Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion

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1 Insurance: Mathematics and Economics 28 (21) 13 2 Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion Hanspeter Schmidli Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, 21 Copenhagen, Denmark Received 1 September 1998; received in revised form 1 August 2; accepted 19 September 2 Abstract We consider a risk model described by an ergodic stationary marked point process. The model is perturbed by a Lévy process with no downward jumps. The (modified) ladder height is defined as the first epoch where an event of the marked point process leads to a new maximum. Properties of the process until the first ladder height are studied and results of Dufresne and Gerber [Insurance: Math. Econ. 1 (1991) 51, Furrer [Scand. Actuarial J. (1998) 59, Asmussen and Schmidt [Stochastic Process. Appl. 58 (1995) 15 and Asmussen et al. [ASTIN Bull. 25 (1995) 49 are generalized. 21 Elsevier Science B.V. All rights reserved. MSC: M1; M13 Subj. Class: C6 Keywords: Perturbed risk model; Lévy process; Ladder heights; Marked point process; Markov modulated risk model 1. Introduction Consider a stationary and ergodic marked point process (smpp) M = (σ i,u i,m i ) on a probability space (Ω, F,P)with event times <σ 1 <σ <σ 1 <σ 2 < and marks (U i,m i ) (, ) E. Here E is a Polish space with Borel σ -algebra E and M i is interpreted as an environmental variable. Let { i=1 I <σi t if t>, N t = i= I t<σ i if t. We call M a compound Poisson model if E ={}, i.e. there are no environmental marks, and...,σ 1 σ 2,σ σ 1, σ,σ 1,σ 2 σ 1,... are i.i.d. exponentially distributed random variables and (U i ) is an i.i.d. sequence of positive random variables independent of (σ i ). In this paper, we consider the process N t S t = U i t + ηb (α) i=1 t, (1) Tel.: ; fax: address: schmidli@math.ku.dk (H. Schmidli) /1/$ see front matter 21 Elsevier Science B.V. All rights reserved. PII: S ()62-7

2 14 H. Schmidli / Insurance: Mathematics and Economics 28 (21) 13 2 where η>is some constant and (B t (α) ) a spectrally positive α-stable Lévy motion independent of M. This is a process with independent and stationary increments, such that the increments follow a stable law (see Section 2). Spectrally positive means that there are no downward jumps. If η =, we call (S t ) an unperturbed risk model. Because we will need that E[ S t < we assume 1 <α 2. In the special case α = 2 the process (B t (2) ) is a Brownian motion with E[(B t (2) ) 2 = 2t. It will be important for our approach that (S t ) has no downward jumps. The reader should note that we prove Theorems 1 and 2 for any perturbation that is a Lévy process (B t ) with no downward jumps and E[B t =. But the distribution H of the maximum of (ηb t t : t ) will then not be given by (4). The case α = 2 is widely discussed in the literature (see, for instance, Gerber, 197; Dufresne and Gerber, 1991; Veraverbeke, 1993; Furrer and Schmidli, 1994; Schmidli, 1995). Furrer (1997, 1998) considers the case where 1 <α<2and M is a compound Poisson process. In Schlegel (1998), a more general perturbation is considered. Let λ = E[N 1 be the intensity of the claim arrivals and µ = λ 1 E[ N 1 i=1 U i be the mean value of a typical claim. We assume the net profit condition ρ = λµ 1. This implies that lim t S t =. Let m t = sup{s s : s<t}, τ + = inf{σ k : k>,s σk >m σk }, L c = m τ+, L d = S τ+ L c, Z + = L c S (τ+ ) and M + = M N(τ+ ). τ + is then the first time where a jump of the unperturbed model leads to a new maximum of the process (S t ). We will call τ + the first (modified) ladder epoch. Note that inf{t >:S t > } = almost surely, and therefore a ladder epoch in the classical sense cannot be defined. L c is then the part of the ladder height due to the perturbation, L d the part due to the jump of the unperturbed model. We furthermore denote by U + = L d + Z + the height of the jump leading to a new ladder height. This also gives an interpretation of Z +. M + denotes the environmental state at the ladder epoch. Note that L d, Z +, U + and M + are not defined if τ + =, whereas L c is well defined. If τ + < a second ladder epoch τ (2) + (τ +, can be defined. We denote the number of finite ladder epochs by K and use the superscript (k) to denote the random variables corresponding to the kth ladder epoch. If α = 2 and M is a compound Poisson model, Dufresne and Gerber (1991) showed that K has a geometric distribution with parameter ρ, and L c and L d are independent with densities f Lc (l c ) = η 2 e l c/η 2, f Ld (l d ) = µ 1 P [U i >l d. (2) This shows that L d has the same distribution as in the unperturbed case (see, for instance, Rolski et al., 1999), and L c has the same distribution as the maximum of (ηb t (2) t : t ). Let G denote the distribution function of L d and let H α denote the distribution function of the maximum of (ηb t (α) t : t ). Then it follows that in the case α = 2 and M is a compound Poisson model [ P sup S t u = (1 ρ) ρ n (G n Hα (n+1) )(u). (3) t n= Furrer (1998) proved that (3) also holds for 1 <α<2 as long as M is a compound Poisson process. His approach did, however, not show whether or not H α and G still can be interpreted as the distribution functions of L c and L d. We will prove that in this paper. In fact, we will prove that the random variables considered at the first ladder epoch have this property whenever M is an smpp. Our approach will moreover indicate that the compound Poisson model is the only one where the ladder heights can be split in this way. The stationarity is the property that makes our approach work. If M is not a compound Poisson model, then the process will not be anymore in its stationary state at the first ladder epoch τ + and therefore L c and L d at the next ladder epoch will have a different form and be dependent on the previous ladder heights if E {}. In Section 4 we will apply the results to prove that ruin in the stationary perturbed risk model in a Markovian environment is more likely than in the perturbed classical risk model with the same claim arrival intensity λ and the same typical claim size distribution F (x) = λ 1 E[ N 1 i=1 I U i x.

3 H. Schmidli / Insurance: Mathematics and Economics 28 (21) Preliminaries Stable distributions are the only distributions that can be obtained as weak limits of normalized sums of i.i.d. random variables. The logarithm of its characteristic function ˆF(r) = E[e irx must then be of the form { σ α r α (1 iβ sign(r) tan(πα/2)) + iµr if α 1, log ˆF(r) = σ r (1 + iβ2/π sign(r) ln r ) + iµr if α = 1, where <α 2, β 1, σ and m R. This is the normal distribution (with variance 2σ 2 )ifα = 2, in which case β does not have any influence. The parameter µ is a location parameter, σ a scale parameter and β a shape parameter. Definition 1. A cadlag process (B t (α) ) is called a (standard) α-stable Lévy motion if 1. B (α) =. 2. (B t (α) ) has independent increments. 3. For s<t, B t (α) and µ =. B (α) s has a stable distribution with parameters α (, 2, σ = (t s) 1/α, β [ 1, 1 For α = 2, we obtain the Brownian motion with E[(B (2) 1 )2 = 2. From the theory of Lévy processes (see, for instance, Furrer, 1997), it follows that for β = 1 the process has no jumps downwards. We call an α-stable Lévy motion with β = 1 spectrally positive. In this case for 1 < α 2 the distribution H α of the maximum of (ηbt α t : t ) is given by (ax α 1 ) n 1 H α (x) = Γ(1 + (α 1)n), (4) where n= a = cos(απ/2) η α, see Furrer (1998). For our applications, we consider (B t (α) ) as a process on R, i.e. ( B t (α) : t ) is an independent copy of (B t (α) : t ). In recent work, Asmussen and Schmidt (1995) showed the result corresponding to Theorem 1 for unperturbed risk processes. Let P denote the Palm probability measure and let (σi,u i,m i ) denote the marked point process starting at a typical claim epoch. For the definition of Palm probabilities, see for instance Rolski et al. (1999), Baccelli and Brémaud (1987), Franken et al. (1982) or König and Schmidt (1992). Proposition 1. If M is ergodic, η = (i.e. L c = a.s.) and ρ 1, then P [M + F,L d y,z + z, τ + < = λ for every z, y,f E. z+y P [U x,m F dx The special case F = E shows that in the unperturbed case the distribution of the first ascending ladder height is independent of the law of the smpp. Because the perturbed case η> is very similar to the unperturbed case one expects such a result to hold for any ergodic smpp. The main tool used by Asmussen and Schmidt (1995) is the following lemma, which also can be found in König and Schmidt (1992) or Rolski et al. (1999). Denote by Θ t the shift operator, i.e. the shifted process M Θ t has events at the points σ i t with marks (U i,m i ).

4 16 H. Schmidli / Insurance: Mathematics and Economics 28 (21) 13 2 Lemma 1. We have [ [ E φ(m,σ k ) = λe k= φ(m Θ t,t)dt R where φ is an arbitrary non-negative measurable functional., 3. Main result We start with the following lemma. Lemma 2. Let a,b be positive numbers and m t = inf{ηb s (α) s : s [,t}. Then [ P [ m t a,ηb t (α) t b + m t dt = ap sup ηb t (α) t b. (5) t Proof. Let τ a = inf{t :ηb t (α) t = a}. Then we have to find [ τa c(a) = E I (α) ηb dt. t t b+ m t If follows readily that c(a + a ) = c(a) + c(a ). Thus, c(a) = ac(1). Consider the process (X t ) defined by X t = ηb t (α) t m t. Then c(1) is the mean time the process (X t ) spends above the level b until time τ 1. Note that E[τ 1 = 1 (see, for instance, Furrer, 1997), and X τ1 =. Moreover, because of the independent and stationary increments lim n τ n /n = E[τ 1 = 1. Because (X t ) is a stationary ergodic process we have by the strong law of large numbers 1 c(1) = lim n n τn 1 I Xs b ds = lim t t t I Xs b ds = lim t P [X t b. The latter expression is the probability that X t exceeds the level b under the stationary initial measure for (X t ). Inverting the time it follows as in Asmussen and Petersen (1989) that lim P [X t b = P [sup ηb (α) t t t b = 1 H α (b). t We can now prove the main result of this paper. Theorem 1. Assume that M is ergodic,1<α 2 and that ρ 1. Then P [τ + <,M + F,L c l c,l d l d,z + z = λ(1 H α (l c )) for every l c,l d,z and F E. In particular, P [τ + < = λµ = ρ. Moreover, l d +z P [U x,m F dx (6) P [τ + =,L c l c = (1 ρ)(1 H α (l c )). (7) Proof. Let us define the functional φ(m,s)= P [M + F,L c l c,l d l d,z + z, τ + = s M.

5 H. Schmidli / Insurance: Mathematics and Economics 28 (21) Note that [ [ P [M + F,L c l c,l d l d,z + z, τ + < = E φ(m,σ k ) = E φ(m,σ k ). (8) By Lemma 1 and (8) where P [M + F,L c l c,l d l d,z + z, τ + < = λ k=1 p(t) dt, k= p(t) = P [M F,sup{S u S t : u ( t,)} l c, = S sup{s u : u ( t,)} +l d, sup{su : u ( t,)} S z, S σ n sup{sv : v ( t,σ n)} for all N t <n<. Let now (U, M) = (U,M ). Consider the process ( S t ) defined by S t = S t. Denote by m t = inf{ S s : s [,t}. Then it is easy to see that S = U and that p(t) = P [M F, S t m t l c, m t l d,u m t z, S σn inf S u for N σ n u t t <n<. Consider the condition { S σn inf S u for N σ n u t t <n<}. (9) This condition is fulfilled in the interval [, σ 1. Thereafter, the condition is not fulfilled until the first epoch t σ 1 where S t = S ( σ 1 ) and so on. If we cut out all intervals in which (9) is not fulfilled then the pieces left follow, by the strong Markov property of the Lévy motion and the fact that (B t (α) ) only admits positive jumps, the same law as (ηb t (α) t). Thus p(t) dt = = P [M F,ηB (α) t t m t l c, m t l d U, m t zdt P [M F,l d U m t z, ηb (α) t t l c + m t dt, where m t = inf{ηb s s : s [,t}. Interchanging the order of integration the integrand will be zero until the first time where m t = z. Thus, we can cut out this piece too giving p(t) dt = P [M F,l d + z U m t,ηb (α) t t l c + m t dt. We condition on U and M and assume U>l d + z. By Lemma 2 this proves the first part of the theorem. Consider the event {τ + >n,sup t<σn S t l c } and let I n = I τ+ >n. Consider the process ( S t ) defined by S t = S σn S (σn t) which is again a perturbed marked point process except that the starting point is S = U n.we cut out the piece of the process until the first time inf{t >: S t = } where the process reaches zero. Then we leave the process as it is until the next jump caused by the unperturbed process, σ n σ k say. Then we cut out the piece until inf{t >σ n σ k : S t = S (σn σ k ) }. Then leave the process as it is until the next jump caused by the unperturbed process, and so on. Denote the process obtained in this way by ( B t (α) t), which has the same law as t). Note that the piece containing S σn is cut out if and only if I n =. Let the time length left between and (B (α) t = B (α) A n n,t ) converge to a process ( B t (α) σ n be A n. Then A n will converge to infinity as n. The process ( B n,t (α) ) defined as B n,t (α) follows the same law as (B t (α) ). Letting n the sample paths of ( B (α) B (α) (A n t) ). ( B (α) t )

6 18 H. Schmidli / Insurance: Mathematics and Economics 28 (21) 13 2 can be obtained directly from (S t ) by cutting out the pieces of (S t ) between σ n and the last time before the jump where (S t ) was at the level S σn. We had to go via the construction above because the last time before σ n where (S t ) was at a certain level is not a stopping time. It follows readily that ( B t (α) : t ) and (I n : n 1) are independent. Indeed, replacing ( B t (α) : t ) by an independent copy and going backward in the construction (not changing the pieces finally cut out) yields a perturbed risk model following the same law as (S t ).Wehave [ P [τ + >n, sup S t l c = P I n = 1, sup B t,n (α) t l c. t<σ n t A n Thus, letting n, [ P [τ + =,L c l c = P [τ + = P sup B t (α) t l c t< proving (7). Remark. Note that in the proof of (7) we did not use the stationarity of M. Thus, P [τ + =,L c l c = P [τ + = (1 H α (l c )) (1) for any risk model perturbed by a Lévy process with no downward jumps and zero mean value with lim t S t =. The above theorem shows that (2) remains valid for any smpp if α = 2. We also have generalized Proposition 1 to the perturbed case. Let us now consider the compound Poisson case. The lack of memory property of the exponential distribution assures that the smpp is in the stationary state at time τ +. Corollary 1. Let M be a compound Poisson model. Then K, (L (n) c ) and (L (n) d ) are independent, the number of ladder heights K has a geometric distribution with parameter ρ, L c has distribution function H α and L d is absolutely continuous with density µ 1 P [U i >x. This corollary leads to the Pollaczek Khinchin type formula (3) obtained by Dufresne and Gerber (1991), and Furrer (1998). Let us consider the joint distribution of (L c,l d,u +,Z +,M + ). Theorem 2. Assume that M is ergodic and that ρ 1. Then the joint (defective) distribution of (L c,l d,u +, Z +,M + ) can be described as follows: 1. P [τ + < = ρ. 2. L c is independent of the random variables (L d,u +,Z +,M + ) and of the event {τ + < }, and has the distribution function H α (i.e. L c has the same distribution as the maximum of an α-stable Lévy motion with drift 1). 3. The conditional distribution of (U +,M + ) given τ + < is obtained from the Palm distribution P of (U,M ) by the change of measure given by the likelihood ratio µ 1 U, i.e. E[g(U +,M + ) τ + < = 1 µ E [U g(u,m ) for every non-negative measurable function g. 4. The conditional distribution of (Z +,L d ) given U +, M +, τ + < is that of (U + V,U + (1 V)) where V is uniformly distributed on (, 1) and independent of U +,M +,τ + <. Proof. This follows from Theorem 1 in the same way as Theorem 2 of Asmussen and Schmidt (1995).

7 H. Schmidli / Insurance: Mathematics and Economics 28 (21) The perturbed risk process in a Markovian environment Let (M t ) be an irreducible time homogeneous Markov chain in continuous time with state space {1, 2,...,p}, intensity matrix Λ = (Λ ij ) and stationary initial distribution π = (π 1,...,π p ). Let λ i be the claim arrival intensity and F i be the claim size distribution if M t = i. This risk process was considered by Janssen (198), Janssen and Reinhard (1985), Asmussen (1989) in the unperturbed case, and by Schmidli (1995) in the perturbed case with α = 2. The intensity of the smpp is then given by λ = p π i λ i, i=1 and the claim size distribution of a typical claim becomes F (x) = 1 λ p π i λ i F i (x). i=1 Without loss of generality we can assume that λ 1 λ 2 λ p. (11) We denote by ψ(u) = P [sup{s t : t>} >u the ruin probability of the (perturbed) risk process in a Markovian environment. In a recent paper, Asmussen et al. (1995) compared the unperturbed risk process in a Markovian environment with the unperturbed standard compound Poisson risk process with claim arrival intensity λ and claim size distribution F. They showed under the assumptions (11), 1 F 1 (x) 1 F 2 (x) 1 F p (x) for all x, (12) and Λ jn Λ kn for all j,k,l with j k, and l j or l>k (13) n l n l that ψ(u) ψ (u) where ψ (u) is the ruin probability of the standard compound Poisson risk process. The main tool in their proof was Proposition 1. It is therefore not surprising that the corresponding result also holds for the perturbed risk process. For the rest of this section we denote by (St ) the (perturbed) standard compound Poisson risk process with claim arrival intensity λ and claim size distribution F, and by ψ (u) its ruin probability. Proposition 2. Assume that the conditions (11) (13) hold. Then ψ (u) ψ(u). Proof. It follows as in Lemma 2.1 of Asmussen et al. (1995) that ψ i (u) ψ j (u) for any 1 i j p where ψ i (u) = P [sup{s t : t>} >u M = i denotes the ruin probability if the process starts in state i. Let furthermore G (x) = P [L c + L d x and G i (x) = P [L c + L d x M + = i. Note that, by Theorem 2, ψ (u) = (1 ρ)(1 H α (u)) + ρ(1 G (u)) + ρ ψ (u x)dg (x), p ψ(u) = (1 ρ)(1 H α (u)) + ρ(1 G (u)) + ρ P [M τ+ = i τ + < The assertion follows similar to the proof of (2.2) in Asmussen et al. (1995). u i=1 u ψ i (u x)dg i (x).

8 2 H. Schmidli / Insurance: Mathematics and Economics 28 (21) 13 2 Denote by τ + (i) the epoch of the ith ladder height. As in Asmussen et al. (1995) the following proposition can be proved. Proposition 3. Assume that conditions (11) (13) hold. Then ρ k P [τ + (1) for any k N. <,...,τ(k) + < Acknowledgements The author thanks Hansjörg Furrer for a fruitful discussion on the topic. References Asmussen, S., Risk theory in a Markovian environment. Scandinavian Actuarial Journal, Asmussen, S., Petersen, S.S., Ruin probabilities expressed in terms of storage processes. Advances in Applied Probability 2, Asmussen, S., Schmidt, V., Ladder height distributions with marks. Stochastic Processes and its Applications 58, Asmussen, S., Frey, A., Rolski, T., Schmidt, V., Does Markov-modulation increase the risk? ASTIN Bulletin 25, Baccelli, F., Brémaud, P., Palm probabilities and stationary queues. Lecture Notes in Statistics, Vol. 41. Springer, Berlin. Dufresne, F., Gerber, H.U., Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance: Mathematics and Economics 1, Franken, P., König, D., Arndt, U., Schmidt, V., Queues and Point Processes. Wiley, New York. Furrer, H.J., Risk theory and heavy-tailed Lévy processes. Ph.D. Thesis. ETH Zürich, Zurich. Furrer, H.J., Risk processes perturbed by α-stable Lévy motion. Scandinavian Actuarial Journal, Furrer, H.J., Schmidli, H., Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance: Mathematics and Economics 15, Gerber, H.U., 197. An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuar Tidskrift 53, Janssen, J., 198. Some transient results on the M/SM/1 special semi-markov model in risk and queueing theories. ASTIN Bulletin 11, Janssen, J., Reinhard, J.M., Probabilités de ruine pour une classe de modèles de risque semi-markoviens. ASTIN Bulletin 15, König, D., Schmidt, V., Random Point Processes. Teubner, Stuttgart (in German). Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.L., Stochastic Processes for Insurance and Finance. Wiley, Chichester. Schlegel, S., Ruin probabilities in perturbed risk models. Insurance: Mathematics and Economics 22, Schmidli, H., Cramér Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insurance: Mathematics and Economics 16, Veraverbeke, N., Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance: Mathematics and Economics 13,