b;∞)∫0b‍ϕd(b-x;∞)eρ1xdx,kw,1=0,kw,2=-ϕw(b;∞)∫0b‍ϕd(b-x;∞)eρ1xdx.
Then, the generalized discounted penalty functions are given by
(51)ϕd(u;b)=ϕd(u;∞)-∫0u‍ϕd(u-x;∞)eρ1xdx∫0b‍ϕd(b-x;∞)eρ1xdxϕd(u;∞),ϕw(u;b)=ϕw(u;∞)-∫0u‍ϕd(u-x;∞)eρ1xdx∫0b‍ϕd(b-x;∞)eρ1xdxϕw(u;∞),
which implies that the generalized discounted penalty functions are proportional to the original discounted penalty functions.



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*Source: 759352-2013-12-03.xml*