MUNEYA MATSUI AND THOMAS MIKOSCH

January 20, 2010 PRDICTION IN A POISSON CLUSTR MODL MUNYA MATSUI AND THOMAS MIKOSCH Abstract. We consider a Poisson cluster model which is motivated by insurance applications. At each claim arrival time, modeled by a point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations on the process over a finite interval we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure. 1. Introduction In this paper we consider the model 1.1) Mt) I {Tk 1}L k t T k ), t 1, where 0 < T 1 < T 2 < are the points of a homogeneous Poisson process with intensity λ > 0 and L k ) is a sequence of iid stochastic processes independent of T k ) and such that L k t) 0 a.s. t 0. Writing N for the counting process generated by the points T k, k 1, 2,..., M takes on the form 1.2) Mt) N1) L k t T k ), t 1. Processes of this type are related to Poisson shot noise which has found a multitude of applications in rather different areas. For example, in an insurance context, T k may describe the arrival of a claim in an insurance portfolio and L k t T k )) t Tk is the corresponding payment process from the insurer to the insured starting at time T k. Alternatively, L k t T k )) t Tk can be the counting process for these payments. The main focus of this paper is on applications in an insurance context. arly on, shot noise processes have been used for modeling purposes in many fields of applied probability such as bunching in traffic Bartlett 1), computer failure times Lewis 17) and earthquake aftershocks Vere Jones 28). More recently, shot noise processes have been used for modeling large computer networks such as the Internet; see for example Konstantopoulos and Lin 12, Kurtz 13 for some early work. In the context of the workload of large computer networks, shot noise processes arise as aggregated versions of the ON/OFF or infinite source Poisson models, also known as M/G/ model; see for example Levy and Taqqu 16, Pipiras and Taqqu 24, Mikosch et al. 22 and for further extensions Faÿ et al. 2, Mikosch and Samorodnitsky 23. Other applications include finance Samorodnitsky 25, Klüppelberg and Kühn 8) and physics Giraitis et al. 3). Most papers mentioned above aim at an asymptotic theory for the shot noise process, deriving Gaussian or Lévy process limits, or at the asymptotics of the extremes of such processes; see also Heinrich and Schmidt 4, Hsing and Teugels 5, Klüppelberg and Mikosch 9, 10, Klüppelberg et al. 11, Lane 14, 15, McCormick 20, Stegeman 27. 1991 Mathematics Subject Classification. Primary 60K30, Secondary 60G25 60G55. Key words and phrases. Poisson cluster model, prediction, conditional expectation, claims reserving, insurance, Lévy process, shot noise. Thomas Mikosch s research is partly supported by the Danish Research Council FNU) Grants 272-06-0442 and 09-072331. Muneya Matsui s research is supported by a JSPS Research Fellowship for Young Scientists. 1

2 M. MATSUI AND T. MIKOSCH The focus of this paper is not on asymptotic properties of the process M defined in 1.1) but on precise results about the prediction of the increments Mt, t + s Mt + s) Mt), t 1, s > 0, i.e., we will calculate Mt, t + s F t for some suitable σ-fields F t, where we do not necessarily assume that Mt, t + s has finite variance. Results of this kind are surprisingly explicit due to the Poisson structure underlying the process M. The particular structure of the process M and the prediction problem are motivated by reserving problems in insurance. Here the points T k 1 describe the arrivals of claims in a portfolio in a given period, 1 year say, and Mt) is the number or amount of payments for the claims arriving in 0, 1 and being paid off in 0, t, t 1. Correspondingly, Mt, t + s is the number or amount of payments executed in the interval t, t + s, s > 0. Problems of this kind are related to claims reserving. Traditionally, claims reserving has dealt with a statistical model assuming conditions of the type Mt + s) Mt) f t,s Mt) for suitable constants f t,s and then one estimates these constants based on several years of data available socalled chain ladder). We refer to Chapter 11 in Mikosch 21 for an introduction to the topic. Jessen et al. 6 consider a stochastic process based model with Poisson components which allows one to derive explicit expressions for the prediction of payment numbers and amounts and the corresponding prediction errors. This model requires that time series of annual observations on payment numbers and amounts are available; the structure of this model is significantly simpler than 1.1). In the present paper we follow a different path which was already mentioned in Section 11.3.3 of 21. There the model 1.1) was considered for iid homogeneous Poisson processes L k, k 1, 2,..., and explicit expressions for the prediction Mt, t + s Mt) were obtained. First, in Section 2, we show that this approach can be extended to general Lévy processes L k. The homogeneous Poisson case is again a benchmark Section 2.3). In this case we can also give a recursive algorithm for determining the prediction. In Section 3 we consider modifications of the model 1.2). Instead of a Lévy process L k we allow for Lévy processes which are truncated at a random level or which take into account a delay in reporting of a claim. In Section 3 we also consider different σ-fields F t and calculate the corresponding predictions Mt, t + s F t and their conditional or unconditional prediction errors. Depending on F t the prediction of Mt, t + s and the prediction error often get a structure which is simpler than for Mt, t + s Mt). 2. Prediction in a Poisson cluster process with Lévy clusters In this section we consider the model 1.1) and we assume that an activity process L k starts at the Poisson point T k 0, 1. In this context, L k ) is a sequence of iid Lévy processes with generic element L; we refer to Sato 26 for the definition and properties of Lévy processes. Then Mt, t + s, t 1, s > 0, can be interpreted as the measurement of activities initiated in 0, 1 which are still alive in t, t + s. In the insurance context mentioned in the Introduction, Mt, t + s has the interpretation as the number or amount of payments for claims that occurred in one year say. Other interpretations are possible as well. For example, Mt, t + s may describe the workload to be managed by a large computer network for sources that started an activity such as sending packets to other sources) in the interval 0, 1. The Lévy process condition on L k is assumed for convenience; in this case we will get explicit expressions for the prediction of Mt, t+s given Mt), t 1; see Section 2.2. We start by an analysis of the moment structure of M. 2.1. The first and second moments of M. The following result is elementary. Lemma 2.1. Assume the model 1.2) with iid Lévy processes L k, k 1, 2,..., and a homogeneous Poisson process N with intensity λ > 0.

1) Assume that µ L1) exists and is finite. Then PRDICTION IN A POISSON CLUSTR MODL 3 Mt) λ µ t 0.5), t 1. 2) Assume that σ 2 VarL1)) is finite. Then, for 1 t 1 t 2, In particular, 1 CovMt 1 ), Mt 2 )) λ σ 2 t 1 0.5) + λ µ 2 t 1 s)t 2 s) ds. VarMt)) λ σ 2 t 0.5) + λ µ 2 t 2 t + 1/3), t 1. We see that the process M is over-dispersed in the sense that lim t VarMt))/Mt). 2.2. Prediction of future increments. In this section we derive explicit expressions for the quantities Mt, t + s Mt) for t 1, s > 0. We assume the general conditions of Lemma 2.1 and also that µ L1) is finite. We start with a simple observation. By the definition 1.2) of Mt), t 1, the σ-field generated by Mt) is contained in the σ-field generated by L i t T i )) and T i ). Therefore, writing L i t, t+s L i t + s) L i t) for the increments of L i, for t 1 and s > 0, 2.1) Mt, t + s Mt) µ s I {Tk 1}L k t T k, t + s T k L i t T i )), T i ) Mt) I {Tk 1} L k t T k, t + s T k L k t T k ), T k Mt) I {Tk 1} Mt) µ s N1) Mt). In the first step we used dominated convergence and in the last one the stationary independent increments of the Lévy process L k. Thus we are left to calculate N1) Mt) for t 1. We mention at this point that, in an insurance context, the number N1) of claims arriving in the interval 0, 1 is in general not observable at time t 1. It usually takes a much longer time period than just one year before the claim number for the first year 0, 1 is known so-called Incurred But Not Reported or IBNR effect). Therefore the quantity N1) Mt), t 1, has the intuitive meaning that one gathers information about the payments in the period 0, t, represented by the quantity Mt), in order to gather information about the generally unknown) claim number N1). It is our aim to express the prediction of Mt, t+s given Mt) as explicitly as possible. Motivated by the previous calculations, we will determine M A t, t + s Mt, t + s Mt) A, t 1, s > 0, for any Borel set A. Later, in Section 2.3, we will specify the set A. The quantity M A t, t + s is the conditional first moment of Mt, t + s. In order to get an idea of the conditional prediction error we are also interested in the conditional second moment of Mt, t + s. Both moments can easily be derived from the characteristic function of Mt, t + s given Mt). Lemma 2.2. Assume the model 1.2) with iid Lévy processes L k, k 1, 2,..., and a homogeneous Poisson process N with intensity λ > 0. For any Borel set A, the characteristic function of Mt, t + s given {Mt) A} has the form f A x) e i x Mt,t+s Mt) A 0

4 M. MATSUI AND T. MIKOSCH e i x Ls) ) N1) P LR N1) t)) A N1)) P LR N1) t)) A) Here we assume that the denominator does not vanish and r 2.2) R r t) t U i ), t 1, r 0, 1,..., i1 for an iid U0, 1) sequence U i ) such that L, N and U i ) are independent., x R, t 1, s > 0. Remark 2.3. The condition P LR N1) t)) A) > 0 is needed for a proper definition of the conditional characteristic function f A x). From the proof below it follows that this condition is equivalent to P Mt) A) > 0. Proof. The same argument leading to 2.1) yields for x R, t 1, s > 0, e i x Mt,t+s Mt) e i x Mt,t+s L k t T k )), T k ) Mt) 2.3) e i x I {T k 1} L k t T k,t+s T k Mt) L k t T k )), T k ) ) I {Tk >1} + I {Tk 1} e i x Ls) Mt) T k ) ) N1) e i x Ls) Mt). Therefore we will calculate the following quantities for any Borel set A assuming that the denominator does not vanish) P N1) r, ) N1) L kt T k ) A P N1) r Mt) A) N1) ) P L kt T k ) A N1) ) ) P N1) r, L t T k) A N1) ) ), r 0, 1,.... P L t T k) A In the last step we used the independent stationary increments of the iid Lévy processes L i, i 1, 2,..., conditional on T k ). Conditioning on N1) and using the order statistics property of a homogeneous Poisson process, we obtain P N1) ) t T k ) N1) r P R r t) ), r 0, 1,..., where R r is defined in 2.2). We conclude that 2.4) P N1) r Mt) A) P N1) r) P LR rt)) A) P LR N1) t)) A). Now plug 2.4) in 2.3) to obtain the desired expression for f A x).

PRDICTION IN A POISSON CLUSTR MODL 5 Since we know the characteristic function f A x), x R, of Mt, t + s given {Mt) A} we can derive the moments of the prediction M A t, t + s by differentiating f A x) at x 0 sufficiently often. The following result summarizes the analysis of the first and second conditional moments. Theorem 2.4. Assume the model 1.2) with iid Lévy processes L k, k 1, 2,..., and a homogeneous Poisson process N with intensity λ > 0. 1) Assume that µ L1) exists and is finite. Then the prediction M A t, t + s of Mt, t + s given {Mt) A} has the following form for any Borel set A, M A t, t + s µ s N1) P LR N1) t)) A N1)) 2.5), t 1, s > 0. P LR N1) t)) A) 2) Assume that σ 2 VarL1)) is finite. Then the conditional variance of Mt, t + s given {Mt) A} has the following form for any Borel set A, 2.6) VarMt, t + s Mt) A) σ2 µ M A t, t + s + µ s) 2 t 1, s > 0. N1)) 2 P LR N1) t)) A N1)) P LR N1) t)) A) Here we have assumed that the probability P LR N1) t)) A) does not vanish. M A t, t + s) 2, Remark 2.5. The variance of Mt, t + s conditional on Mt) gives one a certain measure for the uncertainty of the prediction M A t, t+s. In general, this conditional variance is difficult to obtain. However, if L is a homogeneous Poisson process we can derive recursive algorithms for determining this quantity; see Section 2.3. It would be desirable to get an expression for the unconditional mean square error Mt, t + s Mt, t + s Mt)) 2 VarMt, t + s Mt)) Mt, t + s) 2 Mt, t + s Mt)) 2. The second moment Mt, t + s) 2 provides an upper bound for the mean square error. It can be derived from Lemma 2.1. The quantity Mt, t + s Mt)) 2 seems not tractable. 2.3. Poisson clusters. In this section we assume that L is a homogeneous Poisson process with intensity γ > 0. In this case we can give explicit expressions for the prediction M m t, t + s Mt, t + s Mt) m, t 1, s > 0, m 0, 1,..., and the conditional prediction error. In what follows, it will be convenient to use the Laplace- Stieltjes transform of any non-negative random variable Y : φ Y z) e z Y, z 0, and its mth derivatives φ m) Y z) 1)m Y m e z Y, m 0, 1, 2,.... Theorem 2.6. Assume that L is a homogeneous Poisson process with intensity γ > 0. Then the prediction of Mt, t + s given {Mt) m} has the form 2.7) M m t, t + s λ γ s φ m) R N1)+1 t) γ) and the conditional mean square error is given by VarMt, t + s Mt) m) φ m), t 1, s > 0, m 0, 1,..., R N1) t)γ)

6 M. MATSUI AND T. MIKOSCH where R r is defined in 2.2). λ γ s) 2 φm) R N1)+2 γ) φ m) R N1) γ) + 1 + γ s) M m t, t + s M m t, t + s) 2, t 1, s > 0, m 0, 1,..., Proof. According to 2.5) and 2.6), we need to evaluate N1)) i P LR N1) t)) m N1)), i 0, 1, 2. We have N1) P LR N1) t)) m N1)) r P N1) r) P LR r t)) m) In a similar way, one calculates λ r1 λ P LR N1) t)) m) γ)m m! r0 λ λr e r! γ R r+1 t)) m e γ R r+1t) m! γ RN1)+1 t)) m e γ R N1)+1t) m! λ γ)m φ m) m! R N1)+1 t) γ). φ m) R N1) t) γ), N1)) 2 P LR N1) t)) m N1)) λ γ)m φ m) m! R N1)+1 t) γ) + λ2 γ) m φ m) m! R N1)+2 t) γ). This concludes the proof. In view of Theorem 2.6 it is crucial to be able to evaluate the derivatives of φ RN1) t). Although we have the representations φ RN1) γ) e λ1 φ t U γ)) and φ t U γ) γ 1 e γt e γ 1), their derivatives are complicated and not necessarily useful. Fortunately, the following proposition yields a recursive way of determining these derivatives. Proposition 2.7. Let l 1, 2,... and φ l) t U γ), φl) R N1)+r t)γ), r 0, 1, 2 be the lth derivatives of φ t U, φ RN1)+r t), r 0, 1, 2. Then the following recursive relations are valid: φ l) t U γ) 1)l γ l 1 Γl + 1, γ t 1)) Γl + 1, γ t) l ) φ l) l 1 R N1) t) γ) λ φ k) k 1 t U γ) φl k) R N1) t) γ) l φ l) l R N1)+1 t) k) γ) λ φ k) t U γ) φl k) R N1) t) γ) k0 l φ l) l R N1)+2 t) k) γ) λ φ k) t U γ) φl k) R N1)+1 t) γ), k0

PRDICTION IN A POISSON CLUSTR MODL 7 where Γα, x) x e y y α 1 dy, x > 0, is the incomplete Gamma function. Proof. Observe that for t 1, 2.8) φ t U γ) t e γu e γ 1)du. Then Leibniz s rule yields for u > 0 and l 0, 1,..., l e γu e γ 1) l) l e k) γu ) k) e γ 1) l k) t k0 e γu 1) 1 u) l u) l e γu. Now, interchanging the integral and the derivative in 2.8), we have for t 1, φ l) t U γ) e γ u 1) 1 u) l u) l e γu du 1) l t t 1 e γu u l du 1) l γ l 1 Γl + 1, γ t 1)) Γl + 1, γ t). This is the desired formula for the derivatives of φ t U. By definition of φ RN1) t) observe that φ R N1) t) λφ t U φ RN1) t). Another application of Leibniz s rule yields the desired formula for the derivatives of φ RN1) t). We also observe that φ RN1) t)+1 φ RN1) t)φ t U and φ RN1) t)+2 φ RN1) t)+1φ t U. Applications of the Leibniz rule yield the desired expressions for the derivatives. Remark 2.8. Assume that the homogeneous Poisson process L k has the arrivals 0 < Γ k1 < Γ k2 <, k 1, 2,.... Let X ki ) k,i1,2,... be an iid sequence of random variables such that ν X 11 exists and is finite. Moreover, we assume that T k ), X ki ), Γ ki ) k,i1,2,... are mutually independent. Then it is possible to define the reward process St) N1) i1 X ki I {Tk +Γ ki t}, t 1. The reward St) can be interpreted as the total amount of payments executed at the times T k +Γ ki t, t 1, i 1, 2,..., for claims arriving in an insurance portfolio at times T k 1. A conditioning argument shows that St, t + s Mt) ν Mt, t + s Mt), t 1, s > 0. The iid-ness condition of the X ki s can be further relaxed. For example, if one looks at the conditional expectations Sl, l + 1 Ml), l 1, 2,..., then one may allow the distribution of the iid payments X ki executed in the interval l, l + 1 to depend on l. The conditional variance VarSt, t + s Mt)) can be calculated as well. Some of these calculations are provided in Section 11.3.3 of Mikosch 21. We omit details.

8 M. MATSUI AND T. MIKOSCH 3. Prediction with different information sets and non-lévy clusters 3.1. Prediction with compound Poisson clusters. In this section we assume the model 1.2) with a sequence L k ) of iid compound Poisson processes. We assume the representation 3.1) L k t) X ki I {Γki t}, t 1, i1 where X ki ) is a double array of iid random variables, independent of the double array Γ ki ) and the Poisson points T k ). The points 0 < Γ k1 < Γ k2 < constitute the homogeneous Poisson process N k with intensity γ > 0 underlying the compound Poisson process L k. Throughout we assume that µ L1) X 11 γ exists and is finite. In Section 2.3 we mentioned that, in an insurance context, it is natural to assume that the number of claims N1) is unobservable at time t 1. It is common that claims get reported long after they were incurred. In this section, we replace the condition Mt) in the prediction of Mt, t + s for t 1 and s > 0 by the number of claims that were incurred in 0, 1 and reported by time t 1. We say that the kth claim is reported if T k + Γ k1 t, i.e., if the first payment has been executed by time t. We write for the corresponding counting process of reported claims 3.2) N 0 t) N1) I {Tk +Γ k1 t} N1) We focus on the calculation of the conditional expectation I {Nk t T k ) 1}, t 1. M l t, t + s Mt, t + s N 0 t) l, l 0, 1,..., and the corresponding prediction error. In view of 3.2) the σ-field generated by N 0 t) is contained in the σ-field F t generated by Γ ki ) k,i 1:Tk +Γ ki t and T k ). Therefore for t 1 and s > 0, M l t, t + s I {Tk 1} L k t T k, t + s T k F t N0 t) l µ s N1) N 0 t) l. The latter conditional expectation will be evaluated below. Since we are also interested in other conditional moments of Mt, t+s we calculate the conditional characteristic function e i xmt,t+s N 0 t) l, x R. Lemma 3.1. The conditional characteristic function of Mt, t + s given {N 0 t) l} is given by 3.3) where 3.4) e i xmt,t+s N 0 t) l e i x Ls) ) l e λ 0t) 1 e i x Ls) ), x R, λ 0 t) λ e γ t e γ 1). γ Proof. We proceed similarly to the proof of Lemma 2.2: e i xmt,t+s N 0 t) e i xmt,t+s F t N 0 t) 3.5) e i x Ls)) N1) N0 t) e i x Ls)) N 0 t) e i x Ls)) N1) N 0 t) N0 t).

PRDICTION IN A POISSON CLUSTR MODL 9 Let Q be a Poisson random measure on the state space 0, 1 0, ) with mean measure ν λ Leb F, where F denotes the distribution of Γ k1. Then N1) and N 0 t) have the Poisson integral representations N1) Qds, dy) 1 s0 t s y0 N 0 t) + N1) N 0 t). 1 Qds, dy) + Qds, dy) s0 yt s Due to the splitting property of the Poisson process and since the integrals above are defined on disjoint subsets of the state space, the random variables N 0 t) and N1) N 0 t) are independent and Poisson distributed with parameters N1) N 0 t) λ 1 0 t s Therefore we conclude from 3.5) that e i xmt,t+s N 0 t) F dy) ds λ e γ t e γ 1) λ 0 t) and N 0 t) λ λ 0 t). γ e i x Ls)) N 0 t) e i x Ls)) N1) N 0 t). The latter relation yields the desired conditional characteristic function. The following result can now be obtained by differentiating the characteristic function 3.3) sufficiently often and then considering the derivatives at zero. Theorem 3.2. Assume that L is a compound Poisson process with underlying Poisson intensity γ. 1) If µ X 11 γ exists and is finite then the prediction of Mt, t + s given {N 0 t) l} has the form 3.6) M l t, t + s µ s λ 0 t) + l, t 1, s > 0, l 0, 1,..., where λ 0 t) is given in 3.4). 2) If in addition σ 2 VarL1)) X 2 11 γ < then VarMt, t + s N 0 t) l) 3.7) σ 2 s l + λ 0 t)) + µ s) 2 λ 0 t), t 1, s > 0, l 0, 1,.... Remark 3.3. Notice that the unconditional prediction error of Mt, t + s given N 0 t) is given by VarMt, t + s N 0 t)) σ 2 s + µ s) 2 ) λ 0 t) + σ 2 s N 0 t) σ 2 s λ + µ s) 2 λ 0 t). Remark 3.4. In a next step we include more information in the condition of the expected value of Mt, t + s, i.e., we focus on Mt, t + s Mt), N 0 t). Both processes Mt) and N 0 t) are assumed to be observable at time t. Then the σ-field G t generated by T k ), Γ ki, X ki )) k,i 1,Tk +Γ ki t is larger than the σ-field generated by Mt) and N 0 t). Therefore Mt, t + s Mt), N 0 t) µ s N1) G t Mt), N 0 t) 3.8) µ s N1) Mt), N 0 t).

10 M. MATSUI AND T. MIKOSCH Qds, dγi ), dx i )), Consider the Poisson random measure Q with points T k, Γ ki ) i1,2,..., X ki ) i1,2,... ) on the state space 0, ) 0, ) R. Then N1), N 0 t) and Mt) have the representation as Poisson integrals with respect to Q: N1) 0,1 0, ) R N 0 t) I {s+γ1 t}s, γ i ), x i ))) Qds, dγ i ), dx i )), 0,1 0, ) R Mt) 0,1 0, ) R i1 x i I {s+γ1 + +γ i ) t}s, γ i ), x i ))) Qds, dγ i ), dx i )). Notice that N1) N 0 t) I {s+γ1 >t}s, γ i ), x i ))) Qds, dγ i ), dx i )). 0,1 0, ) R The support of the integrand in N1) N 0 t) is disjoint from the supports of the integrands in N 0 t) and Mt) and therefore N1) N 0 t) is independent of N 0 t) and Mt). We conclude from 3.8) that Mt, t + s Mt), N 0 t) µ s N1) N 0 t) Mt), N 0 t) + N 0 t) Mt), N 0 t) 3.9) µ s N1) N 0 t) + N 0 t) µ s λ 0 t) + N 0 t). Surprisingly, this is the same formula 3.6) as for Mt, t + s N 0 t). Hence taking into account information additional to N 0 t) does not change the prediction of Mt, t + s. A similar calculation shows that the prediction error remains the same. Also notice that we may conclude from 3.9) that Mt, t + s Mt) µ s λ 0 t) + N 0 t) Mt). The latter relation sheds some light on the prediction formula 2.5). 3.2. Prediction with delay in reporting. In this section we assume that the cluster process starting at T k has the form L k t T k ) R k t T k D k ), where R k is a Lévy process on 0, ) with the convention that R k s) 0 a.s. for s 0 and D k is a positive random variable with distribution F D. We write R R 0 for a generic element of the sequence R k ). We also assume that the iid sequence D k ), the sequence of the claim arrivals T k ) and the iid sequence R k ) k0,1,... of Lévy processes are independent. We interpret D k as the time that elapses between the arrival time T k of the kth claim and its reporting time T k + D k. Thus the inhomogeneous Poisson process Ñt) {k 1 : T k + D k t, T k 0, 1}, t 1, is observable at time t 1, whereas the claim number N1) is not necessarily observable. In what follows, we will give expressions for the prediction of Mt, t + s given Ñt): M l t, t + s Mt, t + s Ñt) l, l 0, 1,..., t 1, s > 0. The key to the derivation of the prediction and the prediction error is again an expression for the characteristic function of Mt, t + s given Ñt). Lemma 3.5. The conditional characteristic function of Mt, t + s given {Ñt) l} is given by 3.10) e i x Mt,t+s Ñt) l ) { l 1 t+s v e i x Rs) exp λ 1 e i x Rt+s v r)) } dv F D dr), v0 rt v

PRDICTION IN A POISSON CLUSTR MODL 11 l 0, 1,..., t 1, s > 0. Proof. We start by calculating the characteristic function of Mt, t + s conditional on T j ) and D j ): e i x Mt,t+s T j ), D j ) e i x R kt T k D k,t+s T k D k I {Tk 1} T j ), D j ) 3.11) N1) e i x Rt T k D k,t+s T k D k T k, D k ) Nt) e i x Rs) e k:t k 1,T k +D k >t e i x Rt+s T k D k ) T k, D k. In the last step we used that Rt T k D k ) 0 a.s. for k such that T k + D k t a.s. Write Q for the Poisson process of the points T k, D k ) on the state space 0, 1 0, ) with mean measure λleb F D. Then the Poisson process Ñ has representation 3.12) Ñt) 1 v0 t v r0 Qdv, dr), t 1. Since the σ-field generated by Ñt) is contained in the σ-field generated by T k) and D k ) we have in view of 3.11), e i x Mt,t+s Ñt) ) Nt) e i x Rs) e k:t k 1,T k +D k >t ) Nt) e i x Rs) e { exp k:t<t k +D k t+s e i x Rt+s T k D k ) T k, D k Ñt) } log e i x Rt+s T k D k ) T k, D k Ñt) ) Nt) e i x Rs) e { 1 t+s v exp log e i x Rt+s v r) } Qdv, dr) Ñt). v0 rt v We observe that the Poisson integrals in the last expression and in 3.12) have disjoint supports, hence they are independent and e i x Mt,t+s Ñt) ) Nt) e i x Rs) e { 1 exp v0 Direct calculation for t 1, s > 0 yields 3.10). v0 rt v t+s v rt v Remark 3.6. Notice that for t 1, s > 0, 1 t+s v 1 e i x Rt+s v r)) dv F D dr) where G F U F D. Therefore { 1 exp λ v0 t+s v rt v log e i x Rt+s v r) } Qdv, dr). t+s 1 e i x Rt+s v r)) } dv F D dr) t 1 e i x Rt+s z) ) Gdz)

12 M. MATSUI AND T. MIKOSCH { Ñt, t+s } exp t + s 1 e i x Rt+s z) ) Gdz)/Ñt, t + s) t e i x Rt+s Z)) Nt,t+s e, where Z is independent of R and has distribution Gdz)/Ñt, t + s on t, t + s. This expression is an alternative formula for the second term in 3.10). Differentiation of the conditional characteristic function at x 0 sufficiently often yields the following result. Theorem 3.7. Assume the model 1.2) with the delayed Lévy processes L k t) R k t D k ), k 1, 2,..., as cluster processes, where by convention R k s) 0 a.s. for s 0, D k ) constitutes an iid sequence of positive random variables with distribution F D and the sequences T k ), D k ) and R k ) are independent. 1) Assume that µ R1) exists and is finite. Then the prediction M l t, t + s of Mt, t + s given {Ñt) l} has the following form for l 0, 1,... 3.13) where J i J i t, s) λ M l t, t + s µ s l + µ J 1, t 1, s > 0, 1 v0 t+s v rt v 2) If in addition σ 2 VarR1)) < then 3.14) t + s r v) i F D dr) dv, i 1, 2. VarMt, t + s Ñt) l) l s σ2 + σ 2 J 1 + µ 2 J 2. Remark 3.8. The prediction formulas 3.13) and 3.6) are rather similar. Both are linear functions of Ñt) l or Nt) l, respectively. This is agreement with the assumptions of the chain ladder which is a standard technique for claims reserving; see Mack 18, 19. A particularly interesting case occurs when the delay in reporting variable D is U0, a) distributed for some a > 0. Then M l t, t + s µ s l + µ λ a 1 1 0 t+s v) a rt v) a t + s r v) dr dv. A comparison of the conditional prediction errors 3.7) and 3.14) shows that both are linear function of l as well. Since Ñt) is Poisson distributed with parameter Ñt) λ 1 v0 t v r0 dv F D dr) λ straightforward calculation yields the unconditional prediction error VarMt, t + s Ñt)) λ σ2 s t t 1 t t 1 F D v) dv, t 1, F D v) dv + σ 2 J 1 + µ 2 J 2. 3.3. Truncated compound Poisson clusters. In this section we consider another modification of the Poisson cluster process. We again consider the model 1.2) but the cluster processes L k are not Lévy processes. We assume that the iid processes L i, i 1, 2,..., have the following structure 3.15) L i t) K i R i t), t 1, where K i is a non-negative random variable and R i is a non-negative compound Poisson process with representation given on the right-hand side of 3.1). Moreover, we assume that K i ) i0,1,... is an iid sequence independent of the iid sequence R i ) i0,1,.... We also write N i for the homogeneous Poisson process with intensity γ > 0 which underlies the compound Poisson process R i. If N i R i

PRDICTION IN A POISSON CLUSTR MODL 13 and K i is a non-negative integer-valued random variable, K i has the interpretation as the total number of payments for the ith claim. In particular, if K i k 0 is a constant integer there are exactly k 0 payments for each claim. As explained before, in practice one is often not informed at time t 1 whether a claim has happened, i.e., the arrival times T k are often unknown until some future instant of time. Therefore we assumed in Section 3.1 that we take into account only those claims for which T k + Γ k1 t, and in Section 3.2 we take into account only those claims for which T k + D k t. Here we will consider prediction of Mt, t + s given N 0 t) defined in 3.2) and R k t T k )) k:tk 1,T k +Γ k1 t. We write H t for the σ-field generated by these quantities which are observable at time t. Here and in what follows we use the notation of Section 3.1. Theorem 3.9. Assume that the iid sequence L i ) has the structure described in 3.15). 1) If K 1 < then the prediction of Mt, t + s given H t, t 1, s > 0, has the form Mt, t + s H t ) λ 0 t) K 0 R 0 s) + K 0 R 0 s) + R k t T k )) K 0 R k t T k ) R k t T k ), k:t k 1,T k +Γ k1 t where λ 0 t) is defined in 3.4). Here R k ) k0,1,..., T k ) and K k ) k0,1,... are independent. 2) If in addition VarK 1 ) < then for t 1, s > 0, VarMt, t + s H t ) λ 0 t) K 0 R 0 s)) 2 + Var K 0 R 0 s) + R k t T k )) K 0 R k t T k ) R k t T k ) ). k:t k 1,T k +Γ k1 t This result again follows by differentiation at zero of the characteristic function of Mt, t + s given H t. Lemma 3.10. The conditional characteristic function of Mt, t + s given H t has the form e i x Mt,t+s H t e i x K 0 R 0 s)+r k t T k )) K 0 R k t T k ) R k t T k ) k:t k 1,T k +Γ k1 t 3.16) e λ 0t) 1 e i x K 0 R 0 s)). Proof. We start by observing that Mt, t + s Kk R k t + s T k ) K k R k t T k ) k:t k 1,T k +Γ k1 t + K k R k t + s T k ) k:t k 1,T k +Γ k1 >t Kk R k t + s T k ) K k R k t T k ) k:t k 1,T k +Γ k1 t + K k R k t T k, t + s T k. k:t k 1,T k +Γ k1 >t The σ-field H t is contained in the σ-field G t generated by T k ), Γ ki, X ki )) k,i 1,Tk +Γ ki t. Therefore e i x Mt,t+s G t

14 M. MATSUI AND T. MIKOSCH k:t k 1,T k +Γ k1 t k:t k 1,T k +Γ k1 >t k:t k 1,T k +Γ k1 t e i x K 0 R 0 s)+r k t T k )) K 0 R k t T k ) ) e i x K N1) N 0 R 0 s) 0 t). e i x K 0 R k t T k,t+s T k T k, R k t T k ) R k t T k ) e i x K 0 R 0 s)+r k t T k )) K 0 R k t T k ) R k t T k ) The first factor is measurable with respect to H t, and H t is independent of N1) N 0 t). Therefore we have e i x Mt,t+s H t e i x K 0 R 0 s)+r k t T k )) K 0 R k t T k ) R k t T k ) k:t k 1,T k +Γ k1 t e ) i x K N1) N 0 R 0 s) 0 t). The latter relation implies the desired result 3.16). Remark 3.11. The unconditional prediction error is given by VarMt, t + s H t ) λ 0 t) K 0 R 0 s)) 2 N1) + I {Nk t T k ) 1}Var K 0 R 0 s) + R k t T k )) K 0 R k t T k ) R k t T k ) ). Using the order statistics property of the homogeneous Poisson process N, we obtain VarMt, t + s H t ) λ 0 t) K 0 R 0 s)) 2 λ 1 0 I {N1 t y) 1}Var K 0 R 0 s) + R 1 t y)) K 0 R 1 t y) R 1 t y) ) dy. Acknowledgment. A major part of this paper was written when Muneya Matsui visited the Department of Mathematics of the University of Copenhagen. He would like to thank for hospitality and for financial support from the FNU Grant 272-06-0442. References 1 Bartlett, M.S. 1963) The spectral analysis of point processes. J. Royal Statist. Soc. Ser. B 25, 264 296. 2 Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G. 2006) Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems Theory Appl. 54, 121 140. 3 Giraitis, L., Molchanov, S.A. and Surgailis, D. 1993) Long memory shot noises and limit theorems with application to Burgers s equation. In: New Directions in Time Series Analysis, Part II, pp. 153 176. IMA Vol. Math. Appl. 46, Springer, New York. 4 Heinrich, L. and Schmidt, V. 1985) Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Probab. 17, 709 730. 5 Hsing, T. and Teugels, J.L. 1989) xtremal properties of shot noise processes. Adv. Appl. Probab. 21, 513 525. 6 Jessen, A.H., Mikosch, T. and Samorodnitsky, G. 2009) Prediction of outstanding payments in a Poisson cluster model. Available under www.math.ku.dk/ mikosch/preprint/prediction/prediction260309.pdf 7 Kallenberg, O. 1983) Random Measures. Third edition. Akademie-Verlag, Berlin.

PRDICTION IN A POISSON CLUSTR MODL 15 8 Klüppelberg, C. and Kühn, C. 2004) Fractional Brownian motion as a weak limit of Poisson shot noise processes with applications to finance. Stoch. Process. Appl. 113, 333 351. 9 Klüppelberg, C. and Mikosch, T. 1995) xplosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125 147. 10 Klüppelberg, C. and Mikosch, T. 1995) Delay in claim settlement and ruin probability approximations. Scand. Actuar. J. 154 168. 11 Klüppelberg, C., Mikosch, T. and Schärf, A. 2003) Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9, 467 496. 12 Konstantopoulos, T. and Lin, Si-Jian 1998) Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28, 215 243. 13 Kurtz, T.G. 1997) Limit theorems for workload input models. In: Kelly, F.P., Zachary, S. and Ziedins, I. ds.) Stochastic Networks. Theory and Applications, pp. 119 139. Oxford University Press, Oxford, UK. 14 Lane, J.A. 1984) The central limit theorem for the Poisson shot-noise process. J. Appl. Probab. 21, 287 301. 15 Lane, J.A. The Berry sseen bound for the Poisson shot-noise. Adv. Appl. Probab. 19, 512 514. 16 Levy, J.B. and Taqqu, M.S. 2000) Renewal reward processes with heavy-tailed interrenewal times and heavytailed rewards. Bernoulli 6, 23 44. 17 Lewis, P.A. 1964) A branching Poisson process model for the analysis of computer failure patterns. J. Royal Statist. Soc. Ser. B 26, 398 456. 18 Mack, T. 1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23, 213 225. 19 Mack, T. 1994) Which stochastic model is underlying the chain ladder method? Insurance: Mathematics & conomics 15, 133 138. 20 McCormick, W.P. 1997) xtremes for shot noise processes with heavy tailed amplitudes. J. Appl. Probab. 34, 643 656. 21 Mikosch, T. 2009) Non-Life Insurance Mathematics. An Introduction with the Poisson Process. 2nd dition. Springer, Heidelberg. 22 Mikosch, T., Resnick, S.I., Rootzén, H. and Stegeman, A. 2002) Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12, 23 68. 23 Mikosch, T. and Samorodnitsky, G. 2007) Scaling limits for cumulative input processes. Math. Oper. Res. 32, 890 919. 24 Pipiras, V. and Taqqu, M.S. 2000) The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6, 607 614. 25 Samorodnitsky, G. 1996) A class of shot noise models for financial applications. In: Athens Conference on Applied Probability and Time Series Analysis, Vol. I, pp. 332 353. Lecture Notes in Statistics 114, Springer, New York. 26 Sato, K.-i. 1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, UK. 27 Stegeman, A. 2002) xtremal behavior of heavy-tailed on-periods in a superposition of ON/OFF processes. Adv. Appl. Probab. 34, 179 204. 28 Vere Jones, D. 1970) Stochastic models for earthquake occurrences. J. Royal Statist. Soc. Ser. B, 32, 1 62. Department of Mathematics, Keio University, 3-14-1 Hiyoshi Kohoku-ku, Yokohama 223-8522, Japan -mail address: mmuneya@gmail.com Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark -mail address: mikosch@math.ku.dk