# Approximation for the Finite-Time Ruin Probability of a General Risk Model with Constant Interest Rate and Extended Negatively Dependent Heavy-Tailed Claims

Authors: Yang Yang; Xin Ma; Jin-guan Lin
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

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 Wang (2008).


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## Body

## 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.s) with common distribution F such that F¯(x)=1-F(x)=P(X1>x)>0 for all x>0; the claim arrival process {N(t),t≥0} is a general counting process, namely, a nonnegative, nondecreasing, right continuous, and integer-valued stochastic process with 0<EN(t)=λ(t)<∞ for all large t>0. The times of the successive claims are denoted by {τn,n≥1}. The total amount of premiums accumulated up to time t≥0, denoted by C(t) with C(0)=0 and C(t)<∞ almost surely for every t>0, is another nonnegative and nondecreasing stochastic process. Assume that {Xn,n≥1}, {N(t),t≥0} and {C(t),t≥0} are mutually independent. Let δ>0 be the constant interest rate (i.e., after time t one dollar becomes eδt dollars), and let x≥0 be the initial capital reserve of an insurance company. Then, the total discounted reserve up to time t≥0, denoted by D(t,x), can be written as (1.1)D(t,x)=x+∫0te-δsC(ds)-∑n=1N(t)Xne-δτn.
For a finite time T>0, the finite-time ruin probability is defined by (1.2)Ψ(x,T)=P(D(t,x)<0,forsome0≤t≤T)=P(sup⁡t∈[0,T](∑n=1N(t)Xne-δτn-∫0te-δsC(ds))>x),
while the ultimate ruin probability is defined by (1.3)Ψ(x)=Ψ(x,∞)=P(D(t,x)<0,forsomet≥0).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 Wang [1]. In particular, if C(t)=ct, t≥0, with c>0 a deterministic constant and {N(t),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 functions f(x) and g(x), we write f(x)~g(x) if limf(x)/g(x)=1; write f(x)≲g(x) if limsupf(x)/g(x)≤1 and f(x)=o(g(x)) if limf(x)/g(x)=0. The indicator function of an event A is denoted by 1A.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 limsupV¯(xy)/V¯(x)<∞ for any y>0. A distribution V is said to be long tailed, denoted by V∈ℒ, if limV¯(x+y)/V¯(x)=1 for any y>0. A distribution V is said to be subexponential, denoted by V∈𝒮, if Vn*¯(x)~nV¯(x) for any n≥2, where Vn* denotes the n-fold convolution of itself. A distribution V is said to be regularly varying tailed, denoted by ℛ-α,α>0, if limV¯(xy)/V¯(x)=y-α for any y≥1. A proper inclusion relationship holds that (2.1)R-α⊂L∩D⊂S⊂L,
see, for example, Cline [2] or Embrechts and Omey [3]. For a distribution V, denote the upper Matuszewska index of the distribution V by (2.2)JV+=-lim⁡y→∞log⁡V¯*(y)log⁡ywithV¯*(y)=liminf⁡x→∞V¯(xy)V¯(x),y>1.
In the terminology of Bingham et al. [4], the quantity JV+ is actually the upper Matuszewska index of the function 1/V¯(x),  x≥0, as also pointed out in Tang and Tsitsiashvili [5]. Additionally, denote LV=limy↘1V¯*(y) (clearly, 0≤LV≤1 ). The presented definitions yield that the following assertions are equivalent: (2.3)(i)V∈D,(ii)V¯*(y)>0forsomey>1,(iii)LV>0,(iv)JV+<∞.The asymptotic behavior of the ruin probability in the classical risk model has been extensively investigated in the literature. Klüppelberg and Stadtmüller [6] considered 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  4.1(ii) of his paper. Complementary discussions on the ultimate ruin probability can be found in Kalashnikov and Konstantinides [8], Konstantinides et al. [9], Tang [10], among others.In this paper, we are interested in the finite-time ruin probability. In this aspect, Tang [11] established an asymptotic result in the classical risk model: under the condition F∈𝒮, he obtained that for every T>0 for which λ(T)>0, (2.4)Ψ(x,T)~∫0-TF¯(xeδt)λ(dt).
Recently, Wang [1] derived some important and interesting results in two independent risk models. One is the delayed renewal risk model, in which (2.4) holds if F∈𝒮; another is the general risk model, in which (2.4) also holds if F∈ℒ∩𝒟. We are interested in the latter, for example, the general risk model, and restate Theorem   2.2 of Wang [1] here.Theorem 2.1.
In the independent general risk model introduced in Section1, assume that the claim sizes  {Xn,n≥1} are independent and identically distributed nonnegative r.v.s with common distribution  F∈ℒ∩𝒟. Assume that for any  T>0 with  λ(T)-λ(0)>0, there exists some constant  η=η(T)>0 such that
(2.5)E(1+η)N(T)<∞.
Then, (2.4) holds.In the present paper, we aim to deal with the extended negatively dependent general risk model to get a similar result underF∈𝒟. Simultaneously, the condition (2.5) can be weakened to (2.8) below.We call r.v.s{ξn,n≥1} are extended negatively dependent (END) if there exists some positive constant M such that both (2.6)P(⋂k=1n{ξk>yk})≤M∏k=1nP(ξk>yk),(2.7)P(⋂k=1n{ξk≤yk})≤M∏k=1nP(ξk≤yk)
hold for each n≥1 and all y1,…,yn. This dependence structure was introduced by Liu [12]. Recall that r.v.s {ξn,n≥1} are called upper negatively dependent (UND) if (2.6) holds with M=1, they are called lower negatively dependent (LND) if (2.7) holds with M=1, and they are called negatively dependent (ND) if both (2.6) and (2.7) hold with M=1. These negative dependence structures were introduced by Ebrahimi and Ghosh [13] and Block et al. [14]. Clearly, ND r.v.s must be END r.v.s., and Example  4.1 of Liu [12] shows that the END structure also includes some other dependence structures.Motivated by the work of Wang [1], under the END structure, we formulate our main result as follows.Theorem 2.2.
In the dependent general risk model introduced in Section1, assume that the claim sizes  {Xn,n≥1} are END nonnegative r.v.s with common distribution  F∈𝒟 and finite mean  μ. Assume that for any  T>0 with  λ(T)-λ(0)>0, there exists some constant  p>JF+ such that
(2.8)E(N(T))p<∞.
Then, it holds that
(2.9)LF∫0-TF¯(xeδt)λ(dt)≲Ψ(x,T)≲LF-1∫0-TF¯(xeδt)λ(dt).
Furthermore, if F∈ℒ∩𝒟, then (2.4) holds.The rest of the present paper consists of two sections. We give some lemmas and the proof of Theorem2.2 in Section 3. In Section 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,M and a always 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 class 𝒟 and the END structure. The first lemma is a combination of Proposition  2.2.1 of Bingham et al. [4] and Lemma  3.5 of Tang and Tsitsiashvili [15].Lemma 3.1.
If a distributionV∈𝒟, then (i)
for anyγ>JV+, there exist positive constants  a and  b such that  V¯(y)/V¯(x)≤a(y/x)-γ holds for all  x≥y≥b and(ii)
it holds for everyγ>JV+ that  x-γ=o(V¯(x)).By direct verification, END r.v.s have the following properties similar to those of ND r.v.s; see Lemma  3.1 of Liu [12]. 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 any  n≥1, there exists some positive constant  M such that  E(∏k=1nξk)≤M∏k=1nEξk.
(ii)   If r.v.s{ξn,n≥1} are END and  {fn(·),n≥1} are either all monotone increasing or all monotone decreasing, then  {fn(ξn),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 distribution  V and  μV+=Eξ11{ξ1≥0}<∞. Then, for any  θ>0,   x>0 and  n≥1, there exists some positive constant  M such that
(3.1)P(∑k=1nξk>x)≤nV¯(θx)+M(eμV+nx)θ-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 write Snξ=∑k=1nξk,   n≥1. If μV+=0, then ξn is almost surely nonpositive for each n≥1, implying P(Snξ>x)=0 for any positive x, and thus (3.1) holds.
Let, in the following,μV+>0. For any fixed θ>0 and positive integer n, define
(3.2)ξ̃n=min⁡{ξn,θx},ξ̃n+=max⁡{ξ̃n,0}=ξn1{0≤ξn≤θx}+θx1{ξn>θx}.
According to Lemma 3.2(ii), {ξ̃n,n≥1} and {ξ̃n+,n≥1} are still END r.v.s, respectively. Denote S̃nξ=∑k=1nξ̃k, n≥1. Clearly,
(3.3)P(Snξ>x)=P(Snξ>x,max⁡1≤k≤nξk>θx)+P(Snξ>x,max⁡1≤k≤nξk≤θx)≤nV¯(θx)+P(S̃nξ>x).
It remains to estimate the second summand in (3.3). For a positive h, by Lemma 3.2(ii), {ehξ̃n+,n≥1} are END nonnegative r.v.s. Hence, using identity
(3.4)Eehξ̃1+=∫0θx(ehu-1)V(du)+(ehθx-1)V¯(θx)+1,
by Markov inequality and Lemma 3.2(i) we have
(3.5)P(S̃nξ>x)≤e-hxEehS̃nξ≤e-hxEeh∑k=1nξ̃k+≤e-hxM(Eehξ̃1+)n=Me-hx(∫0θx(ehu-1)V(du)+(ehθx-1)V¯(θx)+1)n.
Since 1+u≤eu for all u∈ℝ and (ehu-1)/u is strictly increasing in u>0, from (3.5), we obtain
(3.6)P(S̃nξ>x)≤Mexp⁡{n∫0θxehu-1uuV(du)+n(ehθx-1)V¯(θx)-hx}≤Mexp⁡{nehθx-1θx(∫0θxuV(du)+θxV¯(θx))-hx}≤Mexp⁡{nehθx-1θxμV+-hx}.
Choose h=(θx)-1log(x(μV+n)-1+1), which is positive. For such h, by (3.6), we have
(3.7)P(S̃nξ>x)≤Mexp⁡{1θ-1θlog⁡(xμV+n+1)}≤Mexp⁡{1θlog⁡eμV+nx}.
The last estimate and (3.3) imply the desired estimate (3.1). The lemma is proved.Lemma 3.4.
In the dependent general risk model introduced in Section1, assume that the claim sizes  {Xn,n≥1} are END nonnegative r.v.s with common distribution  F∈𝒟. Let  Z be an arbitrary nonnegative r.v. and assume that  {Xn,n≥1},  {N(t),t≥0} and  Z are mutually independent. Then, for any  T>0 and any positive integer  n0,
(3.8)LF∑k=1n0∑j=1kP(Xje-δτj>x,N(T)=k)≲∑k=1n0P(∑j=1kXje-δτj>x+Z,N(T)=k)≲LF-1∑k=1n0∑j=1kP(Xje-δτj>x,N(T)=k).
Furthermore, if F∈ℒ∩𝒟, then
(3.9)∑k=1n0P(∑j=1kXje-δτj>x+Z,N(T)=k)∼∑k=1n0∑j=1kP(Xje-δτj>x,N(T)=k).We remark that ifF is consistently varying tailed (see the definition in Chen and Yuen [18]), then by conditioning (3.9) easily follows from Theorem  3.2 of Chen and Yuen [18]. Note that this case is in a broader scope, since there is no need to assume independence between (τ1,…,τn0) and Z.Proof.
We follow the line of the proof of Lemma  3.6 of Wang [1] with some modifications in relation to the properties of the class 𝒟 and the END structure. Clearly, for each k=1,…,n0,
(3.10)P(∑j=1kXje-δτj>x+Z,N(T)=k)=∫{0≤t1≤⋯≤tk≤T,tk+1>T}∫0-∞P(∑j=1kXje-δtj>x+z)×P(Z∈dz,τ1∈dt1,…,τk+1∈dtk+1).
We first show the upper bound. For any fixed l>0,
(3.11)P(∑j=1kXje-δtj>x+z)≤P(⋃j=1k{Xje-δtj>x+z-l})+P(∑j=1kXje-δtj>x+z,max⁡1≤j≤kXje-δtj≤x+z-l):=I1+I2.
By F∈𝒟, for any 0<θ<1 and each k=1,…,n0, we have uniformly for all t1,…,tk∈[0,T] and z∈[0,∞),
(3.12)I1≤∑j=1kF¯(θ(x+z)eδtj)≲LF-1∑j=1kF¯((x+z)eδtj),
by firstly letting x→∞ then θ↗1. We note that {Xn,n≥1} are END r.v.s. Then, by F∈𝒟, there exists some positive constant M=M(n0) such that for sufficiently large x, each k=1,…,n0, all t1,…,tk∈[0,T] and z∈[0,∞),
(3.13)I2=P(∑j=1kXje-δtj>x+z,x+zk<max⁡1≤j≤kXje-δtj≤x+z-l)≤P(⋃i=1k{∑j≠iXje-δtj>l,Xie-δti>x+zk})≤∑i=1k∑j≠iP(Xje-δtj>lk-1,Xie-δti>x+zk)≤M∑i=1k∑j≠iF¯(leδtjk-1)F¯((x+z)eδtik)≤MF¯(ln0-1)∑j=1kF¯((x+z)eδtj).
Since l can be arbitrarily large, it follows that
(3.14)limsup⁡l→∞limsup⁡x→∞sup⁡t1,…,tk∈[0,T],z∈[0,∞)I2∑j=1kF¯((x+z)eδtj)=0.
Hence, from (3.10)–(3.14), we obtain for each k=1,…,n0,
(3.15)P(∑j=1kXje-δτj>x+Z,N(T)=k)≲LF-1∑j=1k∫{0≤t1≤⋯≤tk≤T,tk+1>T}∫0-∞F¯((x+z)eδtj)×P(Z∈dz,τ1∈dt1,…,τk+1∈dtk+1)=LF-1∑j=1kP(Xje-δτj>x+Z,N(T)=k)≤LF-1∑j=1kP(Xje-δτj>x,N(T)=k).
As for the lower bound for (3.10), since {Xn,n≥1} are END r.v.s, we have for sufficiently large x and each k=1,…,n0,
(3.16)P(∑j=1kXje-δtj>x+z)≥P(⋃j=1k{Xje-δtj>x+z})≥∑j=1kF¯((x+z)eδtj)-∑1≤i<j≤kP(Xie-δti>x+z,Xje-δtj>x+z)≥∑j=1kF¯((x+z)eδtj)-M∑1≤i<j≤kF¯((x+z)eδti)F¯((x+z)eδtj)=(1-o(1))∑j=1kF¯((x+z)ertj)
holds uniformly for all t1,…,tk∈[0,T] and z∈[0,∞). By F∈𝒟 and Fatou's lemma, we have for any θ̃>1 and all j=1,2,…,
(3.17)liminf⁡1F¯(x)P(Xj>x+ZeδT)=liminf⁡∫0-∞F¯(x+zeδT)F¯(x)P(Z∈dz)≥∫0-∞liminf⁡F¯(θ̃x)F¯(x)P(Z∈dz)=F*¯(θ̃)⟶LF,θ̃↘1,
which means
(3.18)P(Xj>x+ZeδT)≳LFF¯(x).
Similar to (3.15), from (3.10), (3.16), and (3.18), we obtain for each k=1,…,n0,
(3.19)P(∑j=1kXje-δτj>x+Z,N(T)=k)≳∑j=1kP(Xje-δτj>x+Z,N(T)=k)≥∑j=1k∫{0≤t1≤…≤tk≤T,tk+1>T}P(Xj>xeδtj+ZeδT)P(τ1∈dt1,…,τk+1∈dtk+1)≳LF∑j=1k∫{0≤t1≤…≤tk≤T,tk+1>T}F¯(xeδtj)P(τ1∈dt1,…,τk+1∈dtk+1)=LF∑j=1kP(Xje-δτj>x,N(T)=k).
The desired relation (3.8) follows now from (3.15) and (3.19).
IfF∈ℒ∩𝒟, (3.9) follows by using the properties of the class ℒ to establish analogies of relations (3.12) and (3.17). This ends the proof of the lemma.Proof of Theorem2.2.
We use the idea in the proof of Theorem  2.2 of Wang [1] (e.g., Theorem 2.1 of this paper) to prove this result. Clearly, F∈𝒟 and μ<∞ imply JF+≥1. By (2.8), we have for any ϵ>0, there exists some positive integer n1=n1(T,ϵ) such that
(3.20)E(N(T))p1{N(T)>n1}≤ϵ.
To estimate the upper bound ofΨ(x,T), we split it into two parts as
(3.21)Ψ(x,T)≤P(∑j=1N(T)Xje-δτj>x)=(∑k=1n1+∑k=n1+1∞)P(∑j=1kXje-δτj>x,N(T)=k):=I3+I4.
According to Lemma 3.4 of this paper and Lemma  3.5 of Wang [1], we have for sufficiently large x,
(3.22)I3≤(1+ϵ)LF-1∑k=1n1∑j=1kP(Xje-δτj>x,N(T)=k)≤(1+ϵ)LF-1∑j=1∞P(Xje-δτj>x,N(T)≥j)=(1+ϵ)LF-1∑j=1∞P(Xje-δτj>x,τj≤T)=(1+ϵ)LF-1∫0-TF¯(xeδt)λ(dt).
By Lemma 3.3, F∈𝒟, Lemma 3.1(ii), (3.20), and p>JF+≥1, there exists some positive constant M such that for sufficiently large x,
(3.23)I4≤∑k=n1+1∞P(∑j=1kXj>x)P(N(T)=k)≤F¯(p-1x)∑k=n1+1∞kP(N(T)=k)+M(eμ)px-p∑k=n1+1∞kpP(N(T)=k)≤MF¯(x)(EN(T)1{N(T)>n1}+E(N(t))p1{N(T)>n1})=MϵF¯(x).
By Lemma 3.1(i), for any γ>JF+, there exists some positive constant a such that for sufficiently large x,
(3.24)∫0-TF¯(xeδt)λ(dt)≥a-1F¯(x)∫0-Te-γδtλ(dt)≥a-1e-γδT(λ(T)-λ(0))F¯(x),
which, combining (3.23) and λ(T)-λ(0)>0, implies
(3.25)I4≤Mϵ∫0-TF¯(xeδt)λ(dt).
From (3.21), (3.22), and (3.25), we derive the right-hand side of (2.9).
As for the lower bound ofΨ(x,T), by Lemma 3.4, we have for the above given ϵ>0 and sufficiently large x,
(3.26)Ψ(x,T)≥P(∑j=1N(T)Xje-δτj>x+∫0Te-δsC(ds))≥∑k=1n1P(∑j=1kXje-δτj>x+∫0Te-δsC(ds),N(T)=k)≥(1-ϵ)LF∑k=1n1∑j=1kP(Xje-δτj>x,N(T)=k)=(1-ϵ)LF(∑j=1∞P(Xje-δτj>x,τj≤T)-∑k=n1+1∞∑j=1kP(Xje-δτj>x,N(T)=k)):=(1-ϵ)LF(∫0-TF¯(xeδt)λ(dt)-I5).
Analogously to the estimate for I4, we have for sufficiently large x,
(3.27)I5≤F¯(x)EN(T)1{N(T)>n1}≤Mϵ∫0-TF¯(xeδt)λ(dt).
From (3.26) and (3.27), we obtain the left-hand side of (2.9).
IfF∈ℒ∩𝒟, then (2.4) follows by using (3.9) in the proof of (3.22) and (3.26).


## 4. Numerical Calculations

In this section, we perform some numerical calculations to check the accuracy of the asymptotic relations obtained in Theorem2.2. The main work is to estimate the finite-time ruin probability defined in (1.2).We assume that the claim sizes{Xn,n≥1} come from the common Pareto distribution with parameter κ=1, β=2, (4.1)F(x;κ,β)=1-(κκ+x)β,x≥0,
which belongs to the class ℒ∩𝒟, and {(X2n-1,X2n),n≥1} are independent replications of (X1,X2) with the joint distribution (4.2)FX1,X2(x,y)=-1αln⁡(1+(e-αF(x)-1)(e-αF(y)-1)e-α-1),
with parameter α=1, where the joint distribution FX1,X2(x,y) is constructed according to the Frank Copula. It has been proved in Example  4.2 of Liu [12] that X1 and X2 are END r.v.s. Since {(X2n-1,X2n),n≥1} are independent copies of (X1,X2), the r.v.s {Xn,n≥1} are END as well.Assume that the claim arrival processN(t) is the homogeneous Poisson process with intensity parameter λ. Clearly, such an integer-valued process N(t) satisfies the condition (2.8). Choose λ=0.1. The total amount of premiums is simplified as C(t)=ct with the premium rate c=500, and the constant interest rate δ=0.02. Here, we set the time T as T=10 and the initial capital reserve x=500,103,2×103,5×103, respectively. We aim to verify the accuracy of relation (2.4). The procedure of the computation of the finite-time ruin probability Ψ(x,T) in Theorem 2.2 is listed here.Step 1.
Assign a value for the variablex and set l=0.Step 2.
Divide the close interval[0,T] into m=1000 pieces, and denote each time point as ti, i=1,…,m.Step 3.
For eachti, generate a random number ni from the Poisson distribution P(λti), and set ni as the sample size of the claims.Step 4.
Generate the accident arrival time{τki,k=1,…,ni} from the uniform distribution U(0,ti) and the claim sizes {Xki,k=1,…,ni} from (4.1) and (4.2).Step 5.
Calculate the expressionD below for each ti and denote them as {Di,i=1,…,m}:
(4.3)Di=∑k=1niXkie-rτki-∫0tie-rsC(ds),i=1,…,m,
where r and C(t) have been defined and their values have also been assigned.Step 6.
Select the maximum value from{Di,i=1,…,m}, and denote it as D*, compare D* with x; if D*>x, then the value of l increases 1.Step 7.
Repeat Step2 through Step 6, N=109 times.Step 8.
Calculate the moment estimate of the finite-time ruin probability,l/N.Step 9.
Repeat Step1 through Step 8 ten 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 from Table 1, the larger x becomes, the smaller the difference between the analog value and the theoretical result is. Therefore, the approximate relationship in Theorem 2.2 is reasonable.Table 1
Comparison between the analog value and the theoretical result in Theorem2.2.
x(×103)Theoretical resultAnalog value0.53.2846e-63.8120e-6(16.1%)18.2270e-79.1100e-7(10.7%)22.0586e-72.2300e-7(8.3%)53.2956e-83.5000e-8(6.2%)



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*Source: 852852-2011-07-14.xml*