Ruin probability in a one-sided linear model with constant interest rate

Statistics and Probability Letters 80 (2010) 662–669

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Ruin probability in a one-sided linear model with constant interest rateI Jiangyan Peng ∗ , Jin Huang School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China

article

abstract

info

Article history: Received 3 August 2009 Received in revised form 24 December 2009 Accepted 24 December 2009 Available online 2 January 2010

This paper investigates the ruin probability of a generalized renewal model with a constant interest rate, in which a one-sided linear model is used for the dependent claim process. An explicit asymptotic formula and an exponential upper bound are obtained for the ruin probability. © 2010 Elsevier B.V. All rights reserved.

1. Introduction 1.1. The model Consider a generalized renewal model, in which the claims Xn , n ≥ 1, form a sequence of not necessarily independent, not necessarily identically distributed, and nonnegative random variables (r.v.’s), while their inter-arrival times Tn , n ≥ 1, form another sequence of independent, identically distributed (i.i.d.), nonnegative and not-degenerate-at-zero r.v.’s with common distribution function (d.f.) H. We assume Pn that the sequences {Xn ; n ≥ 1} and {Tn ; n ≥ 1} are mutually independent. The locations of the successive claims σn = i=1 Ti , n ≥ 1, constitute a renewal counting process N (t ) = # {n = 1, 2, . . . : σn ≤ t } ,

t ≥ 0,

with the renewal function EN (t ) = n=1 P {σn ≤ t } for t ≥ 0. Let {C (t ); t ≥ 0} with C (0) = 0 be a nonnegative and nondecreasing stochastic process, denoting the total amount of premiums accumulated up to time t. Let r > 0 be the constant interest rate (that is, after time t a capital x becomes xert ), and let x ≥ 0 be the initial surplus of the insurance company. Then U (t ), the total surplus up to time t, satisfies the equation

P∞

U (t ) = xert +

Z

t

er (t −s) C (ds) −

0

∞ X

Xn er (t −σn ) I(σn ≤t ) ,

t ≥ 0,

(1.1)

n=1

where IA denotes the indicator function of an event A. Assume that the total discounted amount of premiums is finite, that is, ∞

Z

e−rs C (ds) < ∞

e C =

almost surely.

0

The ruin probability for this model is defined by

ψ(x) = P {U (t ) < 0 for some t ≥ 0 | U (0) = x} .

I Supported by National Natural Science Foundation of China (10871034).

∗

Corresponding author. Tel.: +86 0 13679010682. E-mail addresses: [email protected], [email protected] (J. Peng).

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.12.024

(1.2)

J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

663

1.2. Review on related results Throughout this paper, C represents a generic positive constant, which may vary with the context. Hereafter, all limit relations are for x → ∞ unless stated otherwise. For two positive functions a(·) and b(·), we write a(x) . b(x) if a(x) a(x) a(x) lim sup b(x) ≤ 1; a(x) & b(x) if lim inf b(x) ≥ 1; and a(x) ∼ b(x) if lim b(x) = 1. We say that a d.f. F on [0, ∞) belongs to the subexponential class S , denoted by F ∈ S , if lim

x→∞

F n∗ (x)

= n,

F (x)

for any (or, equivalently, for some) n ≥ 2,

where F n∗ denotes the n-fold convolution of F . It is well known that each subexponential d.f. F belongs to the class L of long-tailed distributions, denoted by F ∈ L, in the sense that lim

F (x − y) F ( x)

x→∞

= 1,

for any (or, equivalently, for some) y 6= 0.

Moreover, the class S covers the class R of distributions with regular variations and the class ERV of distributions with extended regular variations. By definition, a d.f. F ∈ R if there is some α > 0 such that lim

x→∞

F (xy) F (x)

= y−α ,

for any y > 0.

Another useful subclass A of the class S was introduced by Konstantinides et al. (2002). By definition, a d.f. F ∈ A if F ∈ S and lim sup

F (xy) F (x)

x→∞

< 1,

for some y > 1.

(1.3)

Since (1.3) is satisfied by almost all useful distributions with unbounded supports, the class A almost coincides with the class S . In conclusion, R ⊂ ERV ⊂ A ⊂ S ⊂ L (see Embrechts et al. (1997) for more details of heavy-tailed distributions). The asymptotic behavior of the ruin probability ψ(x) defined in (1.2) has been discussed by several authors. Under conditions that the claims are identically distributed with B ∈ R−α for some α > 0 and N (·) is a homogeneous Poisson process with intensity λ > 0, Klüppelberg and Stadtmüller (1998) proved that

λ

ψ(x) ∼

B(x). rα Asmussen (1998) extended the study to a larger class of heavy-tailed distributions and obtained that

ψ(x) ∼

λ r

∞

Z x

B(y) y

dy.

(1.4)

(1.5)

Recently, Chen and Ng (2007) considered the case that claims are pairwise negatively dependent with common d.f. B ∈ ERV, and proved that

ψ(x) ∼

∞

Z

B(xers )dEN (s).

(1.6)

0

However, the assumption B ∈ ERV unfortunately excludes some important distributions such as lognormal and Weibull distributions. Under the condition B ∈ S , Hao and Tang (2008) obtained a result similar to (1.6) for the tail probability of discounted aggregate claims. In addition to asymptotic formulas for ruin probabilities, another analytic method commonly used in risk theory is the derivation of inequalities for ruin probabilities presented by Asmussen (2000). Further, ruin probabilities in many continuous time risk models can be reduced to those in embedded discrete time risk models. On the other hand, discrete time risk models themselves are also interesting stochastic models both in theory and in application. With respect to inequalities for the ruin probability ψ(x) in the embedded discrete time risk model of {U (t ); t ≥ 0} in (1.1), Cai (2004) considered the case of identically distributed claims when N (·) is a homogeneous Poisson process and C (t ) = ct with the constant premium rate c > 0. We would like to remark that all the above methods heavily rely on the i.i.d. assumption on the claims except what Chen and Ng (2007) used, while the method of Chen and Ng (2007) also relies on the assumption that the claim-size distribution belongs to the subclass ERV ⊂ A. In this paper, we use a one-sided linear process to model the dependent claims Xn satisfying Xn =

n X

ϕn−j εj + ϕn x0 ,

n = 1, 2, . . . ,

(1.7)

j=1

where {εn }n≥1 is a sequence of i.i.d. nonnegative r.v.’s with generic random variable (r.v.) ε and common d.f. F , and x0 ≥ 0

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J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

is an initial value. The coefficients ϕi ≥ 0 are assumed to satisfy 0<

∞ X

ϕi = ϕ < ∞.

(1.8)

i=0

It is important to point out that a one-sided linear model is often used in time series analysis and includes the ARMA model and the fractional ARIMA model (see Brockwell and Davis (1991)). This paper is organized as follows. In Section 2, we derive an asymptotic formula for the ruin probability ψ(x) when F ∈ A so that lognormal and heavy-tailed Weibull distributions are included. In Section 3, we derive an exponential upper bound for the ruin probability when the claim process follows an AR model and moment generating function of ε exists in an appropriate region. 2. Asymptotic formulas 2.1. Main results Theorem 2.1. Let {Xn ; n ≥ 1}, {Tn ; n ≥ 1} and {C (t ); t ≥ 0} be mutually independent, and {Xn ; n ≥ 1} be a one-sided linear process (1.7) with an initial value x0 . If F ∈ A and P {T1 > η} = 1 for some η > 0, then

ψ(x) ∼

(

∞

Z

P

ε

0

∞ X

) ϕi e

−r σ i

> xe

rs

dEN (s),

(2.1)

i =0

where σ0 = 0 by convention. Remark 2.1. The technical assumption on the distribution of T1 is reasonable for real applications since we may choose η as close to zero as we wish. Now we discuss some special cases of Theorem 2.1. Corollary 2.1. (1) If F ∈ R−α for some α > 0, then (2.1) reduces to

ψ(x) ∼ F (x)

Ee−r α T1 1 − Ee−r α T1

E

∞ X

!α ϕi e−r σi

.

(2.2)

i=0

In particular, if ϕ0 = 1, ϕi = 0, i 6= 0, and N (·) is a homogeneous Poisson process with intensity λ > 0, then (2.2) coincides with (1.4). (2) When ϕ0 = 1 and ϕi = 0, i 6= 0, (2.1) coincides with (1.6), which is equivalent to (1.5) (Hao and Tang, 2008) as t → ∞, i.e., P

( ∞ X

) Xn e

−r σn

>x ∼

Z

∞

F (xers )dEN (s).

0

n =1

In particular, if N (·) is a homogeneous Poisson process with intensity λ > 0, then (2.1) coincides with (1.5). Example 2.1. Suppose that {εn ; n ≥ 1} is a sequence of i.i.d. nonnegative r.v.’s with generic r.v. ε and common d.f. F , and the claim process {Xn ; n ≥ 1} has a dependent AR(1) structure, i.e., Xn satisfies X n = φ X n − 1 + εn ,

n = 1, 2, . . . ,

(2.3)

where 0 ≤ φ < 1 and X0 = x0 ≥ 0 are constants, and φ P = 1 by convention. This process is the most commonly used model n n −j in time series analysis. Here, we rewrite (2.3) as Xn = εj + φ n x0 , n ≥ 1, which satisfies (1.7) since {φ i ; i ≥ 0} j =1 φ satisfies (1.8). 0

Hence, under the conditions of Theorem 2.1, we have the ruin probability ψ(x) for a large initial capital x.

R∞ 0

P {ε

P∞

i=0

φ i e−r σi > xers }dEN (s) as the asymptotic estimate of

2.2. Several lemmas The first lemma is due to Proposition 2.2.1 of Bingham et al. (1987). Lemma 2.1. For F ∈ A, there exist positive constants p, C0 , and x0 such that the inequality F (xy) F (x)

≤ C 0 y −p

holds uniformly for xy ≥ x ≥ x0 .

(2.4)

J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

665

The following lemma will play a crucial role in the proof of Theorem 2.1. Lemma 2.2. Let {εn ; n ≥ 1} and {σn ; n ≥ 1} be defined as those in Theorem 2.1. Suppose that F ∈ A and P {T1 > η} = 1 for some η > 0. Then, P

( ∞ X

εj

∞ X

j=1

) ϕi e−r σi+j > x ∼

( ∞ X

εj

∞ X

j=1

≥

P∞

i=0

) ϕi e

−r σi+j

∞ X

) ϕi e−r σi+j > x .

(2.5)

i =0

( N [

>x ≥P

εj

j =1

−

! ( εj

P

j =N +1

j =1

εj

P

∞ X

ϕi e−r (σi+j −σj ) , j ≥ 1. We first derive the lower bound for P {

i =0

∞ X

(

j=1

i =0

Proof. Write ϕ(σj ) = fixed integer N ≥ 1, P

∞ X

∞ X

∞ X

j=1

P∞

i =0

ϕi e−r σi+j > x}. For any

!) ϕi e

−r σi+j

>x

i=0

) ϕi e

εj

P∞

−r σi+j

( X

>x −

i=0

εk

P

∞ X

ϕi e

−r σi+k

> x, ε l

) ϕi e

−r σi+l

>x .

(2.6)

i=0

i=0

1≤k6=l≤N

∞ X

Note that condition (1.8) holds and {εn }n≥1 is a sequence of i.i.d. nonnegative r.v.’s and is independent of {σn }n≥1 . Then, for 1 ≤ k 6= l ≤ N,

( P

εk

∞ X

ϕi e

−r σi+k

> x, ε l

∞ X

i=0

) ϕi e

−r σi+l

>x

( ≤ P εk

i=0

∞ X

ϕi e

−r σi+1

∞ X

> x, ε l

i =0

( = P εk

) ϕi > x

i=0

∞ X

) ( ϕi e

−r σi+1

∞ X

> x P εl

i =0

( = o P ε1

) ϕi > x

i=0

∞ X

)! ϕi e

−r σi+1

>x

.

(2.7)

i=0

Applying Lemma 2.1, for all large enough x, we have ∞ X

P εj e−rTj e−r σj−1 ϕ(σj ) > x





j =N +1

∞ Z X

=

j =N +1

≤ C0

∞

ϕ

Z

 F

η

∞ X

∞

Z

(j−1)η

xer (tj +sj−1 )



e ϕ

0

P ϕ σj ∈ de ϕ P σj−1 ∈ dsj−1 P Tj ∈ dtj










P εj e−rTj ϕ(σj ) > x Ee−pr σj−1 = C0 P ε1 e−r σ1 ϕ(σ1 ) > x







∞ X



(Ee−pr σ1 )j ,

(2.8)

j =N

j =N +1

where the last equality follows from the facts that εj , Tj and ϕ(σj ) are mutually independent and ϕ(σj ), j ≥ 1, are identically distributed r.v.’s. Therefore, ∞ P

lim lim sup

N →∞

x→∞

P εj e−r σj ϕ(σj ) > x





j =N +1

∞ P

≤ lim C0

P εj e−r σj ϕ(σj ) > x



N →∞



∞ X (Ee−pr σ1 )j = 0.

(2.9)

j =N

j=1

Substituting (2.7) and (2.9) into (2.6), we have P

( ∞ X

εj

j=1

∞ X

) ϕi e−r σi+j > x &

i =0

∞ X

( P

εj

j=1

∞ X

) ϕi e−r σi+j > x .

(2.10)

i =0

Next we prove the upper bound by using the approach in Hao and Tang (2008), but there are many changes in this proof because the coefficients ϕi , i ≥ 0, bring much trouble. Since P {T1 > η} = 1, it follows that for arbitrarily some integer N > 1 such that N η ∈ {t : P (σ1 ≤ t ) > 0} = Γ , P

( ∞ X j=1

εj

∞ X i =0

) ϕi e

−r σi+j

>x ≤P

( N X j =1

εj

∞ X i=0

ϕi e

−r σi+j

+

∞ X j =N +1

εj e

−r (j−N )η

∞ X i=0

) ϕi e

−r σi+N

>x .

(2.11)

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J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

Consider P {

εj e−r (j−K )η > x} = P {

P∞

j =K +1

P∞

j =1

εj e−rjη > x} ∼

P∞

j =1

F (xerjη ) for any integer K (Corollary 3.1 in Chen

et al. (2005)). Then, Lemma 2.1 implies that there is some constant D > 0 such that P { j=N +1 εj e−r (j−N )η > x} ≤ DF (x) for the fixed integer N and all x ≥ 0. Let Wη be a new r.v. independent of {εn }n≥1 and {σn }n≥1 with a tail satisfying P {Wη > x} = min{DF (x), 1} for x ≥ 0. Thus, for some ρ > 0,

P∞

P

( ∞ X

εj

∞ X

j =1

) ϕi e

−r σi+j

>x

( N X

≤P

εj

j =1

i =0

∞ X

ϕi e

−r σi+j

∞ X

+ Wη

) ϕi e

−r σi+N

>x

i =0

i=0

(

N [

= P { J > x} = P J > x,

)

(

(Ti ≥ ρ) + P J > x,

i=1

N \

) (Ti < ρ) .

(2.12)

i=1

Note that independent r.v.’s T1 , . . . , TN are associated (see Esary et al. (1967) for more details). We rewrite N X

J =

 εj 

N X

ϕk−j e−r σk +

k=j

j=1

∞ X

ϕk−j e

−r σN +

k P

Ti

i=N +1

!



 + Wη e−r σN ϕ0 +

k=N +1

∞ X

ϕk−N e

k P

−r

i=N +1

Ti

 .

k=N +1

Since {Ti }1≤i≤N is independent of {εi }1≤i≤N , {Wη } and {TN +i }i≥1 , association is preserved under conditioning. Thus, we have

( P

J > x,

N [

) (Ti ≥ ρ)

" ( = E P J > x,

i=1

N [

)# (Ti ≥ ρ) | ε1 , . . . , εN , Wη , TN +1 , . . .

i=1

)# ( N [ ≤ E P J > x | ε1 , . . . , ε N , W η , T N + 1 , . . . P (Ti ≥ ρ) | ε1 , . . . , εN , Wη , TN +1 , . . . "



i =1

= P {J > x} P

( N [

) (Ti ≥ ρ) .

(2.13)

i =1

Let Yi , 1 ≤ i ≤ n, be n independent r.v.’s distributed by Bi respectively and a d.f. B0 ∈ S such that Bi (x) ∼ bi B0 (x) holds forP some bi > 0. Similar (2003), any 0 < a ≤ b < ∞, Pn to the proof of Proposition 5.1 in Tang and Tsitsiashvili P∞we get−rfor n σi+j n P { i=1 ci Yi > x} ∼ , 1 ≤ j ≤ N, and G be i=1 Bi (x/ci ) holds uniformly for (c1 , . . . , cn ) ∈ [a, b] . Let Zj = i=0 ϕi e the d.f. of (Z1 , . . . , ZN ). Then, for the fixed integer N,

( P

J > x,

N \

) (Ti < ρ)

N X

Z ≤

i=1

∼

P

{ϕ0 e−rN ρ ≤zN ≤···≤z1 ≤ϕ } j =1 " Z N X {ϕ0 e−rN ρ ≤zN ≤···≤z1 ≤ϕ }

=

! εj zj + Wη zN > x dG(z1 , . . . , zN ) #

P (εj zj > x) + P (Wη zN > x) dG(z1 , . . . , zN )

j =1

N n X

o

n

o

P εj Zj I{ϕ0 e−rN ρ ≤ZN ≤···≤Z1 ≤ϕ } > x + P Wη ZN I{ϕ0 e−rN ρ ≤ZN ≤···≤Z1 ≤ϕ } > x .

(2.14)

j =1

Plugging (2.13) and (2.14) into (2.12), and letting ρ → ∞, we get

P

( ∞ X j =1

εj

∞ X

) ϕi e

−r σi+j

>x .

i =0

N X   P εj Zj > x + P Wη ZN > x .

(2.15)

j =1

Similar to the proof of (2.8), we have for some M ∈ (η, N η] ∩ Γ and all large enough x,

lim lim sup

N →∞

x→∞

R∞Rϕ



xers



P {ϕ (σN ) ∈ de ϕN } P {σN ∈ ds} Nη 0 e ϕN  rt  ≤ lim lim sup R R M ϕ N →∞ x→∞ P {ε1 Z1 > x} F xe P {ϕ (σ ) ∈ de ϕ } P {σ ∈ dt }

P Wη ZN > x





η

0

DF

CEe−rp(σN −M )

≤ lim lim sup N →∞

x→∞

1

e ϕ1

P {σ1 ≤ M }

Rϕ 0

Rϕ 0

F F



xe

e ϕN



1

 rM

xerM

e ϕ1



1

P {ϕ (σN ) ∈ de ϕN }

P {ϕ (σ1 ) ∈ de ϕ1 }

= 0.

(2.16)

J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

667

From (2.15) and (2.16), we obtain that P

( ∞ X

∞ X

εj

j=1

) ϕi e

−r σi+j

>x .

∞ X

( εj

P

∞ X

j=1

i =0

) ϕi e

−r σi+j

>x .

(2.17)

i =0

This completes the proof of Lemma 2.2.



2.3. Proof of Theorem 2.1 From (1.1), we have ∞ X

x−

Xn e−r σn ≤ e−rt U (t ) ≤ x + e C−

∞ X

n=1

Xn e−r σn I(σn ≤t ) ,

t ≥ 0.

n =1

Furthermore, (1.7) and (1.8) imply that ∞ X

∞ X

Xi e−r σi =

i =1

εj

j=1

∞ X

ϕi e−r σi+j + x0

i =0

∞ X

ϕi e−r σi <

∞ X

i=1

εj

j =1

∞ X

ϕi e−r σi+j + x0 ϕ.

i=0

By (1.2), we have P

( ∞ X

∞ X

εj

j=1

ϕi e

) ( ) ∞ ∞ X X − r σ > x +e C ≤ ψ(x) ≤ P εj ϕi e i+j > x − x0 ϕ .

−r σi+j

i =0

j =1

(2.18)

i=0

Note that, by Corollary 2.5 in Cline and Samorodnitsky (1994), the d.f. of εj i=0 ϕi e−r σi+j is long tailed for each j ≥ 1 since P∞ −r σi+j F ∈ S ⊂ L and 0 < ≤ ϕ . For any fixed integer N ≥ 1 and any fixed L 6= 0, we have i=0 ϕi e

P∞

N X

( εj

P

)

∞ X

j=1

ϕi e

−r σi+j

N X

>x−L ∼

i =0

( εj

P

j =1

∞ X

) −r σi+j

ϕi e

>x .

(2.19)

i=0

Combining (2.9) and (2.19), we have ∞ X

( P

)

∞ X

εj

j=1

ϕi e

−r σi+j

∞ X

>x−L ∼

i =0

( εj

P

j =1

∞ X

) −r σi+j

ϕi e

>x .

(2.20)

i=0

Applying Lemma 2.2 to (2.20) yields that for any fixed L 6= 0, P

( ∞ X

εj

∞ X

j=1

) ϕi e

−r σi+j

>x−L ∼

∞ X

i =0

( P

εj

j =1

∞ X

) −r σi+j

ϕi e

>x .

(2.21)

i=0

By Fatou’s lemma and (2.21), we obtain that

( P

∞ P

εj

j =1

lim inf

∞ P

x→∞

∞ P

ϕi e

−r σi+j

) > x +e C

i =0



P εj

j =1

∞ P

ϕi e

−r σi+j

>x



( ∞

Z

P

εj

j=1

= lim inf x→∞

∞ P

∞ P

0

i=0

∞ P

P

∞

Z ≥

lim inf 0

x→∞



P εj

εj

j=1

j =1

>x+y

∞ P

ϕi e

−r σi+j

>x

P (e C ∈ dy)



i=0

∞ P

∞ P

ϕi e

i=0

j =1

(

) −r σi+j

∞ P

) ϕi e

−r σi+j

>x+y

i=0



P εj

∞ P

ϕi e

−r σi+j

>x



P (e C ∈ dy) = 1.

(2.22)

i=0

Putting (2.21) and (2.22) together with (2.18) gives that

ϕ(x) ∼

∞ X j =1

( P

εj

∞ X

) ϕi e−r σi+j > x .

(2.23)

i=0

−r (σi+j −σj ) −r σi Note that the same d.f. as that of , j ≥ i=0 ϕi e P∞ i=0 ϕ−i re(σ −σhas i+j j ) > x} by conditioning on σ , j ≥ 1, we obtain (2.1). P {εj e−r σj ϕ e  j i=0 i

P∞

P∞

1. Calculating the probabilities

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J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

3. Integral equations and upper bounds In this section, we turn our attention to the ruin probability ψ(x) for the case that the moment generating function of ε in (1.7) exists. Assume that the claim process {Xn ; n ≥ 1} in (1.7) has a dependent AR(1) structure, i.e., Xn satisfies (2.3). Let {Un ; n ≥ 0} be the embedded discrete time process of {U (t ); t ≥ 0} at claim times σn , n ≥ 1, namely, Un = U (σn ), n ≥ 1, U0 = x. Let C (t ) = ct with the constant premium rate c > 0. Then, (1.1) implies Un = x

n Y

erTi +

n  X c

r

i =1

i =1

(erTi − 1) − Xi

n  Y

erTj ,

n = 1, 2, . . . .

(3.1)

j=i+1 1−φ i

Assume also that the net-profit condition is true, that is, cET1 > EXi . This condition is equivalent to cET1 > 1−φ E ε + φ i x0 for all i ≥ 1. Since ruin can occur only at claim times, the ruin probability ψ(x) in the continuous time model (1.1) reduces to

(

) ∞ [ ψ(x, x0 ) = P (Ui < 0)

(3.2)

i=1

in the embedded discrete time model (3.1) with an initial surplus x and an initial Sn claim amount x0 . Furthermore, define the probability that ruin occurs before or at the nth claim time by ψn (x, x0 ) = P { i=1 (Ui < 0)}. Clearly, ψ1 (x, x0 ) ≤ ψ2 (x, x0 ) ≤ · · · and limn→∞ ψn (x, x0 ) = ψ(x, x0 ). Theorem 3.1. Assume that x ≥ φ x0 and there exists R > 0 satisfying Ee

h i −R cr (erT1 −1)−φ x0 −(1+φ)ε1

= 1.

(3.3)

Then,

ψ(x, x0 ) ≤ ξ EeR(1+φ)ε1 Ee−R[xe ≤ ξe ≤e , R ∞ R(1+φ)y −Rx

where ξ −1 = inft ≥0

rT1 + c (erT1 −1)−φ x ] 0 r

(3.4)

−Rx

e

t

(3.5)

dF (y)/eR(1+φ)t F (t ) .

Proof. By using the approach in Theorem 2.1 of Cai (2004), we have

ψn+1 (x, x0 ) =

∞

Z



F xert1 + 0

∞

Z

Z

c r

 (ert1 − 1) − φ x0 dH (t1 )

xert1 + cr (ert1 −1)−φ x0

+ 0

  c ψn xert1 + (ert1 − 1) − φ x0 − y, φ x0 + y dF (y)dH (t1 ).

(3.6)

r

0

For any s ≥ 0, we have eR(1+φ)s F (s) F (s) = R ∞ e−R(1+φ)s R(1+φ)y dF (y) e s

≤ ξ e−Rs

Z

∞

∞

Z

eR(1+φ)y dF (y)

s

eR(1+φ)y dF (y) ≤ ξ e−Rs EeR(1+φ)ε1 .

(3.7)

s

Then, for any x0 ≥ 0 and x ≥ φ x0 , (3.1) and (3.7) imply

ψ1 (x, x0 ) =

∞

Z



F xert1 + 0

c r

h i  −R xerT1 + cr (erT1 −1)−φ x0 (ert1 − 1) − φ x0 dH (t1 ) ≤ ξ EeR(1+φ)ε1 Ee .

Under an inductive hypothesis, we assume for any x0 ≥ 0 and x ≥ φ x0 ,

ψn (x, x0 ) ≤ ξ EeR(1+φ)ε1 Ee−R[xe

rT1 + c (erT1 −1)−φ x ] 0 r

.

(3.8)

≥ 1 and 0 ≤ φ < 1. Thus, according to (3.8) and (3.3), we can verify that for 0 ≤ y ≤ xe + (e − 1) − φ x0 ,  c rt1 c rt1 rT1 c rT1 xert1 + (ert1 − 1) − φ x0 − y, φ x0 + y ≤ ξ EeR(1+φ)ε1 Ee−R[(xe + r (e −1)−φ x0 −y)e + r (e −1)−φ(φ x0 +y)] rT1

rt1

Note that e

ψn

c r

rt1



r

≤ ξ EeR(1+φ)ε1 Ee−R[(xe = ξ e−R[xe

rt1 + c (ert1 −1)−φ x −y)+ c (erT1 −1)−φ(x +y)] 0 0 r r

rt1 + c (ert1 −1)−φ x −(1+φ)y] 0 r

.

(3.9)

J. Peng, J. Huang / Statistics and Probability Letters 80 (2010) 662–669

669

Substituting (3.7) and (3.9) into (3.6), we get

ψn+1 (x, x0 ) ≤ ξ EeR(1+φ)ε1 Ee

h i −R xerT1 + cr (erT1 −1)−φ x0

.

Hence, for n = 1, 2, . . . , (3.8) holds. And (3.4) follows by letting n → ∞ in (3.8). Since erT1 ≥ 1, (3.5) is satisfied.



Remark 3.1. The above net-profit condition is only a sufficient condition for the ruin probability being less than 1. Moreover, if cET1 > (1 + φ)E ε + φ x0 and R > 0 in (3.3) exists, then R is the unique positive solution of (3.3). Acknowledgements The authors are grateful to the referee for many valuable comments and very helpful suggestions. The authors also thank Dr. Xuemiao Hao for his helpful discussions and sending them his paper. References Asmussen, S., 1998. Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 (2), 354–374. Asmussen, S., 2000. Ruin Probabilities. World Scientific, Singapore. Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press, Cambridge. Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods, 2nd ed. Springer-Verlag, New York. Cai, J., 2004. Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Process. Appl. 112 (1), 53–78. Chen, Y.Q., Ng, K.W., 2007. The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insurance Math. Econom. 40 (3), 415–423. Chen, Y.Q., Ng, K.W., Tang, Q., 2005. Weighted sums of subexponential random variables and their maxima. Adv. Appl. Probab. 37 (2), 510–522. Cline, D.B.H., Samorodnitsky, G., 1994. Subexponentiality of the product of independent random variables. Stochastic Process. Appl. 49 (1), 75–98. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, Heidelberg. Esary, J.D., Proschan, F., Walkup, D.W., 1967. Association of random variables, with applications. Ann. Math. Statist. 38 (5), 1466–1474. Hao, X., Tang, Q., 2008. A uniform asymptotic estimate for discounted aggregate claims with subexponential tails. Insurance Math. Econom. 43 (1), 116–120. Klüppelberg, C., Stadtmüller, U., 1998. Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuar. J. (1), 49–58. Konstantinides, D., Tang, Q., Tsitsiashvili, G., 2002. Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31 (3), 447–460. Tang, Q., Tsitsiashvili, G., 2003. Randomly weighted sums of subexponential random variables with application to the ruin probability. Extremes 6 (3), 171–188.