Approximation for the Finite-Time Ruin Probability of a General Risk Model with

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Volume 2011, Article ID 852852,14pages doi:10.1155/2011/852852

Research Article

Approximation for the Finite-Time Ruin Probability of a General Risk Model with

Constant Interest Rate and Extended Negatively Dependent Heavy-Tailed Claims

Yang Yang,

^{1, 2}

Xin Ma,

³

and Jin-guan Lin

²

1School of Mathematics and Statistics, Nanjing Audit University, Nanjing 210029, China

2Department of Mathematics, Southeast University, Nanjing 210096, China

3Golden Audit College, Nanjing Audit University, Nanjing 210029, China

Correspondence should be addressed to Yang Yang,[email protected] Received 6 March 2011; Revised 3 May 2011; Accepted 9 May 2011

Academic Editor: P. Liatsis

Copyrightq2011 Yang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a general continuous-time risk model with a constant interest rate. In this model, claims arrive according to an arbitrary counting process, while their sizes have dominantly varying tails and fulfill an extended negative dependence structure. We obtain an asymptotic formula for the finite-time ruin probability, which extends a corresponding result of Wang2008.

1. The Dependent General Risk Model

In this paper, we consider the finite-time ruin probability with constant interest rate in a dependent general risk model. In this model, the claim sizes{Xn, n ≥ 1}form a sequence of identically distributed, not necessarily independent, and nonnegative random variables r.v.swith common distributionFsuch thatFx 1−Fx PX1 > x> 0 for allx > 0;

the claim arrival process{Nt, t≥0}is a general counting process, namely, a nonnegative, nondecreasing, right continuous, and integer-valued stochastic process with 0 < ENt λt<∞for all larget >0. The times of the successive claims are denoted by{τn, n≥1}. The total amount of premiums accumulated up to timet≥0, denoted byCtwithC0 0 and Ct<∞almost surely for everyt >0, is another nonnegative and nondecreasing stochastic process. Assume that{Xn, n ≥1},{Nt, t≥ 0}and{Ct, t ≥0}are mutually independent.

Letδ > 0 be the constant interest ratei.e., after timetone dollar becomese^δtdollars, and

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letx ≥ 0 be the initial capital reserve of an insurance company. Then, the total discounted reserve up to timet≥0, denoted byDt, x, can be written as

Dt, x x _t

0

e^−δsCds−^Nt

n1

Xne^−δτⁿ. 1.1

For a finite timeT >0, the finite-time ruin probability is defined by

Ψx, T PDt, x<0, for some 0≤t≤T

P

sup

t∈0,T

_Nt

n1

X_ne^−δτⁿ − _t

0

e^−δsCds

> x

, 1.2

while the ultimate ruin probability is defined by

Ψx Ψx,∞ PDt, x<0, for somet≥0. 1.3

If the claim sizes{Xn, n≥1}are independent r.v.s, the model is called the independent general risk model, which was introduced by Wang1. In particular, ifCt ct,t≥0, with c >0 a deterministic constant and{Nt, t≥0}is a Poisson process, then the model reduces to the classical one.

2. Introduction and Main Result

Hereafter, all limit relationships hold forx tending to∞unless otherwise stated. For two positive functionsfxandgx, we writefx ∼ gxif limfx/gx 1; writefx ^º gxif lim supfx/gx≤1 andfx ogxif limfx/gx 0. The indicator function of an eventAis denoted by1A.

In risk theory, heavy-tailed distributions are often used to model large claim amounts.

They play a key role in insurance and finance. We will restrict the claim-size distributionF to be heavy tailed. A distribution V is said to be dominatedly varying tailed, denoted by V ∈ D, if lim supVxy/Vx <∞for anyy >0. A distributionV is said to be long tailed, denoted byV ∈ L, if limVxy/Vx 1 for anyy > 0. A distributionV is said to be subexponential, denoted byV ∈ S, ifV^n∗x ∼nVxfor anyn ≥2, whereV^n∗denotes the n-fold convolution of itself. A distributionV is said to be regularly varying tailed, denoted byR−α, α >0, if limVxy/Vx y^−αfor anyy≥1. A proper inclusion relationship holds that

R_−α ⊂ L ∩ D ⊂ S ⊂ L, 2.1

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see, for example, Cline2or Embrechts and Omey3. For a distributionV, denote the upper Matuszewska index of the distributionV by

J_V −_ylim

→ ∞

logV_∗ y

logy withV_∗ y

lim inf_x

→ ∞

V xy

Vx , y >1. 2.2

In the terminology of Bingham et al.4, the quantityJ_V is actually the upper Matuszewska index of the function 1/Vx, x ≥ 0, as also pointed out in Tang and Tsitsiashvili 5.

Additionally, denote L_V lim_y1V_∗y clearly, 0 ≤ L_V ≤ 1 . The presented definitions yield that the following assertions are equivalent:

iV ∈ D, iiV_∗ y

>0 for some y >1, iiiL_V >0, ivJ_V <∞.

2.3 The asymptotic behavior of the ruin probability in the classical risk model has been extensively investigated in the literature. Kl ¨uppelberg and Stadtm ¨uller6considered the ultimate ruin probability for the case of regularly-varying-tailed claim sizes. Using the reflected random walk theory, Asmussen 7 extended the study to a larger class of heavy-tailed distributions; see Corollary 4.1ii of his paper. Complementary discussions on the ultimate ruin probability can be found in Kalashnikov and Konstantinides 8, Konstantinides et al.9, Tang10, among others.

In this paper, we are interested in the finite-time ruin probability. In this aspect, Tang 11established an asymptotic result in the classical risk model: under the conditionF ∈ S, he obtained that for everyT >0 for whichλT>0,

Ψx, T∼ _T

0−F

xe^δt λdt. 2.4

Recently, Wang1derived some important and interesting results in two independent risk models. One is the delayed renewal risk model, in which2.4holds ifF ∈ S; another is the general risk model, in which2.4also holds ifF ∈ L ∩ D. We are interested in the latter, for example, the general risk model, and restate Theorem 2.2 of Wang1here.

Theorem 2.1. In the independent general risk model introduced inSection 1, assume that the claim sizes {Xn, n ≥ 1} are independent and identically distributed nonnegative r.v.s with common distributionF∈ L ∩ D. Assume that for anyT >0 withλT−λ0>0, there exists some constant ηηT>0 such that

E

1η_NT

<∞. 2.5

Then,2.4holds.

In the present paper, we aim to deal with the extended negatively dependent general risk model to get a similar result under F ∈ D. Simultaneously, the condition2.5can be weakened to2.8below.

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We call r.v.s{ξn, n≥1}are extended negatively dependentENDif there exists some positive constantMsuch that both

P _n

k1

ξ_k> y_k

≤Mⁿ

k1

P

ξ_k> y_k

, 2.6

P _n

k1

ξk≤yk

≤Mⁿ

k1

P

ξk ≤yk

2.7

hold for each n ≥ 1 and ally₁, . . . , y_n. This dependence structure was introduced by Liu 12. Recall that r.v.s{ξn, n ≥1}are called upper negatively dependentUNDif2.6holds withM 1, they are called lower negatively dependentLNDif2.7holds withM 1, and they are called negatively dependentNDif both2.6and2.7hold withM1. These negative dependence structures were introduced by Ebrahimi and Ghosh13and Block et al.

14. Clearly, ND r.v.s must be END r.v.s., and Example 4.1 of Liu12shows that the END structure also includes some other dependence structures.

Motivated by the work of Wang1, under the END structure, we formulate our main result as follows.

Theorem 2.2. In the dependent general risk model introduced inSection 1, assume that the claim sizes{Xn, n ≥ 1}are END nonnegative r.v.s with common distributionF ∈ Dand finite meanμ.

Assume that for anyT >0 withλT−λ0>0, there exists some constantp > J_Fsuch that

ENT^p<∞. 2.8

Then, it holds that L_F

_T

0−F

xe^δt λdt^ºΨx, T^ºL⁻¹_F _T

0−F

xe^δt λdt. 2.9

Furthermore, ifF∈ L ∩ D, then2.4holds.

The rest of the present paper consists of two sections. We give some lemmas and the proof ofTheorem 2.2inSection 3. InSection 4, we perform some numerical calculations to verify the approximate relationship in our main result.

3. Proof of Main Result and Some Lemmas

In the sequel, Mand aalways represent some finite and positive constants whose values may vary in diﬀerent places. In this section, we start by giving some lemmas to show some properties of the classDand the END structure. The first lemma is a combination of Proposition 2.2.1 of Bingham et al.4and Lemma 3.5 of Tang and Tsitsiashvili15.

Lemma 3.1. If a distributionV ∈ D, then

ifor anyγ > J_V, there exist positive constantsaandbsuch thatVy/Vx≤ay/x^−γ holds for allx≥y≥band

iiit holds for everyγ > J_Vthatx^−γ oVx.

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By direct verification, END r.v.s have the following properties similar to those of ND r.v.s; see Lemma 3.1 of Liu12. For some refined properties of END r.v.s, one can refer to Chen et al.16. The following lemma can also be found in Lemma 2.2 of Chen et al.16.

Lemma 3.2. i If r.v.s{ξn, n≥1}are nonnegative and END, then for anyn≥1, there exists some positive constantMsuch that E_n

k1ξ_k≤M_n

k1Eξ_k.

ii If r.v.s{ξn, n≥ 1}are END and{fn·, n≥1}are either all monotone increasing or all monotone decreasing, then{fnξn, n≥1}are still END.

The following two lemmas play important roles in the proof of our main result.

Lemma 3.3. Let{ξn, n≥ 1}be identically distributed and END r.v.s with common distributionV andμ_V Eξ11{ξ1≥0} <∞. Then, for anyθ >0,x >0 andn≥1, there exists some positive constant Msuch that

P _n

k1

ξ_k> x

≤nVθx M eμ_Vn

x θ⁻¹

. 3.1

Proof. Following the proof of Lemma 2.3 of Tang 17, we employ a standard truncation argument to prove this lemma. For simplicity, we writeS^ξn_n

k1ξk,n≥1. Ifμ_V 0, thenξn

is almost surely nonpositive for eachn≥ 1, implying PS^ξn > x 0 for any positivex, and thus3.1holds.

Let, in the following,μ_V >0. For any fixedθ >0 and positive integern, define

ξnmin{ξn, θx}, ξ_n max

ξ_n,0

ξ_n1_{0≤ξn≤θx}θx1_{ξn>θx}. 3.2

According toLemma 3.2ii,{ξn, n≥1}and{ξ_n, n≥1}are still END r.v.s, respectively. Denote S^ξ_n_n

k1ξk,n≥1. Clearly,

P

S^ξ_n> x P

S^ξ_n> x,max

1≤k≤nξk > θx

P

S^ξ_n> x,max

1≤k≤nξk≤θx

≤nVθx P

S^ξ_n> x .

3.3

It remains to estimate the second summand in 3.3. For a positive h, by Lemma 3.2ii, {e^h^ξⁿ, n≥1}are END nonnegative r.v.s. Hence, using identity

Ee^h^ξ¹ _θx

0

e^hu−1 Vdu

e^hθx−1 Vθx 1, 3.4

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by Markov inequality andLemma 3.2iwe have

P

S^ξ_n> x ≤e^−hxEe^h^S^ξⁿ

≤e^−hxEe^hⁿ^k1^ξ^k

≤e^−hxMEe^h^ξ¹ⁿ Me^−hx

_θx

0

e^hu−1 Vdu

e^hθx−1 Vθx 1 _n

.

3.5

Since 1u ≤ e^u for allu ∈^Ê ande^hu−1/uis strictly increasing inu > 0, from3.5, we obtain

P

S^ξ_n> x ≤Mexp

n _θx

0

e^hu−1

u uVdu n

e^hθx−1 Vθx−hx

≤Mexp

n e^hθx−1 θx

_θx

0

uVdu θxVθx

−hx

≤Mexp

n e^hθx−1

θx μ_V −hx

.

3.6

Chooseh θx⁻¹logxμ_Vn⁻¹1, which is positive. For suchh, by3.6, we have

P

S^ξ_n> x ≤Mexp 1

θ− 1 θlog

x μ_Vn1

≤Mexp 1

θlogeμ_Vn x

.

3.7

The last estimate and3.3imply the desired estimate3.1. The lemma is proved.

Lemma 3.4. In the dependent general risk model introduced inSection 1, assume that the claim sizes {Xn, n ≥ 1}are END nonnegative r.v.s with common distributionF ∈ D. LetZ be an arbitrary nonnegative r.v. and assume that{Xn, n≥1},{Nt, t≥0}andZare mutually independent. Then, for anyT >0 and any positive integern₀,

LF n0

k1

k j1

P

Xje^−δτ^j> x, NT k ^ºⁿ⁰

k1

P

⎛

⎝^k

j1

Xje^−δτ^j> xZ, NT k

⎞

⎠

ºL⁻¹_F ⁿ⁰

k1

k j1

P

Xje^−δτ^j> x, NT k .

3.8

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Furthermore, ifF∈ L ∩ D, then

n0

k1

P

⎛

⎝^k

j1

X_je^−δτ^j > xZ, NT k

⎞

⎠∼ⁿ⁰

k1

k j1

P

X_je^−δτ^j> x, NT k . 3.9

We remark that ifFis consistently varying tailedsee the definition in Chen and Yuen 18, then by conditioning3.9easily follows from Theorem 3.2 of Chen and Yuen18. Note that this case is in a broader scope, since there is no need to assume independence between τ1, . . . , τ_n₀andZ.

Proof. We follow the line of the proof of Lemma 3.6 of Wang1with some modifications in relation to the properties of the classDand the END structure. Clearly, for eachk1, . . . , n₀,

P

⎛

⎝^k

j1

Xje^−δτ^j > xZ, NT k

⎞

⎠

{0≤t1≤···≤tk≤T,tk1>T}

_∞

0−P

⎛

⎝^k

j1

Xje^−δt^j > xz

⎞

⎠×PZ∈dz, τ₁∈dt₁, . . . , τ_k1∈dt_k1. 3.10

We first show the upper bound. For any fixedl >0,

P

⎛

⎝^k

j1

Xje^−δt^j > xz

⎞

⎠≤P

⎛

⎝^k

j1

Xje^−δt^j > xz−l⎞

⎠

P

⎛

⎝^k

j1

Xje^−δt^j > xz,max

1≤j≤kXje^−δt^j≤xz−l

⎞

⎠:I₁I₂. 3.11

ByF∈ D, for any 0< θ <1 and eachk1, . . . , n₀, we have uniformly for allt₁, . . . , t_k∈0, T andz∈0,∞,

I₁≤^k

j1

F

θxze^δt^j ^ºL⁻¹_F ^k

j1

F

xze^δt^j , 3.12

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by firstly lettingx → ∞ then θ 1. We note that {Xn, n ≥ 1} are END r.v.s. Then, by F∈ D, there exists some positive constantMMn0such that for suﬃciently largex, each k1, . . . , n₀, allt₁, . . . , tk∈0, Tandz∈0,∞,

I₂P

⎛

⎝^k

j1

Xje^−δt^j > xz,xz k <max

1≤j≤kXje^−δt^j ≤xz−l

⎞

⎠

≤P

⎛

⎝^k

i1

⎧⎨

⎩

j /i

X_je^−δt^j> l, X_ie^−δtⁱ > xz k

⎫⎬

⎭

⎞

⎠

≤^k

i1

j /i

P

X_je^−δt^j > l

k−1, X_ie^−δtⁱ> xz k

≤M^k

i1

j /i

F le^δt^j

k−1

F

xze^δtⁱ k

≤MF l

n0−1 _k

j1

F

xze^δt^j .

3.13

Sincelcan be arbitrarily large, it follows that

lim sup

l→ ∞ lim sup

x→ ∞ sup

t1,...,tk∈0,T, z∈0,∞

I2

_k

j1F

xze^δt^j 0. 3.14

Hence, from3.10–3.14, we obtain for eachk1, . . . , n₀,

P

⎛

⎝^k

j1

X_je^−δτ^j > xZ, NT k

⎞

⎠^ºL⁻¹_F ^k

j1

{0≤t1≤···≤tk≤T,tk1>T}

_∞

0−F

xze^δt^j

×PZ∈dz, τ₁∈dt₁, . . . , τ_k1∈dt_k1

L⁻¹_F ^k

j1

P

X_je^−δτ^j > xZ, NT k

≤L⁻¹_F ^k

j1

P

Xje^−δτ^j > x, NT k .

3.15

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As for the lower bound for3.10, since{Xn, n ≥ 1}are END r.v.s, we have for suﬃciently largexand eachk1, . . . , n₀,

P

⎛

⎝^k

j1

Xje^−δt^j > xz

⎞

⎠≥P

⎛

⎝^k

j1

Xje^−δt^j> xz⎞

⎠

≥^k

j1

F

xze^δt^j −

1≤i<j≤k

P

X_ie^−δtⁱ> xz, X_je^−δt^j> xz

≥^k

j1

F

xze^δt^j −M

1≤i<j≤k

F

xze^δtⁱ F

xze^δt^j

1−o1^k

j1

F

xze^rt^j

3.16

holds uniformly for allt₁, . . . , tk ∈ 0, Tandz ∈ 0,∞. ByF ∈ Dand Fatou’s lemma, we have for anyθ > 1 and allj 1,2, . . .,

lim inf 1

FxP

Xj> xZe^δT lim inf _∞

0−

F

xze^δT

Fx PZ∈dz

≥ _∞

0−lim inf F

θx

Fx PZ∈dz

F_∗

θ −→LF, θ1,

3.17

which means

P

X_j > xZe^δT ^²L_FFx. 3.18

Similar to3.15, from3.10,3.16, and3.18, we obtain for eachk1, . . . , n₀,

P

⎛

⎝^k

j1

Xje^−δτ^j> xZ, NT k

⎞

⎠

²

k j1

P

Xje^−δτ^j > xZ, NT k

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≥^k

j1

{0≤t1≤...≤tk≤T,tk1>T}P

X_j> xe^δt^jZe^δT Pτ₁∈dt₁, . . . , τ_k1∈dt_k1

²LF

k j1

{0≤t1≤...≤tk≤T,tk1>T}F

xe^δt^j Pτ1 ∈dt1, . . . , τ_k1∈dt_k1

LF

k j1

P

Xje^−δτ^j> x, NT k .

3.19 The desired relation3.8follows now from3.15and3.19.

IfF∈ L ∩ D,3.9follows by using the properties of the classLto establish analogies of relations3.12and3.17. This ends the proof of the lemma.

Proof ofTheorem 2.2. We use the idea in the proof of Theorem 2.2 of Wang 1 e.g., Theorem 2.1of this paperto prove this result. Clearly, F ∈ Dand μ < ∞ implyJ_F ≥ 1.

By2.8, we have for any >0, there exists some positive integern₁n₁T, such that

ENT^p1{NT>n1}≤. 3.20

To estimate the upper bound ofΨx, T, we split it into two parts as

Ψx, T≤P

⎛

⎝^NT

j1

Xje^−δτ^j > x

⎞

⎠

_n

1

k1

^∞

kn11

P

⎛

⎝^k

j1

X_je^−δτ^j > x, NT k

⎞

⎠:I₃I₄.

3.21

According toLemma 3.4of this paper and Lemma 3.5 of Wang1, we have for suﬃciently largex,

I₃≤1L⁻¹_F ⁿ¹

k1

k j1

P

Xje^−δτ^j > x, NT k

≤1L⁻¹_F ^∞

j1

P

Xje^−δτ^j > x, NT≥j

1L⁻¹_F ^∞

j1

P

X_je^−δτ^j > x, τ_j≤T 1L⁻¹_F

_T

0−F

xe^δt λdt.

3.22

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By Lemma 3.3,F ∈ D, Lemma 3.1ii,3.20, andp > J_F ≥ 1, there exists some positive constantMsuch that for suﬃciently largex,

I₄≤ ^∞

kn11

P

⎛

⎝^k

j1

Xj> x

⎞

⎠PNT k

≤F

p⁻¹x ^∞

kn11

kPNT k M

eμ_p x^−p ^∞

kn11

k^pPNT k

≤MFx

ENT1{NT>n1}ENt^p1{NT>n1} MFx.

3.23

ByLemma 3.1i, for anyγ > J_F, there exists some positive constantasuch that for suﬃciently largex,

_T

0−F

xe^δt λdt≥a⁻¹Fx _T

0−e^−γδtλdt

≥a⁻¹e^−γδTλT−λ0Fx,

3.24

which, combining3.23andλT−λ0>0, implies

I₄≤M _T

0−F

xe^δt λdt. 3.25

From3.21,3.22, and3.25, we derive the right-hand side of2.9.

As for the lower bound ofΨx, T, byLemma 3.4, we have for the above given >0 and suﬃciently largex,

Ψx, T≥P

⎛

⎝^NT

j1

Xje^−δτ^j > x _T

0

e^−δsCds

⎞

⎠

≥ⁿ¹

k1

P

⎛

⎝^k

j1

X_je^−δτ^j > x _T

0

e^−δsCds, NT k

⎞

⎠

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≥1−L_Fⁿ¹

k1

k j1

P

X_je^−δτ^j > x, NT k

1−LF

⎛

⎝^∞

j1

P

X_je^−δτ^j > x, τ_j≤T − ^∞

kn11

k j1

P

X_je^−δτ^j > x, NT k

⎞

⎠

: 1−LF

_T

0−F

xe^δt λdt−I₅

.

3.26

Analogously to the estimate forI₄, we have for suﬃciently largex, I₅≤FxENT1{NT>n1}

≤M _T

0−F

xe^δt λdt.

3.27

From3.26and3.27, we obtain the left-hand side of2.9.

IfF∈ L ∩ D, then2.4follows by using3.9in the proof of3.22and3.26.

4. Numerical Calculations

In this section, we perform some numerical calculations to check the accuracy of the asymptotic relations obtained in Theorem 2.2. The main work is to estimate the finite-time ruin probability defined in1.2.

We assume that the claim sizes{Xn, n≥1}come from the common Pareto distribution with parameterκ1,β2,

F x;κ, β

1− κ

κx _β

, x≥0, 4.1

which belongs to the classL ∩ D, and{X2n−1, X2n, n ≥ 1}are independent replications of X1, X₂with the joint distribution

F_X₁_,X₂ x, y

−1 αln

1

e^−αFx−1

e^−αFy−1 e^−α−1

, 4.2

with parameterα1, where the joint distributionF_X₁_,X₂x, yis constructed according to the Frank Copula. It has been proved in Example 4.2 of Liu12thatX₁andX₂ are END r.v.s.

Since{X2n−1, X2n, n≥1}are independent copies ofX1, X2, the r.v.s{Xn, n ≥1}are END as well.

Assume that the claim arrival processNtis the homogeneous Poisson process with intensity parameterλ. Clearly, such an integer-valued process Nt satisfies the condition 2.8. Choose λ 0.1. The total amount of premiums is simplified as Ct ct with

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Table 1: Comparison between the analog value and the theoretical result inTheorem 2.2.

x×10³ Theoretical result Analog value

0.5 3.2846e−6 3.8120e−616.1%

1 8.2270e−7 9.1100e−710.7%

2 2.0586e−7 2.2300e−78.3%

5 3.2956e−8 3.5000e−86.2%

the premium ratec 500, and the constant interest rateδ 0.02. Here, we set the timeT asT 10 and the initial capital reservex 500,10³,2×10³,5×10³, respectively. We aim to verify the accuracy of relation2.4. The procedure of the computation of the finite-time ruin probabilityΨx, TinTheorem 2.2is listed here.

Step 1. Assign a value for the variablexand setl0.

Step 2. Divide the close interval0, Tintom1000 pieces, and denote each time point asti, i1, . . . , m.

Step 3. For eachti, generate a random numberni from the Poisson distributionPλti, and setnias the sample size of the claims.

Step 4. Generate the accident arrival time{τ_kⁱ, k 1, . . . , n_i}from the uniform distribution U0, tiand the claim sizes{X_kⁱ, k1, . . . , ni}from4.1and4.2.

Step 5. Calculate the expressionDbelow for eachtiand denote them as{Di, i1, . . . , m}:

Diⁿⁱ

k1

Xⁱ_ke^−rτ^kⁱ − _t_i

0

e^−rsCds, i1, . . . , m, 4.3

whererandCthave been defined and their values have also been assigned.

Step 6. Select the maximum value from{Di, i 1, . . . , m}, and denote it asD^∗, compareD^∗ withx; ifD^∗> x, then the value oflincreases 1.

Step 7. RepeatStep 2throughStep 6,N10⁹times.

Step 8. Calculate the moment estimate of the finite-time ruin probability,l/N.

Step 9. RepeatStep 1throughStep 8ten times and get ten estimates. Then, choose the median of the ten estimates as the analog value of the finite-time ruin probability.

For diﬀerent value ofx, the analog value and the theoretical result of the finite-time ruin probability are presented in Table 1, and the percentage of the error relative to the theoretical result is also presented in the bracket behind the analog value. It can be found that fromTable 1, the largerxbecomes, the smaller the diﬀerence between the analog value and the theoretical result is. Therefore, the approximate relationship inTheorem 2.2is reasonable.

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Acknowledgments

The authors would like to thank the two referees for their useful comments on an earlier version of this paper. The revision of this work was finished during a research visit of the first author to Vilnius University. He would like to thank the Faculty of Mathematics and Informatics for its excellent hospitality. Research supported by National Natural Science Foundation of Chinano. 11001052, China Postdoctoral Science Foundation20100471365, National Science Foundation of Jiangsu Province of Chinano. BK2010480, Natural Science Foundation of the Jiangsu Higher Education Institutions of China no. 09KJD110003, Postdoctoral Research Program of Jiangsu Province of China no. 0901029C, and Jiangsu Government Scholarship for Overseas Studies, Qing Lan Project.

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