Key words. renewal risk models, surplus dependent premiums, boundary value problems, Green s operators, asymptotic expansions,

Transcription

1 EXACT AND ASYMPTOTIC RESULTS FOR INSURANCE RISK MODELS WITH SURPLUS-DEPENDENT PREMIUMS HANSJÖRG ALBRECHER, CORINA CONSTANTINESCU, ZBIGNIEW PALMOWSKI, GEORG REGENSBURGER, AND MARKUS ROSENKRANZ Abstract. In this paper we develop a symbolic techniqe to obtain asymptotic expressions for rin probabilities and disconted penalty fnctions in renewal insrance risk models when the premim income depends on the present srpls of the insrance portfolio. The analysis is based on bondary problems for linear ordinary differential eqations with variable coefficients. The algebraic strctre of the Green s operators allows s to develop an intitive way of tackling the asymptotic behavior of the soltions, leading to exponential-type expansions and Cramér-type asymptotics. Frthermore, we obtain closed-form soltions for more specific cases of premim fnctions in the compond Poisson risk model. Key words. renewal risk models, srpls dependent premims, bondary vale problems, Green s operators, asymptotic expansions, AMS sbject classifications. 9B3, 34B27, 34B5. Introdction. The stdy of level crossing events is a classical topic of risk theory and has trned ot to be a fritfl area of applied mathematics, as depending on the model assmptions often sbtle applications of tools from real and complex analysis, fnctional analysis, asymptotic analysis and also algebra are needed see e.g. [4] for a recent srvey. In classical insrance risk theory, the collective renewal risk model describes the amont of srpls Ut of an insrance portfolio at time t by Nt Ut = + c t X k,. where c represents a constant rate of premim inflow, Nt is a renewal process that conts the nmber of claims incrred dring the time interval, t] and X k k is a seqence of independent and identically distribted i.i.d. claim sizes with distribtion fnction F X and density f X also independent of the claim arrival process Nt. Let τ k k be the i.i.d. seqence of interclaim times. One of the crcial qantities to investigate in this context is the probability that at some point in time the srpls in the portfolio will not be sfficient to cover the claims, which is called the probability k= Department of Actarial Science, Faclty of Bsiness and Economics, University of Lasanne, Extranef Bilding, CH-5 Lasanne, Switzerland, and Swiss Finance Institte hansjoerg.albrecher@nil.ch. Institte for Financial and Actarial Mathematics, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom c.constantinesc@liverpool.ac.k. Partially spported by the Swiss National Science Fondation Project /. Mathematical Institte, University of Wroclaw, Pl. Grnwaldzki 2/4, Wroclaw, Poland zbigniew.palmowski@gmail.com. Spported by the Ministry of Science and Higher Edcation grant NCN 2//B/HS4/982. INRIA Saclay Île de France, Project DISCO, L2S, Spélec, 992 Gif-sr-Yvette Cedex, France georg.regensbrger@ricam.oeaw.ac.at. Spported by the Astrian Science Fnd FWF: J33- N8. School of Mathematics, Statistics & Actarial Science, University of Kent, Cornwallis Bilding, Canterbry, Kent CT2 7NF, United Kingdom M.Rosenkranz@kent.ac.k.

2 2 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ of rin ψ = P T < U =, where U = is the initial capital in the portfolio and T = inf {t : Ut < U = }. A related, more general qantity is the expected disconted penalty fnction, which penalizes the rin event for both the deficit at rin and the srpls before rin, Φ = E e δt wut, UT T< U =, where δ is a discont rate and the penalty wx, y is a bivariate fnction. In risk theory literatre, Φ is often referred to as the Gerber-Shi fnction, see [8]. The classical collective risk model is based on the assmption of a constant premim rate c. However, it is clear that it will often be more realistic to let premim amonts depend on the crrent srpls level. In this case, the risk process. is replaced by Ut = + t Nt pus ds X k. Hence, in between jmps claims the risk process moves deterministically along the crve ϕ, t, which satisfies the partial differential eqation k= ϕ t = p ϕ ; ϕ, =. There are only a few sitations for which exact expressions for ψ are known for srpls-dependent premims. One sch case is the Cramér-Lndberg risk model where Nt is a homogeneos Poisson process with intensity λ and the linear premim fnction p = c + ε, which has the interpretation of an interest rate ε on the available srpls. In the case of exponential claims, it was already shown by [22] that the probability of rin then has the form ψ = λε λ/ε µ λ/ε c λ/ε exp µc/ε + λε λ/ε Γ µc ε, λ ε Γ µc+ε ε, λ ε,.2 where Γη, x = x tη e t dt is the incomplete gamma fnction for extensions to finite-time rin probabilities, see [, 2] and [3]. In fact, for the Cramér-Lndberg risk model with exponential claims and general monotone premim fnction p, one has the explicit expression ψ = γ λ px exp {λqx µx} dx,.3 where /γ + λ px exp {λqx µx} dx and qx x py dy is assmed finite for x > see [23]. Since for srpls-dependent premims the probabilistic approach based on random eqations does not work, and also the sal analytic methods lead to difficlties becase the eqations become too complex, it is a challenge to derive explicit soltions beyond the one given above.

3 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 3 In this paper we will employ a method based on bondary problems and Green s operators to derive closed-form soltions and asymptotic properties of ψ and Φ nder more general model assmptions. For that prpose we will employ the algebraic operator approach developed in [2]. However, since that approach was restricted to linear ordinary differential eqations LODEs with constant coefficients, we will have to extend the theory to tackle the variable-coefficients eqations that occr in the present context. In Section 2 we derive the bondary problem for the Gerber-Shi fnction Φ in a renewal risk model with claim and interclaim distribtions having rational Laplace transform. For solving it, we employ a new symbolic method, described in Section 3. This allows to constrct integral representations for the soltion of inhomogeneos LODEs with variable coefficients, for given initial vales, nder a stability condition. In Section 4 we derive a general asymptotic expansion for the disconted penalty fnction in the renewal model framework. Sbseqently, Section 5 is dedicated to the more specific case of compond Poisson risk models with exponential claims, for which we have second-order LODEs. More specifically, in 5. we derive exact soltions for a generic premim fnction p. Frther, in 5.2, we consider some interesting particlar cases of p. In 5.3 we identify the necessary conditions a premim fnction shold satisfy sch that the asymptotic analysis is possible and the assmptions necessary for the asymptotic reslts in Section 4 are validated. We will end by giving concrete examples of sch premim fnctions and their asymptotics. Throghot the paper we will assme that Ut a.s. This assmption is satisfied for example when p > EX/Eτ + ς for some ς > and sfficiently large ; see e.g. [4]. 2. Deriving the bondary problem. Assme that the distribtion of the interclaim time of the renewal process Nt has rational Laplace transform. For simplicity of notation, we assme frther that the rational Laplace transform has a constant nmerator. Then its density f τ satisfies a LODE with constant coefficients L τ d dt f τ t = 2. and homogeneos initial conditions f τ k t = k =,..., n 2, where L τ x = x n + α n x n + + α. Using the method of [6], we can then derive an integro-differential eqation for Φ L τ p d d δ Φ = α Φ y df X y + ω, 2.2 where L τ is the adjoint operator of L τ defined throgh L τ x = L τ x = x n + α n x n + + α. Assme now that the claim size distribtion also has a rational Laplace transform, so that its density f X satisfies another sch LODE L X d dy f Xy = 2.3 with initial conditions f k X x = k =,..., m 2, where L X x = x m + β m x m + + β.

4 4 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ Then the integro-differential eqation 2.2 becomes a LODE with variable coefficients of order m + n, namely T Φ = g 2.4 with differential operator d T = L X L τ p d d d δ α β 2.5 and right-hand side g = α L X d d ω, where ω w, y f Xy dy. For δ = and w =, Eqation 2.4 redces to the well-known eqation for the probability of rin. The eqations hold for sfficiently reglar fnctions p. In the special case p c one recovers the LODE with constant coefficients whose characteristic polynomial is of degree n + m and corresponds to Lndberg s eqation. It is known that, for δ >, this polynomial has m soltions σ i, with negative real part, and n soltions ρ i, with positive real part; see for example [4] and [3]. In [2], we have derived Φ = γ e σ + + γ m e σm + Gg, 2.6 where the γ i are determined by the initial conditions and Gg m n c ij e σi ξ + e ρj ξ e σi e ρjξ gξ dξ 2.7 defines the Green s operator for the inhomogeneos LODE 2.4 with homogeneos bondary conditions, where c ij = m k=,k i σ i σ k n k=,k j ρ j ρ k ρ j σ i. The bondary conditions for 2.4 consist of the initial conditions Φ k k =,..., m, determined from the integro-differential eqation, and the stability condition Φ =, provided by the model assmptions. In analogy to the constant coefficients case, we assme the existence of a fndamental system for eqation 2.4 with m stable soltions s i and n nstable soltions r j. Here a soltion f is called stable if f and nstable if f as. We write t,..., t m+n for the complete seqence of soltions s,..., s m, r,..., r n, and we assme frthermore that the sccessive Wronskians w k W [t,..., t k ], for k =,..., m + n are all nonzero on the half-line R + = [,. Under these assmptions, the algebraic operator approach developed for the constant coefficients case [2] will be extended to the srpls-dependent premim case in Section 3, and the general soltion of 2.4 then has the form Φ = γ s + + γ m s m + Gg, where the γ i are determined by the initial vales and Gg is again the Green s operator for the inhomogeneos LODE 2.4 with homogeneos bondary conditions,

5 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 5 bt this time with non-constant p. As a conseqence, the representation 2.7 is no longer valid, and we will derive a new explicit expression that generalizes it Theorem 3.4. Let s complete this section with a remark abot how to check that the fndamental system has stable and nstable soltions. Roghly speaking, this amonts to an asymptotic analysis of the soltions of the homogeneos eqation. According to [7, Ch.5], one can identify conditions on p that garantee the existence of sch a fndamental system. These conditions specify the strctre of the coefficients, namely: either they converge sfficiently fast to constants in this case one speaks of almost constant coefficients or they diverge to infinity. The canonical form of 2.4 indicates of corse that the former case applies for or setting here. However, the speed of convergence of the coefficients depends crcially on the premim fnction p. For instance, we will show in Example 5.4 that for p = c e ε/, the LODE with almost constant coefficients converges to the LODE with constant coefficients given in [2]. 3. Green s operator approach. In the previos section we have seen that the core task for compting the Gerber-Shi fnction Φ is to determine the Green s operator G for the inhomogeneos LODE 2.4 with homogeneos bondary conditions consisting of the initial conditions Φ k = k =,..., m and the stability condition Φ =. In this section we will present a symbolic method that allows to constrct G for a generic LODE with variable coefficients and homogeneos bondary conditions. In other words, we consider bondary problems of the general type { T Φ = g, 3. Φ = Φ = = Φ m = and Φ =, where T D m+n + c m+n D m+n + + c D + c is a linear differential operator with variable coefficients and leading coefficient normalized to nity and D d d. Under the conditions described in Section 2 the soltion of 3. is niqe and depends linearly on the so-called forcing fnction g. Therefore the assignment g Φ is a linear operator: the Green s operator G of 3.. The following fact follows immediately from the theory of ordinary differential eqations. Theorem 3.. The Gerber-Shi fnction eqals Φ = γ s + + γ m s m + Gg, 3.2 where G is the Green s operator for eqation 3., and the constants γ i can be identified from the initial conditions. For describing or new method of constrcting an explicit representation of G, let s recall how this was achieved in [2] for the special case of constant coefficients c i c i. We will se the same notation as there, in particlar the basic operators A =, B = and the definite integral F = A + B =. Employing the basic operators, the crcial idea was to factor the Green s operator as G = n A σ A σm B ρ B ρn, 3.3 where the factor operators are defined by A σ e σx Ae σx and B ρ e ρx Be ρx with σ i and ρ j as described before. So the strategy was to decompose the problem and tackle the stable exponents with the basic operator A, the nstable ones with B.

6 6 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ This idea can be carried over to the general case of 3.. Using the reslts of [9], any Green s operator can be flly broken down to basic operators if one can factor the differential operator T into first-order factors. Having a fndamental system t,..., t m+n = s,..., s m, r,..., r n with sccessive Wronskians w k k =,..., m + n for R +, sch a factorization of T can always be achieved by wellknown techniqes described for example in Eqn. 8 of [5]; see also [7] and [24]. Using this factorization, we can break down G in a way similar to 3.3 except that the A σi mst be replaced by more complicated operators based on A and s i, similarly the B ρj by sitable operators involving B and r j. We assme m, n > throghot for avoiding degenerate cases. Proposition 3.2. The Green s operator of 3. is given by G = G s G r where G s = A s A sm and G r = n B r B rn with A ti = A si = wi w i A wi w i for i m, B tj = B rj m = wj w j B wj w j for m + j m + n, setting w = for convenience. Proof. We employ the factorization T = T rn T r T sm T s, with the first-order operators given by T ti = wi w i D wi w i for i m, T tj = T rj m = wj w j D wj w j for m + j m + n. It is then clear that G = A s A sm B r B rn is a right inverse of T since both A and B are right inverses of D. It remains to show that Φ = Gg satisfies the bondary conditions. Differentiating Φ fewer than m times reslts in an expression whose smmands all have the form h A g for some fnctions h; evalating any sch smmand yields h g =, so the homogeneos initial conditions are indeed satisfied. For showing that the stability condition Φ = is also flfilled we write Φ = A s g with g A s2 A sm G r g. Then Φ = s As g and hence Φ = s s g d = becase s = and the integral is assmed to converge. Note that we assme, in the above proof and henceforth, that all forcing fnctions are chosen so that all infinite integrals have a finite vale this will be the case in all the examples treated here. This is also the reason why the r j are incorporated in B operators rather than in A operators as for the s i. Since we want to focs on the symbolic aspects here, we shall not elaborate these points frther. Spelled ot in detail, we can now write the Green s operator of 3. in the factored form G = w w w C w 2 w C w 3 w w 2 2 C w2 2 3 C m+n 2 w n w m+n C m+n wm+n 2 m+n, 3.4 where C i is A for i m and B for m + i m + n. Althogh this brings s already some way towards a closed form for Φ, we wold like to collapse the m + n integrals of 3.4 into a single integration, jst as we did in [2].

7 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 7 To start with, assme for a moment that we did not have any nstable soltions so that the fndamental system is only s,..., s m. In that case we mst dispense with the stability condition, imposing only the homogeneos initial conditions in 3.. The Green s operator consists only of A operators, withot any occrrence of B. In this simplified case, how can one collapse the m integral operators C,..., C m = A in 3.4 by a linear combination of single integrators mltiplication operators combined with a single A? The answer is given by the sal variation-of-constants formla, which can be rewritten in or operator notation as follows [6, 2]. Proposition 3.3. If s,..., s m is a fndamental system for the homogeneos eqation T Φ =, the Green s operator of 3. is given by G s = s A dm, w m + + s m A dm,m w m, 3.5 where w m is the Wronskian determinant of s,..., s m and d m,i reslts from w m by replacing the i-th colmn by the m-th nit vector. In other words, Φ = Gg is a particlar soltion of T Φ = g, made niqe by imposing the initial conditions Φ = Φ = = Φ m =. In or case, the stability condition Φ = follows becase s i = for all i =,..., m. Bt note that 3.5 is valid for any fndamental system s,..., s m of T, yielding a particlar soltion for the initial vale problem meaning 3. withot the stability condition. Let s now trn to the general case, where the fndamental system t,..., t m+n consists of m stable soltions s,..., s m and n nstable soltions r,..., r n. In that case the Green s operator has a representation analogos to 3.5 except that we need B operators in addition to A operators and we have to inclde definite integrals F for balancing the B against the A operators. Theorem 3.4. Define the constants α i,j = d i,m+j /w m+j 3.6 for j =,..., n and i =,..., m + n; the fnctions a j = α,j s + + α m,j s m for j =,..., n; and the fnctions ã,..., ã n by the recrsion ã = a, ã j = a j α m+,j ã α m+j,j ã j. Then the Green s operator of 3. is given by G = m+n t i C i d i,m+n n where C i is A for i m and B for m + i m + n. ã j F dm+j,m+n, 3.7 The proof of this reslt is given in Appendix A. There is a more explicit way of specifying the seqence of fnctions ã,..., ã n occrring in Theorem 3.4. Proposition 3.5. The fnctions ã j in Theorem 3.4 can be compted by solving the system T ã = a, where T is the lower trianglar matrix with entries α m+k,j for j > k, T jk = for j = k, otherwise,

8 8 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ while ã and a are respectively colmns with entries ã,..., ã n and a,..., a n. Hence we have explicitly ã j = det T j /det T, where T j is the matrix reslting from T by replacing its j-th colmn by a. Proof. We have α m+,j ã + + α m+j,j ã j + ã j = a j, for j >, by the definition of the ã j. Bt this is clearly the j-th row of the matrix T ã, while the recrsion base ã = a provides the first row. The explicit formla is an application of Cramer s rle. In either form, the fnctions ã,..., ã n can be readily compted from the given fndamental system s,..., s m, r,..., r n, and the representation 3.7 provides a closed form for the Green s operator of Asymptotic reslts for the renewal risk model. In the seqel, we will k write k l if lim l = for some fnctions k and l. Assme that both the interclaim distribtion and the claim size distribtion have rational Laplace transform, i.e. their densities satisfy the ODE 2. and 2.3, respectively. Assme that the soltions of Eqation 2.4 are of the form t i βi e yi, i.e. with and t i exp {Aη i }, i =,..., n + m, 4. η i y i + β i, i =,..., n + m 4.2 y m <... < y < y m+ <... < y m+n 4.3 so the η i are not asymptotically eqivalent. Remark 4.. Note that for the premim fnctions p = c+ε, p = c+ +ε and p = c exp ε/, the corresponding t i flfill the conditions For m = n =, a more detailed analysis is presented in Section 5.3. Define the constants h k = γ k n α jk F dm+j,m+n with γ k appearing in 3.2 and α jk as defined in 3.6. For a permtation ϕ on {,..., m + n} we define π i = where sgnϕ denotes the parity of ϕ. ϕi=n+m sgnϕ k i yϕk k ϕ, sgnϕ n+m k= yϕk k Theorem 4.. If g e ν for ν > y, then nder the asymptotic expansion Φ m+ holds, with ϑ i = h i s i i =,..., m and ϑ m+ m+n ϑ i 4.4 π i y i + ν g.

9 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 9 Φ P k This is eqivalent to saying that lim ϑi ϑ k+ =, for k =,..., m. Proof. Note that by , t k i yi keaηi. Using 3.7 and the Leibniz formla for the determinant, after some calclations one gets that expansion 4.4 with ϑ k = l k t k and ϑ m+ = m+n Using l Hôpital s rle completes the proof. π i t i C i g t i. 5. Compond Poisson risk process with exponential claims. Let s now focs on the case of a compond Poisson model exponential interclaim times with mean /λ with exponential claim sizes with mean µ and a generic premim fnction p. The differential eqation in 2.4 has order two in this case, so we expect to have one stable soltion s and one nstable soltion r. In fact, here we can relax the notion of an nstable soltion, allowing any fnction where r = lim r exists and is different from zero so the limit does not necessarily have to be infinity. The reason for this extension is that the basic argment for the ansatz Φ = γ s s + γ r r carries over: Every soltion of 2.4 mst be of the form 5 since r, s forms a fndamental system. Bt then the stability condition Φ = can only be satisfied if γ r = becase we reqire s =. This is why the form 3.2 is still jstified in the special case n = with γ = γ s. Bt note that this argment fails when there are more than two nstable soltions since they can cancel ot nless we take some frther precations e.g. reqiring them to be of the same sign. 5.. Closed-form soltions for generic premim. For a discont factor δ >, the expected disconted penalty fnction satisfies the second order LODE D + µ pd + δ + λ Φ λµ Φ = λd + µ ω. Expanding the operators, the eqation is eqivalent with pd 2 µ p + p λ δ D + δµ Φ = λd + µ ω. Assming that p for all, this is frther eqivalent to D 2 + µ + p p λ + δ D δµ Φ = g, 5. p p with g = λ p D + µ ω. Frthermore, we assme that p is chosen in sch a way that the associated homogeneos soltion has a fndamental system s, r with one stable soltion s and one nstable soltion r with Wronskian w = w 2 = sr s r nonzero on R +. Then the Green s operator for the bondary problem for the Gerber- Shi fnction Φ is given by Theorem 3.4 with s = s and r = r, namely Gg = s rv wv r sv wv + r s s sv wv gv dv. 5.2

10 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ For calclating the fll expression Φ = γ s + Gg 5.3 we have to determine the constant γ. 2.2 at zero, one obtains Evalating the integro-differential eqation c Φ + λ + δ Φ = λ ω and therefore γ = λ ω + c Gg λ + δ s c s = λ ω + c rs r s s λ + δ s c s sv wv gv dv 5.4 for the reqired constant. For δ =, the LODE 5. is of first order in Φ, and its associated homogeneos eqation has an nstable soltion r = and a stable soltion v dy dv s = exp µv + λ py 5.5 pv cf. [4]. For the fndamental system s, r, the Wronskian is jst w = w 2 = s, and the Green s operator 5.2 specializes to Gg = s s v + sv s v s s while the constant in Φ = γ s + Gg is now given by γ = s λω p s sv s v gv dv λs ps. Ths the Gerber-Shi fnction can be written generically as Φ = in terms of s. s λω p s + s λs ps s v + sv s v gv dv s sv s v s s sv s v gv dv sv s v gv dv Remark 5.. For δ = and w =, one has g = and ψ = γ s, recovering.3 for the rin probability Closed-form soltions for some particlar premim strctres. A Linear premim: As discssed in Section, the linear fnction p = c+ε can be interpreted as describing investments of the srpls into a bond with a fixed interest rate ε > ; see for example [22]. For δ > and p = c + ε, we can compte a fndamental system for the second-order LODE D 2 + µ + ε c+ε λ+δ c+ε D δµ c+ε Φ = λ c+ε D + µ ω

11 in the form ON SURPLUS-DEPENDENT PREMIUM RISK MODELS s = U δ λ+δ ε +, ε +, µ + µc λ+δ ε ε + c ε r = M δ λ+δ ε +, ε +, µ + µc ε ε + c λ+δ ε exp µ, exp µ, 5.6 where M and U denote the sal Kmmer fnctions as in [, 3.]. For, the estimate in 3..8 yields s = K ε + c λ/ε exp µ + O ε+c, 5.7 while the estimate in 3..4 yields r = K 2 ε+c δ/ε +O ε+c = K 2 ε+c δ/ε +Oε+c δ/ε, 5.8 where K and K 2 are some constants. Hence s is indeed a stable and r an nstable soltion. Using 3..4 one derives the Wronskian w 2 = ε λ+δ/εε µ + c λ+δ/ε exp µ + µc ε. λ+δ Γ ε ελ+δ Γ δ ε δ Sbstitting these expressions in 5.2, we end p with Γδ/ε+ Gg = µ λ+δ/ε Γδ+λ/+ε ε ε ε + c λ+δ/ε exp µ µc ε U Mv M Uv + M U U Uv gv dv, where U and M are Kmmer fnctions appearing on the right hand side of 5.6. This jointly with 5.4 is sfficient to determine the disconted penalty fnction in 5.3. B Exponential premim: In general, an exponential premim fnction leads to intractable reslts. However, for p = c + e the probability of rin can be worked ot from the expression in Section 5.: ψ = + λ c F λ c, µ; + λ c ; e + 2 e + 2 λ c 2µF + µ, + λ c ; 2 + λ c ; 2 where F a, b; c; z = 2 F a, b; c; z stands for the hypergeometric fnction, see e.g. []. C Rational premim: For a basic rational premim like p = c + +, the exact symbolic form for the probability of rin can be compted p to qadratres, namely ψ = 2 λc + λ/c e cµ λ/c c + c + λ+c2 /c 2 + d + λc + λ/c2 e cµ λ/c c + c + λ+c2 /c 2 + d. D Qadratic premim: For the qadratic fnction p = c + 2, the probability of rin can be determined as ψ = λ e λ arctanx/ c+µx c/ c /c + x 2 dx + λ e λ arctanx/ c+µx c/ c /c + x 2 dx.

12 2 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ 5.3. Asymptotic reslts for generic premim. Assme the LODE Φ n + c n Φ n + + c Φ = 5.9 has complex coefficients c i continos on R + and define its characteristic eqation as y n + c n y n + + c =. 5. Then the asymptotic behavior of soltions of 5.9 for essentially depends on the behavior of the roots y,..., y n of 5. as see e.g. [7, 5.3., p. 25], which will be exploited below Probability of rin. When δ = and w =, the expected disconted penalty is the probability of rin. For this qantity we have the following asymptotic estimate we se the convention p = lim p: Proposition 5... If p = c, where c is constant, then 2. If p =, then ψ µ λ γ exp µ + λ ψ µ λ γ p exp µ + λ Proof. Integration by parts in 5.5 gives s = µ λ exp µv + λ v with s = µ λ ĥµ λ, s = p h = expλ λ dw pw. Letting f = λ dw pw,. dw pw,. dw pw dv λ exp µ + λ dw pw., where ĥ denotes the Laplace transform of exp µv + λ v dw pw dv, one gets ψ = γs = µf f = L X d d f, + p. To prove the first part with γ = λs ps. Note that f = µf of the proposition we need to show that ψ = µ +c f + o. According to or previos observation, µf f µf = +p, which completes the proof sing l Hôpital rle. The second part can be proved similarly Expected disconted penalty. We consider two cases of premim fnctions: P. the premim fnction behaves like a constant at infinity p = c, p = O 2 ; 5. or

13 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 3 P2. the premim fnction explodes at infinity, p = as p = c + l ε i i, c >. 5.2 The first case is satisfied by the rational and exponential premim fnctions. The second case is satisfied by the linear and qadratic premim fnctions see Section 5.2. Consider first the homogenos eqation, 5. with g =, i.e. eqation 2.4 with T given in 2.5 with c = δµ p, c = µ + p p λ + δ p. After tedios calclations one can check that in the case 5. we have lim c c < so that Conditions and 2 of [7, p. 252] are satisfied. In the second case 5.2 we have c lim c 2 = and Conditions and 2 of [7, p. 254] hold. From [7, p.252] we hence know that for both cases 5. and 5.2 the asymptotic behavior of the soltion of 5. is where and ϱ = ϱ 2 = { } t i exp ϱ i t + ϱ i tdt, i =, 2, µ + p p λ+δ p µ + p p λ+δ p + 4 δµ p 2 2 µ + p p λ+δ p + µ + p p λ+δ p + 4 δµ p are the negative and positive soltion, respectively, of the characteristic eqation x 2 + µ + p p λ + δ x δµ =. 5.4 p p Here ϱ and ϱ 2 are defined by ϱ i ϱ i = = 2ϱ i + µ + p p λ+δ p 2 ϱ i µ + p p λ+δ p δµ p

14 4 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ Remark 5.2. Note that if the premim fnction p satisfies conditions 5. and 5.2, the soltions t i will be of the asymptotic form 4., where η i = ϱ i +ϱ i. In order to complete the asymptotic analysis of Φ for large, recall that the Gerber- Shi fnction Φ is given by Φ = γs + Gg, for a normalizing constant γ. Theorem 5.2. Under the assmptions 5. and 5.2 regarding the premim fnction, the asymptotics of the Gerber-Shi fnction are described by with the exception Φ h s + K g, Φ h s + K 2 g for the case 5.2 with l =. Here h = γ sv s v gv dv. Remark 5.3. Moreover, for the particlar examples considered here, the strctre of s and r is indeed that of the form that we had to impose as a condition in the more general framework. Proof. First note that for δ =, Gg =, and ths Φ has the same behavior as the probability of rin ψ = γs. Evalating the expression 5.3 at δ =, s e µ+λ R dv pv µp + p λ leads to the classical reslt regarding the probability of rin.3. For δ, one needs the asymptotic behavior of Gg, which based on 5.2 can be redced to analyzing the asymptotic behavior of q = s since the term s rv wv gvdv r sv wv gvdv, sv s v gv dv behaves as s at infinity. After rewriting q = rv wv gvdv s sv wv gvdv r and expanding the Wronskian, one can apply l Hôpital rle and see that as after some algebra q g r r s s s s g s s r r r r. 5.5 Using Fedoryk s asymptotic expressions 5.3 one more time, one can perform the analysis along the two cases introdced here. It easy to check that in the first case, P, we have s exp { k }, r exp { k 2 }, 5.6

15 λ+δ µ c + where k = as lim s s where K = ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 5 r µ λ+δ c 2 +4 δµ c 2 and k 2 = µ = k r and lim r = k 2, then k+k2 k = k 2k 2 k λ+δ c r µ λ+δ c 2 +4 δµ c 2. Ths, h K g,, 5.7 r δµ c µ λ+δ c µ λ+δ c 2 +4 δµ c. The second case, P2, is more complex, prodcing more intriging asymptotics. One can show that in this case s h β e µ, with β R. Note that for l =, ε = ε one recovers the asymptotics 5.7. One can also check that whereas r r prodcing respectively s lim s = µ, for l =, and r for l >, q g and q g. 5.8 Example 5.3. Consider a compond Poisson risk model with premim fnctions described by assmptions P or P2. Let the penalty be a fnction of the srpls only, wx, y = e νx. Since we are in the exponential claims scenario, g = λµd + µ Ths, for a linear premim fnction, w ye µy dy = λνe ν+µ. Φ = γs + GgΦ h β e µ + λνe ν+µ, with β = λ/ε, whereas for all the other premim fntions in the class considered here, Φ h β e µ + λνe ν+µ,, with β R. Example 5.4. When p = c exp ε/, one has a differential eqation with almost constant coefficients, D 2 + µ ε 2 λ + δ exp ε/d δµ c c exp ε/ Φ = λ exp ε/d + µω. 5.9 c This is an eqation of form 5.9, with coefficients satisfying c k = α k + a k, k =, 2 5.2

16 6 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ + λ+δ c a and a = with α k constant and a k d <. Here a = ε 2 δµ c a, with a = n ε n k= n n! a. From [5, Th. 8., p. 92] see also Problem 32, p.5 there we can hence conclde that the homogeneos eqation has a fndamental system with asymptotics ths a d <, and similarly for a and s = e σ + o and r = e ρ + o, where σ and ρ are soltions of the eqation x 2 + µ λ + δ x δµ c c =, with Reσ < and Reρ >. Note that these soltions coincide with the one of the constant premim case. Conseqently, one has the same asymptotic behavior as when the premim rate is constant. Remark 5.4. The above closed-form soltions were worked ot for the compond Poisson model. In principle, similar closed-form soltions are possible for more general renewal risk models as discssed in Sections 2-4, in which case higherorder differential eqations appear and have a practical meaning. For LODEs with constant coefficients one can often find closed-form soltions for certain fnctions of interest [2]. For eqations with variable coefficients, closed-form soltions can be obtained by or method if explicit fndamental soltions t,..., t m+n of the homogeneos eqation are available as for the second-order examples treated in this paper. Typically this will happen for eqations with inherent symmetries. Bt even if this is not the case, one may always consider nmerical approximations for the fndamental soltions t,..., t m+n and then apply the Green s operator with those approximations inside. Of corse this raises the interesting qestion how the error propagates, a problem somewhat similar to the asymptotic analysis presented earlier. 6. Conclsion. We have provided a symbolic method and a conceptal framework for stdying bondary vale problems with variable coefficients as they appear in modelling the srpls level in a portfolio of insrance contracts in classical risk theory. The approach presented allows a detailed analysis of the asymptotic behavior of the soltions of these eqations nder a set of conditions. For the specific case of the compond Poisson risk model, these conditions were made more explicit in terms of conditions on the form of p. Moreover, several new closed-form soltions were established within this framework. Appendix. Proof of Theorem 3.4. The proof of Theorem 3.4 hinges on the following technical lemma on Wronskian determinants. Lemma A.. We have d i,k+ w k = di,k w k+ w 2 k for i k < m + n. Proof. We have to show d i,k+ w k d i,k+ w k = d i,k w k+. We note that all expressions in this formla are certain minors of the Wronskian matrix W for t,..., t m+n. So let s write W i,...,i l j,...,j l for the minor of W reslting from deleting the colmns indexed i,..., i l and the rows indexed j,..., j l. Then we have w k = W k+ k+, d i,k+ = i+k+ Wk+ i, d i,k = i+k W i,k+ k,k+, with the derivatives d i,k+ = i+k+ Wk i and w k = W k. For the latter, we se the fact that a Wronskian determinant can be

17 ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 7 differentiated if one replaces the last row by its derivative; see for example [, p. 8]. Mltiplying by i+k, it remains to show Wk iw k+ k+ W k+ i W k+ k = W i,k+ k,k+ det W. Bt this is a classical determinant formla of Sylvester; see for example [2, p. 57] or Eqn in [9]. The preceding lemma is the key tool for removing the nested integrals in 3.7. For seeing this, note that it can be read backwards as giving the integral of d i,k w k+ /w 2 k. In conjnction with certain operator identities taken from [8], this allows s to collapse expressions of the form A A or B B or, at the interface of the two blocks, A B. Proof. [Proof of Theorem 3.4] Note that the case n = redces to Proposition 3.3, so we may assme n > in the seqel. We know from Proposition 3.2 that G = n A s A sm B r B rn, sing the notation employed there. Based on this factorization, we prove 3.7 by indction on n. In the base case n =, applying Proposition 3.3 again yields m G = G s B r = s i A dm,i wm+ w m w m B wm w m+ = m s i A d m,i w m+ w 2 m B wm w m+, and Lemma A. gives d m+,i /w m for the expression in parentheses. Now we employ the identity AfB = A f + f B, for arbitrary fnctions f, from [8]. Sbstitting the expression in parentheses for f, this gives so we end p with G = m A dm+,i w m + dm+,i w m B α,i F, s i A dm+,i d w m+ + s m+,i i w m B wm w m+ α,i s i F wm w m+. In the middle, we factor ot i s i d m+,i, which eqals r d m+,m+ as one sees by replacing the last row in w m+ by the first and then expanding along that last row. Bt d m+,m+ = w m, so the middle sm simplifies to r B d m+,m+ /w m+ and may ths be incorporated into the first sm. In the third sm of the above expression, we factor ot i α,i s i = a. Ths we obtain finally G = m+ t i C i d m+,i w m+ ã F dm+,m+ w m+, which is the desired formla 3.7 for n =. Now assme Eqation 3.7 for n; we

18 8 ALBRECHER, CONSTANTINESCU, PALMOWSKI, REGENSBURGER, ROSENKRANZ prove it for n +. Using the indction hypothesis we obtain G = n A s A sm B r B rn B rn+ m+n n d = t i C i,m+n i ã j F dm+j,m+n wm+n+ B wm+n = m+n t i C i di,m+n w B m+n 2 n ã j F w d m+n+ m+j,m+n w B m+n 2. As before, we see that Lemma A., with n + in place of n, can be applied to the expressions in the two parentheses, yielding d m+n+,i / for the former and d m+n+,m+j / for the latter. In addition to the identity for AfB sed for the base case, we need now also the related identities BfB = f B B f and F fb = F f also to be fond in [8]. When we sbstitte for f, these identities take on the form for the first expression and AfB = A dm+n+,i BfB = B dm+n+,i + dm+n+,i B α n+,i F, dm+n+,i F fb = F d m+n+,m+j / α n+,m+j F for the second. We split the first sm above into the two sms m d s i A m+n+,i + dm+n+,i B α n+,i F, n r j dm+n+,m+j B B B dm+n+,m+j wm+n. In the lower-range sm m m s i A dm+n+,i + s i d m+n+,i / B wm+n m α n+,i s i F wm+n we can apply the same determinant expansion as before to obtain m n s i A dm+n+,i d + r n+ r m+n+,m+j j B = m a n+ F t i C i d m+n+,i n wm+n + t m+n+ C m+n+ d m+n+,m+n+ wm+n r j d m+n+,m+j B wm+n a n+ F wm+n ;

19 in the pper-range sm we get n ON SURPLUS-DEPENDENT PREMIUM RISK MODELS 9 r j d m+n+,m+j B wm+n + n t m+j C m+j d m+n+,m+j. Combining the lower-range with the pper-range sm, the first sm within the latter cancels with the second sm within the former, yielding m+n+ t i C i d m+n+,i a n+ F wm+n. Now let s tackle the second sm in the above expression for G, namely = n n ã j F w d m+n+ m+j,m+n w B m+n 2 ã j αn+,m+j F F dm+n+,m+j wm+n n = α n+,m+j ã j F wm+n n ã j F dm+n+,m+j, where the expression in parentheses is a n+ ã n+ by the definition of the ã j. Altogether we obtain now G = = m+n+ m+n+ t i C i d m+n+,i a n+ F t i C i d m+n+,i n ã j F dm+n+,m+j wm+n a n+ ã n+ F wm+n n+ which is indeed 3.7 with n + in place of n. ã j F dm+n+,m+j, In conclding this Appendix, let s also mention that Theorem 3.4 is also valid if B is taken to be the operator b with finite b R rather than b =. The reason x is that the operator identities from [8] are also valid in this case and were actally set p for this case in the first place. Acknowledgments. The athors wold like to thank the Mathematical Institte of Wroclaw, Poland, the Radon Institte for Comptational and Applied Mathematics, Astria, and the University of Lasanne, Switzerland, for accommodating several research visits. REFERENCES [] M. Abramowitz and I.A. Stegn, Handbook of mathematical fnctions with formlas, graphs, and mathematical tables, vol. 55 of National Brea of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C., 964.

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