1. Introduction
Consider the following Sparre Andersen risk model perturbed by a Brownian motion:
()
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
. {
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
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
. 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.
Let
T = 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,
, and this results in the following generalized discounted penalty function:
()
where
δ ≥ 0 is the interest force,
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
()
where
b > 0 and
is a measurable function. Thus,
ϕ123(
u) reduces to the following generalized discounted penalty function for 0 ≤
u <
b:
()
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
()
where
()
are, respectively, the discounted penalty functions caused by a claim and oscillation of the Brownian motion.
Let b → ∞; 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 of 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 that ϕw(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 of
X(
t) is defined as
()
which is finite at least on the positive half axis since
X(
t) does not have positive jumps. Furthermore,
ψ(
β) is convex and lim
β→∞ψ(
β) =
∞. Define the right inverse
()
for each
q ≥ 0.
For
q ≥ 0, the scale function
W(q)(
x) :
ℝ → [0,
∞) is defined as the continuous function on [0,
∞) such that
()
and
W(q)(
x) = 0 for
x < 0. For
q ≥ 0, define the function
Z(q)(
x) by
()
with
Z(q)(
x) = 1 for
x ≤ 0.
For
x ∈
ℝ, define
()
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
()
The
q-potential measure killed on exiting [0,
a] is defined as
()
for
q ≥ 0, where
. By Theorem 8.7 of [
5], we know that
R(q)(
x,
dy;
a) has a density
r(q)(
x,
y;
a) given by
()
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
()
Inverting the Laplace transform (
9) gives
()
where
,
. Then, by (
10), we have
()
By (
16) and (
17), we can reproduce (
12) and (
14) as follows:
()
2.2. Main Results
For q > 0, denote by eq an exponential r.v. with mean 1/q. Then, the interclaim generic variable V can be expressed as , where are mutually independent. Let ℐ be the identity operator.
Now, we first consider the generalized discounted penalty function ϕd(u; b).
Theorem 1. For 0 < u < b, ϕd(u; b) is differentiable at least 2n times and satisfies the following integrodifferential equation:
()
with boundary conditions
()
for
j = 0,1, …,
n − 1.
Proof. Prior to the first claim, the surplus process behaves like the Brownian motion X(t) = u + ct + σB(t) starting from u. By considering whether or not ruin occurs prior to the first claim, we have
()
where
.
Now for j = 1, …, n, let ,
()
and define
()
Then, we have
ϕd(
u;
b) =
ϕd,1(
u;
b).
For j = 1, …, n − 1, we have by the Markov property,
()
Similarly, we have
()
Thus, (
23) becomes
()
while for
j =
n, (
23) reads
()
with the understanding that
ϕd,n+1(
x;
b): =
σd(
x;
b).
Let , . Then, by formulae (18), we have for j = 1, …, n,
()
From (
28), we obtain the boundary conditions
()
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
()
Then, applying the operator (
∂/
∂u +
qj,1ℐ)(
∂/
∂u +
qj,2ℐ) to both sides of (
28) and using previous identities, we can obtain
()
Recursively, we obtain
()
In particular, by (
32), the twice differentiability of
ϕd,n(
u;
b) implies that
ϕd(
u;
b) is 2
n 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 for ϕw(u). Similar to Theorem 1, we have the following.
Theorem 2. Let . 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:
()
with boundary conditions
()
for
j = 0,1, …,
n − 1.
Proof. Similar to (21), we have
()
where
. 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 the 2n 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 restriction 0 <
u <
b to
u > 0 in equations (
19) and (
33) and note by Theorem 1 of Li and Garrido [
1] that
()
Thus, by the general theory of differential equations, we have
()
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:
()
We remark that ϕd(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 size
X is rationally distributed with
()
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
()
Taking Laplace transforms on both sides of (
38) gives
()
where
is a polynomial of degree 2
n − 1 or less. Then (
41) gives
()
By Theorem 2 of [
1], the denominator of (
42) can be factorized as follows:
()
where
,
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
()
where
()
Upon inversion, (
44) gives
()
Finally, the 2
n linearly independent solutions
v1(
u), …,
v2n(
u) can be obtained by specifying the initial conditions
()
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
()
where
ρ1 is the solution of equation
in the right half complex plane.
By the boundary conditions
()
we obtain
()
Then, the generalized discounted penalty functions are given by
()
which implies that the generalized discounted penalty functions are proportional to the original discounted penalty functions.
Acknowledgments
The authors would like to thank the editor and the anonymous referee for their very helpful suggestions and comments. This research is supported by the Fundamental Research Funds for the Central Universities (Project no. CQDXWL-2012-001).
