Exponential asymptotics for perturbed renewal equations

M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 77 where R and ψ are the same as in 1.3 and an explicit recursive algorithm can be given for calculating the coefficients a 1 , . . . , a r as rational functions of the coefficients in the expansions for the characteristic quantities of the perturbed risk processes. The ruin probability Ψ ε u ε = P {inf t ≥0 X ε t −u ε } can be interpreted as a tail probability for the infimum of the risk process X ε t . With this interpretation the asymptotic relation 1.6 takes the form of a large deviation theorem. Depending on whether τ r equals 0 or is positive, the balancing condition 1.5 states that either u ε = oε −r or u ε = Oε −r . Following the standard terminology of large deviation theory we refer to this asymptotic behaviour of u 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 x ε t − sF ε ds, t ≥ 0, 2.1 where for every ε ≥ 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 measure F ε A on [0, ∞ which is not concentrated at 0 and can be improper, i.e., its total variation F ε ∞ can be equal to, less than, or greater than 1. As is well-known, there is a unique measurable and locally bounded solution x ε t of Eq. 2.1 Feller, 1971. We denote the expectation of F ε by m ε : m ε = Z ∞ sF ε ds. 2.2 We assume that the functions q ε t and distributions F ε s satisfy the following continuity conditions at ε = 0: Condition A: 1. F ε converges weakly to the non-arithmetic distribution function F as ε → 0. F is not concentrated at the origin; 2. m ε → m ∞ as ε → 0. Condition B: 1. lim u→0 lim ε→0 sup |v|≤u |q ε t + v − q t | = 0 almost everywhere with respect to Lebesgue measure on [0, ∞; 2. lim ε→0 sup 0≤t≤T |q ε t | ∞ for every T ≥ 0; 3. lim T →∞ lim 0≤ε→0 h P r≥T h sup rh≤t ≤r+1h |q ε t | = 0 for some h 0. 78 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 Note that condition A implies that F ε ∞ → F ∞ as ε → 0, 2.3 but that the limiting total variation F ∞ 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 if F ε = F and q ε = q do not depend on ε and F t is a proper distribution function, i.e., if F ∞ = 1. In particular, condition A reduces to the assumption that F t is a non-arithmetic distribution function with a finite expectation m and B to the assumption that the function q 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 every ε ≥ 0 F ε be a proper distribution and assume that conditions A and B hold. Assume t ε → ∞ as ε → 0. Then: x ε t ε → x ∞ = R ∞ q sds m as ε → 0. 2.4 We denote the moment generating function of a distribution function H by ˆ H : ˆ H ρ = Z ∞ e ρs H ds. 2.5 ˆ H ρ is defined for all ρ ∈ R such that the integral 2.5 converges. We assume that the distributions F ε have finite exponential moments and that the functions q ε are exponentially integrable, that is, we assume that the following conditions are fulfilled: Condition C: There exists a δ 0 such, that 1. lim ε→0 d F ε δ ∞, 2. d F δ ∈ 1, ∞. Condition D: Let δ 0 be the number given by condition C. Then the functions e δt q ε t satisfy condition B. Recall that the distributions F ε are not concentrated at 0. Therefore, if d F ε δ ∞, then d F ε is a non-negative, strictly increasing, continuous function on the interval −∞, δ] such that d F 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 ∞ e ρs F ε ds = 1. 2.6 As a matter of fact, the real root of 2.6 depends continuously on the parameter ε: Lemma 2. If conditions A and C hold, then for all ε small enough there exists a unique real root ρ = ρ ε of Eq. 2.6 and ρ ε → ρ as ε → 0. 2.7 Proof. According to condition C 2 one can choose β δ such that d F β 1. Using condition A we have lim ε→0 d F ε β ≥ lim T →∞ lim ε→0 Z T e sβ F ε ds = lim T →∞ Z T e sβ F ds = d F β 1. 2.8 It follows from 2.8 that d F ε β 1 for all ε small enough. Since d F ε is a non-negative, strictly increasing, continuous function on the interval −∞, β] and d F ε −∞ = 0 there exists a unique real root of the equation d F ε ρ ε = 1 for all such ε and this root satisfies ρ ε ≤ β. M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 79 We still have to show that ρ ε → ρ as ε → 0. Using the fact that ρ ε ≤ β for all ε small enough and condition C we obtain lim T →∞ lim ε→0 Z ∞ T e sρ ε F ε ds ≤ lim T →∞ lim ε→0 Z ∞ T e sβ F ε ds ≤ lim T →∞ e T β−δ lim ε→0 Z ∞ T e sδ F ε ds = 0. 2.9 Assume that there exists a subsequence ε k → 0 as k → ∞ such that ρ ε k ≤ ρ − γ , k = 1, 2, . . . for some γ 0. Then using A we have lim T →∞ lim k→∞ Z T e sρ εk F ε k ds ≤ lim T →∞ lim k→∞ Z T e sρ −γ F ε k ds ≤ = lim T →∞ Z T e sρ −γ F ds = d F ρ − γ 1. 2.10 It follows from 2.9 and 2.10 that 1 = lim k→∞ [ F ε k ρ ε k = lim T →∞ lim k→∞ Z T e sρ εk F ε k ds + Z ∞ T e sρ εk F ε k ds 1. 2.11 This contradiction shows that the subsequence ε k does not exist and so lim ε→0 ρ ε ≥ ρ . Assume now that there exists a subsequence ε k → 0 as k → ∞ such that ρ ε k ≥ ρ + γ , k = 1, 2, . . . for some γ 0. Then using A we have 1 ≥ lim T →∞ lim k→∞ Z T e sρ εk F ε k ds ≥ lim T →∞ lim k→∞ Z T e sρ +γ F ε k ds ≥ = lim T →∞ Z T e sρ +γ F ds = d F ρ + γ 1. 2.12 This contradiction implies that lim ε→0 ρ ε ≤ ρ . The proof of Lemma 2 is complete. The root ρ of Eq. 2.6 for ε = 0 plays a special role. Note first of all that a ρ = 0 if F ∞ = 1, b ρ 0 if F ∞ 1, and c ρ 0 if F ∞ 1. Note also that condition C implies ρ δ. Therefore we can define the following moments of the distributions F ε with exponential weights e ρ s : m ε [n] = Z ∞ s n e ρ s F ε ds, n = 0, 1, . . . 2.13 It follows from conditions A, C and the fact that ρ δ that the moments m ε [n] are finite for all n = 0, 1, . . . and all ε small enough, and that m ε [n] → m [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] = b 0n + b 1n ε + · · · + b kn ε k + oε k for n = 0, . . . , k, where |b ln | ∞, l, n = 0, . . . , k and 0 b 0n ∞, n = 0, . . . , k. Note that b 0n = m [n], n = 0, 1, . . . By the definition of ρ it is clear that b 00 = m [0] = 1. The following theorem is our main result concerning exponential asymptotics for perturbed renewal equations. 80 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 Theorem 1. Let conditions A–E be satisfied. Then: 1. The root ρ ε of Eq. 2.6 has the asymptotic expansion ρ ε = ρ + a 1 ε + · · · + a k ε k + oε k , 2.15 where the coefficients a l 1 ≤ l ≤ k are given by the recursion formula a l = b −1 01  −b l0 − l−1 X i=1 b l−i,1 a i − l X r=2 l X j =2 b l−j,r X n 1 ,... ,n j −1 ∈D rj j −1 Y i=1 a n i i n i   . 2.16 Here D rj is the set of all non-negative, integer solutions of the system n 1 + · · · + n j −1 = r, n 1 + · · · + j − 1n j −1 = j . 2. If t ε → ∞ such that ε r t ε → τ r ∈ [0, ∞ as 0 ε → 0 for some 1 ≤ r ≤ k , the following asymptotic relation holds x ε t ε exp{−ρ + a 1 ε + · · · + a r−1 ε r−1 t ε } → e −τ r a r ˜ x ∞ as 0 ε → 0, 2.17 where ˜ x ∞ = R ∞ e sρ q sds R ∞ se sρ F ds . 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 a 1 and a 2 in the following form: a 1 = − b 10 b 01 , a 2 = − 1 b 01 b 20 + b 11 a 1 + 1 2 b 02 a 2 1 = b 11 b 10 − b 20 b 01 b 2 01 − 1 2 b 02 b 2 10 b 3 01 . Remark 2. The case in which F ε = F and q ε = q do not depend on ε is also covered by Theorem 1. In this case the coefficients b lr = 0 for l, r = 0, . . . , k and oε k ≡ 0 in condition E and hence the coefficients a l = 0 for l = 1, . . . , k in the expansion 2.15. In this case the expansion 2.15 takes the trivial form ρ = ρ + oε k which implies oε k ≡ 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 distribution F . One can always take t ε = ε −1 so the condition balancing the rate of perturbation with the rate of growth of time will automatically be satisfied with r = 1. The statement 2 then takes the form x ε ε −1 e −ρ ε −1 → ˜ x ∞ as ε → 0. 2.18 Proof of Theorem 1. By Lemma 2 1 ε := ρ ε − ρ → 0 as ε → 0. 2.19 The Taylor expansion for the function e s yields e sρ ε = e sρ 1 + s1 ε 1 + · · · + s k 1 ε k k + s k+1 1 ε k+1 e s|1 ε | θ ε ks k + 1 , 2.20 where 0 ≤ θ ε ks ≤ 1, 0 ≤ s ∞. M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 81 Recall that ρ δ and 1 ε → 0. Therefore there exist δ δ such that ρ + |1 ε | ≤ δ for ε small enough, say ε ≤ ε . Integrating 2.20 and taking condition C into account one obtains for ε ≤ ε 1 = Z ∞ e sρ ε F ε ds = m ε [0] + m ε [1]1 ε 1 + · · · + m ε [k]1 ε k k + 1 ε k+1 M k θ ε k . 2.21 Here M k = k + 1 −1 sup ε≤ε Z ∞ s k+1 e sδ F ε ds ∞, 0 ≤ θ ε k ≤ 1. Formula 2.21 can be rewritten in the form m ε [1]1 ε 1 + m ε [2]1 ε 2 2 + · · · + m ε [k]1 ε k k + 1 ε k+1 M k θ ε k = 1 − m ε [0]. 2.22 The difference 1 ε → 0 and the sum of all terms on the left side in 2.22 beginning from the second one is o1 ε . Recall also that b 00 = 1 and therefore the expression on the right-hand side is of order ε. Dividing both sides in 2.22 by m ε [1]ε and evaluating the corresponding limits we obtain using E that 1 ε ε → −b 10 b 01 . This means that 1 ε can be represented in the form 1 ε = a 1 ε + 1 ε 1 , 2.23 where a 1 = −b 10 b 01 and 1 ε 1 = oε. The relation 2.23 reduces to 2.15 in the case k = 1. Substituting the expansions given in condition E and 2.23 into 2.22 one obtains 1 ε 1 = a 2 ε 2 + 1 ε 2 , 2.24 where a 2 = b −1 01 −b 20 − b 11 a 1 − 1 2 b 02 a 2 1 and 1 ε 2 = oε 2 . The relations 2.23 and 2.24 yield the relation 2.15 for the case k = 2. The expression for a 2 given above is exactly formula 2.16 with k = 2. Repeating the above argument we obtain the expansion 2.15 and the formula 2.16 for k 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: b 01 + b 11 ε + · · · a 1 ε + a 2 ε 2 + · · · 1 + b 02 + b 12 ε + · · · a 1 ε + a 2 ε 2 + · · · 2 2 + · · · = −b 10 ε + b 20 ε 2 + · · · . 2.25 Equalizing the coefficients of ε l , l ≥ 1 on the right and left-hand sides of 2.25 we obtain the formula 2.16 for calculating the coefficients a 1 , . . . , a k . This completes the proof of 1. To prove the statement 2 we multiply the renewal equation 2.1 by e tρ ε and transform it to the equivalent form ˜ x ε t = ˜ q ε t + Z t ˜ x ε t − s ˜ F ε ds, t ≥ 0, 2.26 where, ˜ x ε t = e tρ ε x ε t , ˜ q ε t = e tρ ε q ε t , ˜ F ε t = Z t e sρ ε F ε ds. 82 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 Because ρ ε → ρ and ρ ε ≤ β δ for all ε small enough it is obvious that condition A is satisfied for the distribution functions ˜ F ε if conditions A and C are satisfied for the distribution functions F ε . Note also that the distribution functions ˜ F ε are proper for all ε ≥ 0. Also, the mean of the distribution ˜ F ε coincides with the mean with exponential weight m ε [1] for the distribution F ε . Similarly, condition B is satisfied for the functions ˜ q ε t = e tρ ε q ε t if this condition is satisfied for the functions e δt q 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 e tρ q t and the distribution function ˜ F must be used instead of q t and F , respectively, when calculating the constant x ∞, i.e., this constant must be replaced by the constant ˜ x ∞. The limit 2.4, written for the functions ˜ x ε t , takes the form ˜ x ε t ε = x ε t ε e −ρ ε t ε → ˜ x ∞ as ε → 0. 2.27 Under the balancing condition ε r t ε → τ r 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 +a 1 ε+···+a r ε r +oε r t ε ∼ x ε t ε e −ρ +a 1 ε+···+a r ε r t ε → ˜ x ∞ 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 coefficient ρ 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 lim 0ε→0 ε r t ε = τ r ∈ [0, ∞. The stronger form of this condition used in statement 2 allows us to trans- form2.28 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 Let Xt , t ≥ 0 be the standard risk process defined by 1.1 and let Ψ u be the corresponding ruin probability defined in 1.2. It is known see for example Feller, 1971, or Grandell, 1991 that the ruin probability Ψ u, u ≥ 0 satisfies the renewal equation Ψ u = α1 − Gu + α Z u Ψ u − sG ds, u ≥ 0, 3.1 where Gu = 1 µ Z u 1 − Gsds if α = λµ c ≤ 1. The distribution function generating this equation is F u = αGu. The total variation of F is α. Only the case α 1 is non-trivial, since obviously the solution of Eq. 3.1 is Ψ u ≡ 1 if α = 1. Note also that in the case α 1 the ruin probability is still Ψ 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 a δ 0 such, that 1. ˆ Gδ ∞, 2. α ˆ Gδ 1. M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 83 Condition F guarantees the existence of a unique real solution R of the characteristic equation α ˆ GR = α Z ∞ e Rs Gds = 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 has R = 0. Multiplication of Eq. 3.1 by e Ru transforms it into a proper renewal equation for the function Ψ ue Ru . 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 ∞ e Rs 1 − Gs ds R ∞ se Rs Gds . 3.4 Let us now consider the family X ε t , t ≥ 0 of risk processes depending on a perturbation parameter ε ≥ 0. This means that the characteristic quantities c ε and λ ε and the distributions G ε u and so the mean µ ε are functions of ε. As a consequence the ruin probability Ψ ε u also depends on ε. The object of our study is the asymptotic behaviour of Ψ ε u as the variable u → ∞ and the perturbation parameter ε → 0. Consider the renewal equation 3.1 parametrized by ε: Ψ ε u = α ε 1 − G ε u + α ε Z u Ψ ε u − sG ε ds, u ≥ 0, 3.5 where G ε u = 1 µ ε Z u 1 − G ε s ds. To ensure that Ψ ε is a solution of 3.5 we assume Condition G: α ε = λ ε µ ε c ε ≤ 1 for all ε ≥ 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 distributions G ε u ≡ Gu do not depend on the perturbation parameter ε. The perturbation conditions for the parameters c ε and λ ε are formulated in the following way: Condition H: c ε = c + c 1 ε + · · · + c k ε k + oε k , where |c l | ∞, l = 1, . . . , k. Condition I: λ ε = λ + λ 1 ε + · · · + λ k ε k + oε k , where |λ l | ∞, l = 1, . . . , k. Note that the definition of the model includes the assumption that c ε 0 and λ ε 0 for all ε ≥ 0 and in particular that c 0 and λ 0. A possible interpretation could be based on the assumption that the intensity of the claim flow λ = λc is a function of the gross premium rate. Let us assume that: a λc has k derivatives, b 0 λ0+ ≤ ∞, c lim c→∞ λcµc α ≤ 1. Under these assumptions there exists a largest root of the equation λc µ c = α . 3.6 The balance relation connected to the diffusion approximation prompts that the difference c − c can play the role of the parameter ε. Note that due to condition G only values c ≥ c have to be considered. The expansion in condition H takes the form c = c + ε, 3.7 84 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 and the expansion in condition H takes the form λc = λc + λ ′ c ε + · · · + λ k c ε k + oε k . 3.8 The standard case corresponds to a model in which the parameter λc = λ does not depend on c. In this case there is another natural choice of parameter ε = 1 − α = 1 − λµc that leads to the following form of the expansion in condition H: c = λµ 1 − ε = λµ + λµε + · · · + λµε k + oε k . 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 case R 0 corresponds to the Cramér–Lundberg approximation and the case R = 0 to the diffusion type approximation. Theorem 2. Let conditions F–I be satisfied and assume that u ε → ∞ such that ε r u ε → τ r ∈ [0, ∞ as 0 ε → 0 for some 1 ≤ r ≤ k. Then the following asymptotic relation holds: Ψ ε u ε exp{−R + a 1 ε + · · · + a r−1 ε r−1 u ε } → e −τ r a r ψ as 0 ε → 0, 3.10 where: a R is the Lundberg exponent determined by the characteristic Eq. 3.2 with α = α ; b ψ is the limiting constant defined in 3.4; c a 1 , . . . , a k are determined by formulas 2.16 with the coefficients b ln , 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 Gu where α ε = λ ε µc ε . The total variation of F ε is α ε . The forcing function of Eq. 3.5 is q ε u = α ε 1 − Gu. The conditions H and I imply that α e → α 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 moments m ε [n] take the form m ε [n] = λ ε µ c ε Z ∞ s n e Rs Gds = λ ε c ε M n , n = 0, . . . , k, 3.12 where R is the solution of Eq. 3.2 with α = α and the coefficients M n are defined by M n = Z ∞ s n e Rs 1 − Gs ds, n = 0, . . . , k. 3.13 Under conditions H and I, the function λ ε c ε can be expanded in an asymptotic series as λ ε c ε = d + d 1 ε + · · · + d k ε k + oε k , 3.14 where the coefficients d , . . . , d k can be found by equalizing the coefficients of ε n on the left-and right-hand sides of the formal expansion λ + λ 1 ε + · · · = c + c 1 ε + · · · d + d 1 ε + · · · . 3.15 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 85 This yields d = λ c , d n = λ n − P n−1 l=0 d l c n−l c , n = 1, . . . , k. 3.16 After this, the moments m ε [n] can be expanded in an asymptotic series as m ε [n] = d M n + d 1 M n ε + · · · + d k M n ε k + oε k , n = 0, . . . , k. 3.17 Therefore the perturbation condition E holds with b ln = d l M n , 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 coefficients a 1 , . . . , a k in the asymptotic expansion 3.10 equal 0. This happens if the coefficients c 1 , . . . , c k , λ 1 , . . . , λ k in the perturbation conditions H and I equal 0. The first non-zero coefficient in the sequence a 1 , . . . , a k , if such a coefficient exists, can be either positive or negative in case R 0. However, if R = 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 d 1 , . . . , d k exists and is positive. This implies that α ε 1 for ε small enough. This contradicts condition G. Remark 5. The first two coefficients a 1 and a 2 are given by a 1 = λ c 1 − λ 1 c λ c M M 1 , a 2 = −λ 2 λ c + λ 2 1 c − λ 1 λ c 1 + λ 2 c 2 λ 2 c M M 1 − 1 2 λ c 1 − λ 1 c 2 λ 2 c 2 M 2 M 2 M 3 1 , where the moment coefficients M n are determined by formula 3.13.

4. Cramér–Lundberg approximation for risk processes with perturbed claim distributions