Cramér–Lundberg approximation for nonlinearly
perturbed risk processes
qMats Gyllenberg
a,∗, Dmitrii S. Silvestrov
a,b,c aDepartment of Mathematics, University of Turku, Turku, Finland bDepartment of Mathematical Statistics, Umeå University, Umeå, Sweden cDepartment of Mathematics and Physics, Mälardalen University, SE-72123 Västerås, SwedenReceived April 1999; accepted September 1999
Abstract
An extension of the classical Cramér–Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér–Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations. ©2000 Elsevier Science B.V. All rights reserved.
MSC: 60K05; 60K30; 90A46
Keywords: Risk process; Cramér–Lundberg approximation; Diffusion approximation; Large deviations; Renewal equation
1. Introduction
The aim of this paper is to present an extension of the classical Cramér–Lundberg approximation of ruin proba-bilities to a model of nonlinearly perturbed risk processes. In traditional risk theory a risk process
X(t )=ct−
N (t )X
k=1
Zk, t≥0 (1.1)
is used to model the business of an insurance company. In (1.1) the positive constant c is the gross premium rate,
N (t )is a Poisson process (with parameterλ) counting the number of claims on the company in the time-interval
q
This work has been supported by The Academy of Finland and The Swedish Academy of Sciences (Grant 1413).
∗Corresponding author. Tel.:+358-23336567; fax:+358-23336595.
E-mail addresses: matsgyl@utu.fi (M. Gyllenberg), dmitrii.silvestrov@mdh.se (D.S. Silvestrov).
[0, t] andZkis a sequence of non-negative i.i.d. random variables (with a finite meanµ) which are independent on
N (t ).Zkis the amount of thekth claim. An important object of study is the ruin probability
Ψ (u)=P{u+inf
t≥0X(t ) <0} (1.2)
for a company which has an initial capital u. Of special interest is the asymptotics of the ruin probabilities for large values of initial capital u.
The loading claim rate of the company is characterized by a constantα=λµ/c. Only the subcritical caseα <1 is non-trivial (ifα≥1, then the ruin probability is identically equal to 1:Ψ (u)≡1). The classical result, known as the Cramér–Lundberg approximation, gives under certain conditions for the claim distributionG(so-called Cramér type conditions), the asymptotics of the ruin probability in the form
Ψ (u)
e−Ru →ψ as u→ ∞, (1.3)
where the constant R, known as the Lundberg exponent, and the limiting constantψare determined by the constant
αand the claim distributionG.
We refer to the paper by Cramér (1955) for a survey of basic results in the area. Later developments are reviewed in the books by Gerber (1979) and Grandell (1991) and in the paper by Thorin (1982).
The Cramér–Lundberg approximation describes the asymptotics of ruin probabilities for fixed values ofα <1. On the other hand, it does not give a detailed description of the asymptotics for the ruin probabilityΨ (u)for large values of u and values ofαwhich are less than but close to 1. For example, it does not give the asymptotic behaviour of ruin probabilities when the initial capitalu→ ∞and simultaneously the gross premium ratec↓λµ. Here the so-called diffusion approximation gives the answer. The asymptotics depend on the relation between the speeds at which c tends toλµand u tends to infinity. Under the condition thatu→ ∞such that(c−λµ)u→τ1the diffusion
approximation of risk processes yields the asymptotics of the ruin probability in the form
Ψ (u)→e−a1τ1 as u→ ∞, (1.4)
where the constanta1is determined by the parameterλand the claim distribution G.
Taking the asymptotic relations (1.3) and (1.4) as the starting point we formulate the problem in a more general way. We consider a whole familyX(ε)(t ), t ≥ 0 of risk processes depending on a small parameterε ≥ 0. The processX(ε)(t )is considered as a perturbation of the processX(0)(t )and therefore we assume some weak continuity conditions for the characteristic quantitiesc=c(ε),λ=λ(ε)and the distributionsG=G(ε)as functions ofεat point
ε=0. Moreover, we admit nonlinear perturbations which means that the characteristic quantities of the perturbed risk processes are nonlinear functions ofε. However, we restrict our attention to the case of a smooth perturbation and hence assume that these functions have k derivatives atε=0, i.e., these functions can be expanded in a power series with respect toεup to and including the order k. The object of our study is the asymptotic behaviour of the ruin probabilitiesΨ(ε)(u)when the initial capitalu→ ∞and the perturbation parameterε→0.
The balance between the speeds at whichεtends to 0 and the initial capital u tends to∞has a delicate influence upon the results. Without loss of generality it can be assumed thatu=u(ε)is a function of the parameterε. The balance between the rate of perturbation and the rate of growth of the initial capital is characterized by the following balancing condition
εru(ε)→τr as ε→0 (1.5)
for some positive integer r andτr ∈[0,∞).
Under the assumptions described above and some natural Cramér type condition for the claim distributionsG(ε)
we obtain the asymptotic relation
Ψ(ε)(u(ε))
exp{−(R+a1ε+ · · · +ar−1εr−1)u(ε)}
where R andψare the same as in (1.3) and an explicit recursive algorithm can be given for calculating the coefficients
a1, . . . , aras rational functions of the coefficients in the expansions for the characteristic quantities of the perturbed
risk processes.
The ruin probabilityΨ(ε)(u(ε)) = P{inft≥0X(ε)(t ) < −u(ε)}can be interpreted as a tail probability for the
infimum of the risk processX(ε)(t ). With this interpretation the asymptotic relation (1.6) takes the form of a large deviation theorem. Depending on whetherτrequals 0 or is positive, the balancing condition (1.5) states that either
u(ε) = o(ε−r)or u(ε) =O(ε−r). Following the standard terminology of large deviation theory we refer to this asymptotic behaviour ofu(ε)as two different large deviation zones.
Our approach is based on renewal arguments as developed by Feller (1971). We use results obtained by Silvestrov (1976, 1978, 1979) concerning an extension of the renewal theory to a model of perturbed renewal equation and follow the lines of recent works by Silvestrov (1995) and Gyllenberg and Silvestrov (2000a) concerning exponential asymptotics for perturbed renewal equations. The main improvement is that the new version of exponential asymp-totics obtained in the present paper covers not only the case where the total variation of the distribution generating the limiting renewal equation is equal to 1 (see Silvestrov, 1995; Gyllenberg and Silvestrov, 2000a) but also the cases in which this variation is less than or greater than 1. This new result plays a key role in the extension of the Cramér–Lundberg approximation to the model of perturbed risk processes.
The paper is organized as follows. In Section 2 we present the results concerning exponential asymptotics for perturbed renewal equation, which we think are interesting in their own right. We present our main results concerning the Cramér–Lundberg approximation for nonlinearly perturbed risk processes in Sections 3 and 4 and in Section 5 we apply these to some important special cases. In Section 6 we give some concluding remarks.
2. Exponential asymptotics for perturbed renewal equations
Consider the family of renewal equations
x(ε)(t )=q(ε)(t )+
Z t
0
x(ε)(t−s)F(ε)(ds), t ≥0, (2.1)
where for everyε ≥ 0 (a)q(ε)(t )is a measurable and locally bounded (that is, bounded on every finite interval) real-valued function on [0,∞)and (b)F(ε)(s)=F(ε)([0, s])is the distribution function of a finite measureF(ε)(A)
on [0,∞)which is not concentrated at 0 and can be improper, i.e., its total variationF(ε)(∞)can be equal to, less than, or greater than 1. As is well-known, there is a unique measurable and locally bounded solutionx(ε)(t )of Eq. (2.1) (Feller, 1971).
We denote the expectation ofF(ε)bym(ε):
m(ε)=
Z ∞
0
sF(ε)(ds). (2.2)
We assume that the functionsq(ε)(t )and distributionsF(ε)(s)satisfy the following continuity conditions atε=0:
Condition (A):
1. F(ε)converges weakly to the non-arithmetic distribution functionF(0)asε→0.F(0)is not concentrated at the origin;
2. m(ε)→m(0)<∞asε→0.
Condition (B):
1. limu→0limε→0sup|v|≤u|q(ε)(t+v)−q(0)(t )| =0 almost everywhere with respect to Lebesgue measure on
[0,∞);
2. limε→0sup0≤t≤T|q(ε)(t )|<∞for everyT ≥0;
Note that condition (A) implies that
F(ε)(∞)→F(0)(∞) as ε→0, (2.3) but that the limiting total variationF(0)(∞)can be equal to, less than, or greater than 1.
The conditions (A) and (B) reduce to the conditions of the classical renewal theorem (Feller, 1971) ifF(ε)=F(0)
andq(ε)=q(0)do not depend onεandF(0)(t )is a proper distribution function, i.e., ifF(0)(∞)=1. In particular, condition (A) reduces to the assumption thatF(0)(t )is a non-arithmetic distribution function with a finite expectation
m(0)and (B) to the assumption that the functionq(0)(t )is directly Riemann integrable on [0,∞).
The starting point for our investigation is the following lemma (Silvestrov, 1976, 1978, 1979) generalizing the classical renewal theorem to the model of perturbed renewal equations.
Lemma 1. Let for everyε≥0 F(ε)be a proper distribution and assume that conditions (A) and (B) hold. Assume t(ε)→ ∞asε→0. Then:
x(ε)(t(ε))→x(0)(∞)=
R∞ 0 q
(0)(s)ds
m(0) as ε→0. (2.4)
We denote the moment generating function of a distribution function H byHˆ:
ˆ
H (ρ)=
Z ∞
0
eρsH (ds). (2.5)
ˆ
H (ρ)is defined for allρ∈R such that the integral (2.5) converges.
We assume that the distributionsF(ε)have finite exponential moments and that the functionsq(ε)are exponentially integrable, that is, we assume that the following conditions are fulfilled:
Condition (C): There exists aδ >0 such, that 1. limε→0Fd(ε)(δ) <∞,
2. Fd(0)(δ)∈(1,∞).
Condition (D): Letδ >0 be the number given by condition (C). Then the functions eδtq(ε)(t )satisfy condition (B).
Recall that the distributionsF(ε)are not concentrated at 0. Therefore, ifFd(ε)(δ) <∞,thenFd(ε)is a non-negative,
strictly increasing, continuous function on the interval(−∞, δ] such thatFd(ε)(−∞)=0.
As is well-known, the asymptotic behaviour of the solution of the renewal equation depends on the real root of the characteristic equation
d
F(ε)(ρ)= Z ∞
0
eρsF(ε)(ds)=1. (2.6)
As a matter of fact, the real root of (2.6) depends continuously on the parameterε:
Lemma 2. If conditions (A) and (C) hold, then for allεsmall enough there exists a unique real rootρ =ρ(ε)of Eq. (2.6) and
ρ(ε)→ρ(0) as ε→0. (2.7)
Proof. According to condition (C) (2) one can chooseβ < δsuch thatFd(0)(β) >1. Using condition (A) we have
limε→0Fd(ε)(β)≥ lim
T→∞limε→0 Z T
0
esβF(ε)(ds)= lim
T→∞ Z T
0
esβF(0)(ds)=Fd(0)(β) >1. (2.8)
It follows from (2.8) thatFd(ε)(β) > 1 for allεsmall enough. SinceFd(ε)is a non-negative, strictly increasing,
continuous function on the interval(−∞, β] andFd(ε)(−∞) =0 there exists a unique real root of the equation d
We still have to show thatρ(ε)→ρ(0)asε→0. Using the fact thatρ(ε)≤βfor allεsmall enough and condition
It follows from (2.9) and (2.10) that
1=limk→∞F[(εk)(ρ(εk))= lim we can define the following moments of the distributionsF(ε)with exponential weights eρ(0)s:
m(ε)[n]=
Z ∞
0
sneρ(0)sF(ε)(ds), n=0,1, . . . (2.13)
It follows from conditions (A), (C) and the fact thatρ(0)< δthat the momentsm(ε)[n] are finite for alln=0,1, . . .
and allεsmall enough, and that
m(ε)[n]→m(0)[n] as ε→0. (2.14) Note also that all the moments m(ε)[n] are strictly positive for ε ≥ 0 since the distributions F(ε)(t ) are not concentrated at 0.
The following perturbation condition will be crucial in the subsequent analysis:
Condition (E): There exists a positive integer k such that m(ε)[n] = b0n +b1nε+ · · · +bknεk +o(εk)for
n=0, . . . , k, where|bln|<∞,l, n=0, . . . , kand 0< b0n<∞,n=0, . . . , k.
Note that b0n = m(0)[n], n = 0,1, . . . By the definition of ρ(0) it is clear that b00 = m(0)[0] = 1.
Theorem 1. Let conditions (A)–(E) be satisfied. Then:
1. The rootρ(ε)of Eq. (2.6) has the asymptotic expansion
ρ(ε)=ρ(0)+a1ε+ · · · +akεk+o(εk), (2.15)
where the coefficientsal(1≤l≤k) are given by the recursion formula
al=b−011
Before we prove Theorem 1 we make a couple of remarks.
Remark 1. Empty sums in formula (2.16) are of course interpreted as 0. In particular, we get the first two coefficients
a1anda2in the following form: which implies o(εk)≡0 and it does not yield any information. However, the statement (2) yields the well known
(Feller, 1971) exponential asymptotics for solutions of the renewal equation under Cramér type conditions for the distributionF(0). One can always taket(ε)=ε−1so the condition balancing the rate of perturbation with the rate of growth of time will automatically be satisfied (withr=1). The statement (2) then takes the form
x(ε)(ε−1)
e−ρ(0)ε−1 → ˜x
(0)(∞) as ε→0. (2.18)
Proof of Theorem 1. By Lemma 2
1(ε):=ρ(ε)−ρ(0)→0 as ε→0. (2.19) The Taylor expansion for the function esyields
Recall thatρ(0)< δand1(ε)→0. Therefore there existδ0< δsuch thatρ(0)+ |1(ε)| ≤δ0forεsmall enough,
Formula (2.21) can be rewritten in the form
m(ε)[1]1(ε)
The difference1(ε) → 0 and the sum of all terms on the left side in (2.22) beginning from the second one is o(1(ε)). Recall also thatb00 =1 and therefore the expression on the right-hand side is of orderε. Dividing both
sides in (2.22) bym(ε)[1]εand evaluating the corresponding limits we obtain using (E) that1(ε)/ε→ −b10/b01.
This means that1(ε)can be represented in the form
1(ε)=a1ε+1(ε)1 , (2.23)
wherea1= −b10/b01and1(ε)1 =o(ε). The relation (2.23) reduces to (2.15) in the casek=1.
Substituting the expansions given in condition (E) and (2.23) into (2.22) one obtains
1(ε)1 =a2ε2+1(ε)2 , (2.24)
wherea2=b−011(−b20−b11a1−12b02a21)and1(ε)2 =o(ε2). The relations (2.23) and (2.24) yield the relation (2.15)
for the casek=2. The expression fora2given above is exactly formula (2.16) withk=2.
Repeating the above argument we obtain the expansion (2.15) and the formula (2.16) fork >2. However, the formula (2.16) can be obtained in a simpler way when the asymptotic expansion (2.15) has already been proven. From (2.22) we get the following formal equation:
(b01+b11ε+ · · ·)(a1ε+a2ε2+ · · ·)
1! +
(b02+b12ε+ · · ·)(a1ε+a2ε2+ · · ·)2
2! + · · ·
= −(b10ε+b20ε2+ · · ·). (2.25)
Equalizing the coefficients ofεl,l≥1 on the right and left-hand sides of (2.25) we obtain the formula (2.16) for calculating the coefficientsa1, . . . , ak. This completes the proof of (1).
To prove the statement (2) we multiply the renewal equation (2.1) by etρ(ε)and transform it to the equivalent form
Becauseρ(ε)→ρ(0)andρ(ε)≤β < δfor allεsmall enough it is obvious that condition (A) is satisfied for the distribution functionsF˜(ε)if conditions (A) and (C) are satisfied for the distribution functionsF(ε). Note also that the distribution functionsF˜(ε)are proper for allε≥0. Also, the mean of the distributionF˜(ε)coincides with the mean with exponential weightm(ε)[1] for the distributionF(ε). Similarly, condition (B) is satisfied for the functions
˜
q(ε)(t )=etρ(ε)q(ε)(t )if this condition is satisfied for the functions eδtq(ε)(t ).
Now we can apply Lemma 1 to Eq. (2.26). Conditions (A) and (B) are satisfied. According to the remark made above the function etρ(0)q(0)(t )and the distribution functionF˜(0) must be used instead ofq(0)(t )andF(0), respectively, when calculating the constantx(0)(∞), i.e., this constant must be replaced by the constantx˜(0)(∞).
The limit (2.4), written for the functionsx˜(ε)(t ), takes the form
˜
x(ε)(t(ε))= x
(ε)(t(ε))
e−ρ(ε)t(ε) → ˜x
(0)(∞) as ε→0. (2.27)
Under the balancing conditionεrt(ε)→τr as 0< ε→0 the asymptotic relation (2.27) can be rewritten in the
following equivalent form:
˜
x(ε)(t(ε))= x
(ε)(t(ε))
e−(ρ(0)+a
1ε+···+arεr+o(εr))t(ε)
∼ x
(ε)(t(ε))
e−(ρ(0)+a
1ε+···+arεr)t(ε)
→ ˜x(0)(∞) as 0< ε→0. (2.28) The latter asymptotic relation is equivalent to (2.17). Theorem 1 is proven.
Remark 3. The limit (2.27) can be obtained under conditions (A)–(D). However, in this case the coefficientρ(ε)in the normalizing exponent is given only in implicit form as the solution of Eq. (2.6). It is the perturbation condition (E) that permits the expansion (2.15) and hence gives a more explicit representation of the normalizing exponent in
(2.27).
Note also that the asymptotic relation (2.28) can be written down under the weaker balancing condition
lim0<ε→0εrt(ε) = τr ∈ [0,∞). The stronger form of this condition used in statement (2) allows us to
trans-form(2.28) into the form given in (2.17).
3. Cramér–Lundberg approximation for perturbed risk processes with perturbed premium rate and perturbed intensity of claim flow
LetX(t ),t ≥0 be the standard risk process defined by (1.1) and letΨ (u)be the corresponding ruin probability defined in (1.2). It is known (see for example Feller, 1971, or Grandell, 1991) that the ruin probabilityΨ (u),u≥0 satisfies the renewal equation
Ψ (u)=α(1−G(u))+α
Z u
0
Ψ (u−s)G(ds), u≥0, (3.1)
where
G(u)= 1
µ
Z u
0
(1−G(s))ds if α= λµ
c ≤1.
The distribution function generating this equation isF (u)=αG(u). The total variation of F isα. Only the case
α <1 is non-trivial, since obviously the solution of Eq. (3.1) isΨ (u)≡1 ifα=1. Note also that in the caseα >1 the ruin probability is stillΨ (u)≡1 but in this case it does not satisfy (3.1).
The standard condition under which the Cramér–Lundberg approximation is valid is:
Condition (F): There exists aδ >0 such, that 1. G(δ) <ˆ ∞,
Condition (F) guarantees the existence of a unique real solution R of the characteristic equation
αG(R)ˆ =α
Z ∞
0
eRsG(ds)=1. (3.2)
The root R is called the Lundberg exponent. In the subcritical case (α <1) the Lundberg exponent is positive and in the critical case (α=1) one hasR=0.
Multiplication of Eq. (3.1) by eRutransforms it into a proper renewal equation for the functionΨ (u)eRu. Applying the renewal theorem to this equation one obtains the following asymptotic relation known as the Cramér–Lundberg approximation
Ψ (u)
e−Ru →ψ as u→ ∞, (3.3)
where,
ψ=
R∞ 0 e
Rs(1−G(s))ds R∞
0 seRsG(ds)
. (3.4)
Let us now consider the familyX(ε)(t ),t ≥ 0 of risk processes depending on a perturbation parameterε ≥0. This means that the characteristic quantitiesc(ε)andλ(ε)and the distributionsG(ε)(u)(and so the meanµ(ε)) are functions ofε. As a consequence the ruin probabilityΨ(ε)(u) also depends onε. The object of our study is the asymptotic behaviour ofΨ(ε)(u)as the variableu→ ∞and the perturbation parameterε→0.
Consider the renewal equation (3.1) parametrized byε:
Ψ(ε)(u)=α(ε)(1−G(ε)(u))+α(ε)
Z u
0
Ψ(ε)(u−s)G(ε)(ds), u≥0, (3.5)
where
G(ε)(u)= 1
µ(ε) Z u
0
(1−G(ε)(s))ds.
To ensure thatΨ(ε)is a solution of (3.5) we assume
Condition (G):α(ε)=λ(ε)µ(ε)/c(ε)≤1 for allε≥0.
The formulation of the perturbation condition for the claim distribution functions G(ε)(u) requires a special discussion. We postpone the presentation of the corresponding results to the next section. In this section we treat the simpler case in which the claim distributionsG(ε)(u)≡G(u)do not depend on the perturbation parameterε.
The perturbation conditions for the parametersc(ε)andλ(ε)are formulated in the following way:
Condition (H):c(ε)=c(0)+c1ε+ · · · +ckεk+o(εk), where|cl|<∞, l=1, . . . , k.
Condition (I):λ(ε)=λ(0)+λ1ε+ · · · +λkεk+o(εk), where|λl|<∞,l=1, . . . , k.
Note that the definition of the model includes the assumption thatc(ε) > 0 andλ(ε) >0 for allε ≥0 and in particular thatc(0)>0 andλ(0)>0.
A possible interpretation could be based on the assumption that the intensity of the claim flowλ = λ(c) is a function of the gross premium rate. Let us assume that: (a)λ(c) has k derivatives, (b) 0 < λ(0+) ≤ ∞, (c) limc→∞λ(c)µ/c < α(0)≤1. Under these assumptions there exists a largest root of the equation
λ(c(0))µ
c(0) =α
(0). (3.6)
The balance relation connected to the diffusion approximation prompts that the differencec−c(0)can play the role of the parameterε. Note that due to condition (G) only valuesc≥c(0)have to be considered. The expansion in condition (H) takes the form
and the expansion in condition (H) takes the form
λ(c)=λ(c(0))+λ′(c(0))ε+ · · · +λ(k)(c(0))εk+o(εk). (3.8) The standard case corresponds to a model in which the parameterλ(c)=λdoes not depend onc. In this case there is another natural choice of parameterε=1−α=1−λµ/cthat leads to the following form of the expansion in condition (H):
c= λµ
1−ε =λµ+λµε+ · · · +λµε
k+o(εk). (3.9)
The following theorem presents a new type of Cramér–Lundberg and diffusion approximations for risk processes with a perturbed gross premium rate and a perturbed intensity of the claim flow. The theorem covers both types of approximations in a unified form. The caseR > 0 corresponds to the Cramér–Lundberg approximation and the caseR =0 to the diffusion type approximation.
Theorem 2. Let conditions (F)–(I) be satisfied and assume thatu(ε) → ∞such thatεru(ε) → τr ∈ [0,∞)as
0< ε→0 for some 1≤r ≤k. Then the following asymptotic relation holds:
Ψ(ε)(u(ε))
exp{−(R+a1ε+ · · · +ar−1εr−1)u(ε)}
→e−τrarψ as 0< ε→0, (3.10)
where: (a) R is the Lundberg exponent determined by the characteristic Eq. (3.2) withα=α(0); (b)ψis the limiting constant defined in (3.4); (c)a1, . . . , akare determined by formulas (2.16) with the coefficientsbln, l, n=0, . . . , k
determined by formula (3.18).
Proof. The proof is based on an application of Theorem 1. Eq. (3.5) is a particular case of the perturbed renewal
equation (2.1). The distribution function generating the renewal equation (3.5) is F(ε)(u) = α(ε)G(u) where
α(ε)=λ(ε)µ/c(ε). The total variation ofF(ε)isα(ε). The forcing function of Eq. (3.5) isq(ε)(u)=α(ε)(1−G(u)). The conditions (H) and (I) imply that
α(e) →α(0) as ε→0. (3.11)
The limit (3.11) and condition (F) imply that conditions (A)–(D) hold. To apply Theorem 1 we must also show that conditions (F), (H) and (I) imply the perturbation condition (E).
The momentsm(ε)[n] take the form
m(ε)[n]=λ
(ε)µ
c(ε) Z ∞
0
sneRsG(ds)=λ
(ε)
c(ε)Mn, n=0, . . . , k, (3.12)
whereRis the solution of Eq. (3.2) withα=α(0)and the coefficientsMnare defined by
Mn =
Z ∞
0
sneRs(1−G(s))ds, n=0, . . . , k. (3.13)
Under conditions (H) and (I), the functionλ(ε)/c(ε)can be expanded in an asymptotic series as
λ(ε)
c(ε) =d0+d1ε+ · · · +dkε
k+o(εk), (3.14)
where the coefficientsd0, . . . , dkcan be found by equalizing the coefficients ofεnon the left-and right-hand sides
of the formal expansion
This yields
d0=
λ(0)
c(0), dn=
λn−Pnl=−01dlcn−l
c(0) , n=1, . . . , k. (3.16)
After this, the momentsm(ε)[n] can be expanded in an asymptotic series as
m(ε)[n]=d0Mn+d1Mnε+ · · · +dkMnεk+o(εk), n=0, . . . , k. (3.17)
Therefore the perturbation condition (E) holds with
bln=dlMn, l, n=0, . . . , k. (3.18)
A direct application of Theorem 1 to the renewal equation (3.5) completes the proof.
Remark 4. It is possible that all the coefficientsa1, . . . , ak in the asymptotic expansion (3.10) equal 0. This
happens if the coefficients c1, . . . , ck,λ1, . . . , λk in the perturbation conditions (H) and (I) equal 0. The first
non-zero coefficient in the sequencea1, . . . , ak, if such a coefficient exists, can be either positive or negative in case
R >0. However, ifR =0, then condition (G) implies that the first non-zero coefficient has to be positive. To see
this, assume that such a coefficient exists and that it is negative. Then the first non-zero coefficient in the sequence d1, . . . , dkexists and is positive. This implies thatα(ε)>1 forεsmall enough. This contradicts condition (G). Remark 5. The first two coefficientsa1anda2are given by
a1=
λ(0)c1−λ1c(0)
λ(0)c(0)
M0
M1
,
a2=
−λ2λ(0)c(0)+λ21c(0)−λ1λ(0)c1+(λ(0))2c2
(λ(0))2c(0)
M0
M1
−1
2
(λ(0)c1−λ1c(0))2
(λ(0))2(c(0))2
M2M02
M13 , where the moment coefficientsMnare determined by formula (3.13).
4. Cramér–Lundberg approximation for risk processes with perturbed claim distributions
In this section we consider the general model in which not only the premium ratec=c(ε)and the intensity of claim flowλ=λ(ε)but also the claim distributionG(u)=G(ε)(u)(and so the meanµ=µ(ε)) are perturbed. The perturbation conditions (H) and (I) forc(ε)andλ(ε)were formulated in Section 3.
The generating distribution in the renewal equation (3.5) is F(ε)(u) = α(ε)G(ε)(u) and its total variation is
α(ε) =λ(ε)µ(ε)/c(ε). The forcing function isq(ε)(u)=α(ε)(1−G(ε)(u)). We assume the following analogue of condition (A):
Condition (J):G(ε)converges weakly toG(0)asε→0.G(0)is not concentrated at the origin. Let,
K(ε)(β)=
Z ∞
0
eβs(1−G(ε)(s))ds. (4.1)
The following condition is an analogue of (C):
Condition (K): There exists aδ >0 such, that 1. limε→0K(ε)(δ) <∞,
2. λ(0)
The condition (K) guarantees the existence of a unique solution of the characteristic equation
λ(0) c(0)
Z ∞
0
eR(0)s(1−G(0)(s))ds=1. (4.2) We also define the constantψ(0)analogously with (3.4):
ψ(0)=
R∞ 0 e
R(0)s
(1−G(0)(s))ds
R∞ 0 seR
(0)s
G(0)(ds)
. (4.3)
The momentsm(ε)[n] take the form
m(ε)[n]=λ
(ε)
c(ε)M
(ε)[n], n=0, . . . , k, (4.4)
where
M(ε)[n]=
Z ∞
0
sneR(0)s(1−G(ε)(s))ds, n=0, . . . , k. (4.5)
The analogue of the perturbation condition (E) is now formulated in terms of the moment coefficientsM(ε)[n]:
Condition (L): There exists a positive integer k such thatM(ε)[n] = e0n +e1nε+ · · · + eknεk +o(εk)for
n=0, . . . , k, where|eln|<∞, l, n=0, . . . , kand 0< e0n<∞, n=0, . . . , k.
Note that conditions (J) and (K) imply that the moments M(ε)[n] → M(0)[n] asε → 0. This implies that
e0n=M(0)[n], n=0, . . . , k.
The perturbation condition (L) is a very natural one. When one considers a specific class of claim distributions a smooth polynomial perturbation of the parameters involved usually implies condition (L). In most cases it is an easy task to translate the expansions of the parameters into the corresponding expansions for the moment coefficients
M(ε)[n] in (L).
We are now ready to formulate the main result of this paper.
Theorem 3. Let conditions (G)–(L) be satisfied and assume thatu(ε) → ∞such thatεru(ε)→ τr ∈[0,∞)as
0< ε→0 for some 1≤r ≤k. Then the following asymptotic relation holds:
Ψ(ε)(u(ε))
exp{−(R(0)+a
1ε+ · · · +ar−1εr−1)u(ε)}
→e−τrarψ(0) as 0< ε→0, (4.6)
where (a)R(0)is the Lundberg exponent determined from the Eq. (4.2); (b)ψ(0) is the limiting constant defined in (4.3); (c)a1, . . . , ak are determined by formulas (2.16) with the coefficientsbln, l, n=0, . . . , kdetermined by
formula (4.8).
Remark 6. It is possible that all the coefficientsa1, . . . , akin (4.6) are equal to 0. This happens if the corresponding
coefficients in the perturbation conditions (H), (I) and (L) equal 0. The first non-zero coefficient in the sequence a1, . . . , ak, if such a coefficient exists, can be either positive or negative in the caseR(0)>0. However (see Remark
3), in the caseR(0)=0 condition (G) implies that the first non-zero coefficient has to be positive.
Remark 7. The balancing condition used in Theorems 2 and 3 can be weakened and replaced by the condition
lim0<ε→0εru(ε)=τr ∈[0,∞)(see Remark 3).
Proof of Theorem 3. We apply Theorem 1 to Eq. (3.5). As follow from the remarks made above, conditions (H)–(L)
imply the conditions (A)–(D).
be expanded according to condition (L), the required expansion for their productm(ε)[n] also exists, and is given by
m(ε)[n]=b0n+b1nε+ · · · +bknεk+o(εk), n=0, . . . , k, (4.7)
where the coefficientsblncan be calculated from the simple discrete convolution formula
bln= l X
m=0
dmel−m,n, l, n=0, . . . , k. (4.8)
Hence condition (E) holds. Now we can apply Theorem 1 and complete the proof.
5. Some important special cases
To illustrate the content of the asymptotic formula (1.6) (or (3.10)), we consider in this section the two simplest casesr=1 andr=2.
In the caser =1 the balancing condition (1.5) reduces toεu(ε)→ τ1and the asymptotic relation (1.6) can be
rewritten in the form
Ψ(ε)(u(ε))−exp{−Ru(ε)}ψ
exp{−Ru(ε)}ψ →e
−τ1a1−1 as ε→0. (5.1)
Let us first consider the asymptotically subcritical case in which the limiting constantα(0) =λ(0)µ(0)/c(0) <1 and so the Lundberg exponentR >0. The limit on the right-hand side of (5.1) is the limit of the relative error of the Cramér–Lundberg approximation in the large deviation zoneu(ε) =o(ε−1)oru(ε)=O(ε−1). Ifτ1=0 then
u(ε) = o(ε−1)and the asymptotic relative error is 0. Ifτ1 >0 thenu(ε) = O(ε−1)and the asymptotic error is
e−τ1a1−1, which differs from zero. Therefore o(ε−1)is the asymptotic bound for the large deviation zone in which
the Cramér–Lundberg approximation (1.3) has zero asymptotic relative error and does not require any correction. In the large deviation zone of the order O(ε−1)the Cramér–Lundberg approximation requires an asymptotic correction and (5.1) yields the value for the corresponding multiplicative correction e−τ1a1.
Similar comments can be made in the asymptotically critical case in whichα(0) =1 and hence the Lundberg exponentR = 0. In this case ψ = 1. Ifτ1 = 0,thenu(ε) = o(ε−1)and the trivial approximation of the ruin
probability byψ = 1 has zero asymptotic relative error. However ifτ1 > 0 thenu(ε) =O(ε−1)and there is a
non-zero asymptotic error e−τ1a1−1. Therefore o(ε−1)is the asymptotic bound for the large deviation zone where
this trivial approximation gives zero asymptotic relative error. In the large deviation zone of the order O(ε−1)the asymptotic correction e−τ1a1 has to be used in order to ensure zero asymptotic relative error.
In the caser =2 the balancing condition (1.5) reduces toε2u(ε) →τ2and the asymptotic relation (1.6) can be
rewritten in the form
Ψ(ε)(u(ε))−exp{−(R+a1ε)u(ε)}ψ
exp{−(R+a1ε)u(ε)}ψ
→e−τ2a2−1 as ε→0. (5.2)
Again we consider first the asymptotically subcritical caseα(0)<1 andR >0. Now the limit on the right-hand side is the limit of the relative error of the corrected Cramér–Lundberg approximation in two large deviation zones of higher order:u(ε)=o(ε−2)oru(ε)=O(ε−2), respectively.
Ifτ2 = 0, thenu(ε) = o(ε−2)and the asymptotic relative error is 0. Ifτ2 > 0, thenu(ε) = O(ε−2)and the
asymptotic error is e−τ2a2−16=0. Therefore o(ε−2)is the asymptotic bound for the large deviation zone in which
In the asymptotically critical case whenα(0)=1 andR=0 the limit on the right-hand side of (5.2) is the limit of the relative error of the diffusion (see (1.4)) type approximation in two large deviation zonesu(ε) =o(ε−2)or
u(ε) = O(ε−2), respectively. What we found for the subcritical case holds true also for the critical case but for the diffusion type approximation instead of the corrected Cramér–Lundberg approximation. In particular, o(ε−2)is the asymptotic bound for the large deviation zone in which the diffusion type approximation has zero asymptotic relative error and does not require any additional correction. In the large deviation zone of the order O(ε−2)the diffusion type approximation requires an asymptotic correction and (5.2) yields the corresponding asymptotic value of the correction e−τ2a2.
As an example let us consider the classical model of diffusion approximation in which the parameterλdoes not depend onεand in which the claim distributionG(ε)(u)=G(u)is also unperturbed. We take the differencec−c(0)
as the parameterε. Herec(0)=λµ. The expansion in condition (H) takes the formc=c(0)+ε. In this caseR(0) =0,ψ(0) =1. Also,M[n]=R0∞sn(1−G(s))ds =µn+1/n+1, whereµn =
R∞
0 snG(ds).
The formulas given in Remark 5 take the form
a1=
2
λµ2
, a2= −
4 3
µ3
λ2µ3 2
. (5.3)
In the caseεu(ε)→τ1∈[0,∞)Theorem 2 yields the classical formula (1.4) of diffusion approximation:
Ψ(ε)(u(ε))→e−a1τ1 as ε→0. (5.4)
In the caseε2u(ε)→τ2∈[0,∞)Theorem 2 yields the relation
Ψ(ε)(u(ε))
e−a1εu(ε) →
e−a2τ2 as ε→0. (5.5)
Relation (5.5) shows that the diffusion approximation does not require any correction in the large deviation zone
u(ε)=o(ε−2)but it does require the additional correction e−a2τ2 in the large deviation zoneu(ε)=O(ε−2).
The example considered above shows that even under standard conditions of diffusion approximation, when only the parametercis perturbed and the perturbation has the simplest possible linear form, Theorem 2 yields a new improved version of the classical diffusion approximation.
6. Concluding remarks
The main new element of our results is a high order exponential asymptotic expansion which extends in a unified way both the classical Cramér–Lundberg approximation and the diffusion approximation of ruin probabilities to a model of nonlinearly perturbed risk processes. The derived approximations have an optimal asymptotic behaviour of the relative errors which depends on the balance between the order of the perturbation parameterεand the order of growth of the initial capitalu. The approximations are supplemented by an explicit algorithm for calculating the asymptotic corrections.
Our results continue the line of research related to analytical, numerical and simulation approximations of ruin probabilities. We refer here to the papers by Beekman (1969), Iglehart (1969), Grandell and Segerdahl (1971), Thorin (1977), De Vylder (1978, 1996), Wikstad (1983,1971), Asmussen (1985), Asmussen and Klüppelberg (1996), De Vylder and Marceau (1996), Grandell (1998) and the books by Gerber (1979), Asmussen (1987, 1999), Sundet (1991), and Grandell (1991), Beirlant et al. (1996) and Embrechts et al. (1997)).
The results related to asymptotics of perturbed renewal equation, given in the paper, have their own value. For example, Gyllenberg and Silvestrov (1999, 2000a) recently applied such type of asymptotics to the study of quasi-stationary phenomena for Markov type processes, stochastic models of population dynamics and highly reliable queuing systems.
As areas for further research, we mention problems related to the rate of convergence and stability prob-lems, possible extensions of the results to more general models of risk processes as well as numerical studies connected with the asymptotic expansions given in Theorems 1–3. Some numerical studies based on nonlinear exponential asymptotics introduced in the present work can be found in the paper by Gyllenberg and Silvestrov (2000b).
It would also be interesting to study other types of perturbations for claim distributions, for instance those connected to truncation of claims in reinsurance models. Our conjecture is that it may be necessary to consider other models of nonlinearly perturbed renewal equations based on non-polynomial asymptotic expansions. Some promising results in this direction have recently been obtained by Englund and Silvestrov (1997) and Englund (1998).
In conclusion we mention potential statistical applications of exponential asymptotics presented in this paper to models in which the perturbed risk process has parameters which are empirical estimators of the corresponding true parameters.
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