The Distribution of the Interval of the Cox Process with Shot Noise Intensity for Insurance Claims and Its Moments

The distribution of the interval of the Cox process with shot noise intensity for insurance claims and its moments

Angelos Dassios 1 , Ji-Wook Jang 2

1 Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom (e-mail: [email protected])

2 Actuarial Studies, Faculty of Commerce and Economics, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected])

Abstract. Applying piecewise deterministic Markov processes theory, the probability generating function of the Cox process, incorporating with shot noise process as the claim intensity, is obtained. We also derive the Laplace transform of the distribution of the shot noise process at claim jump times, using stationary assumption of the shot noise process at any times. Based on this Laplace transform and from the probability generating function of the Cox process with shot noise intensity, we obtain the distribution of the interval of the Cox process with shot noise intensity for insurance claims and its moments, i.e. mean and variance.

Key words: Piecewise deterministic Markov processes theory; martingale; stopping time; the distribution of the interval the Cox process with shot noise intensity; insurance claims.

1. Introduction

In insurance modeling, the Poisson process has been used as a claim arrival process. Extensive discussion of the Poisson process, from both applied and theoretical viewpoints, can be found in Cramér (1930), Cox and Lewis (1966), Bühlmann (1970), Cinlar (1975), Gerber (1979) and Medhi (1982). However there has been a significant volume of literature that questions the suitability of the Poisson process in insurance modeling (Seal 1983 and Beard et al. 1984). From a practical point of view, there is no doubt that the insurance industry needs a more suitable claim arrival process than the Poisson process that has deterministic intensity.

As an alternative point process to generate the claim arrivals, we can employ the Cox process or a doubly stochastic Poisson process (Cox 1955; Bartlett 1963; Haight 1967; Serfozo 1972; Grandell 1976, 1991, 1997; Brémaud 1981 and Lando 1994). An

1 important book on Cox processes is the book by Benning and Korolev 2002, where the applications in both insurance and finance are discussed. The Cox process provides us with the flexibility to allow the intensity not only to depend on time but also to be a stochastic process. In a recent paper (Dassios and Jang 2003), we demonstrated how the Cox process with shot noise intensity could be used in the pricing of catastrophe reinsurance and derivatives.

It is important to measure the time interval between the claims in insurance. Thus in this paper, we examine the distribution of the interval of the Cox process with shot noise intensity for insurance claims. The result of this paper can be used or easily modified in computer science/telecommunications modeling, electrical engineering and queueing theory.

We start by defining the quantity of interest; this is the doubly stochastic (with a shot- noise intensity) point process of claim arrivals. Then we derive the probability generating function of the Cox process with shot noise intensity using piecewise deterministic Markov processes (PDMP) theory, whose theory was developed by Davis (1984). The piecewise deterministic Markov processes theory is a powerful mathematical tool for examining non-diffusion models. For details, we refer the reader to Davis (1984), Dassios (1987), Dassios and Embrechts (1989), Jang (1998, 2004), Rolski et al. (1999), Dassios and Jang (2003) and Jang and Krvavych (2004). In section 3, we derive the Laplace transform of the distribution of the shot noise process at claim times, using stationary assumption of the shot noise process at any times. Using this Laplace transform within the probability generating function of the Cox process with shot noise intensity, we derive the distribution of the interval of the Cox process with shot noise intensity for insurance claims and its moments, i.e. mean and variance. In section 4 concludes.

2. The Cox process and the shot noise process

The Cox process (or a doubly stochastic Poisson process) can be viewed as a two-step randomisation procedure. A process λt is used to generate another process Nt by acting as its intensity. That is, Nt is a Poisson process conditional on λt which itself is a stochastic process (if λt is deterministic then Nt is a Poisson process). Many alternative definitions of a doubly stochastic Poisson process can be given. We will offer the one adopted by Brémaud (1981).

2 Definition 2.1 Let (,Ω FP,) be a probability space with information structure given by Ft=ℑ{ t , ∈[0, T]} . Let Nt be a point process adapted to F. Let λt be a non- negative process adapted to F such that t λ ds < ∞ almost surely (no explosions). ∫ s 0

If for all 0 ≤t1≤t2 and u ∈ℜ

t2 iu N −N ⎧ ⎫ ()tt21 λ ⎪ iu ⎪ Ee ℑ=texp ⎨ e −1 λsds⎬ (2.1) {}2 ()∫ ⎩⎭⎪ t1 ⎪ λ then Nt is called a ℑt -doubly stochastic Poisson process with intensity λt where ℑt λ is the σ − algebra generated by λ up to time t, i.e. ℑts=≤σλ{ ; st} .

Equation (2.1) gives us k ⎛ t2 ⎞⎛ t2 ⎞ exp⎜− λ ds⎟⎜ λ ds⎟ ⎜ ∫ s ⎟⎜ ∫ s ⎟ t t Pr{}N − N = k λ ;t ≤ s ≤ t = ⎝ 1 ⎠⎝ 1 ⎠ (2.2) t2 t1 s 1 2 k! and ⎛t2 ⎞ Pr τλ>≤tt; s≤t=Pr N−N=0 λ;t≤s≤t=exp −λds (2.3) {}21st2{21ts12}⎜∫ s⎟ ⎝⎠t1 th th where τ k denotes the length of the time interval between the (k −1) and the k claims. Therefore from (2.3), we can easily find that ⎧ t2 ⎫ Pr(τλ≤=tE) exp(−λds) . (2.4) 2 ⎨ t2 ∫ s⎬ ⎩⎭t1

t If we consider the process Λ= λ ds (the aggregated process), then from (2.2) we t∫ s 0 can also easily find that NN−−(1−θ )(Λ−Λ) EE()θ tt21= {et2t1} . (2.5)

The equation (2.5) suggests that the problem of finding the distribution of Nt , the point process, is equivalent to the problem of finding the distribution of Λt , the aggregated process. It means that we just have to find the p.g.f. (probability generating function) of Nt to retrieve the m.g.f. (moment generating function) of Λt and vice versa.

One of the processes that can be used to measure the impact of primary events is the shot noise process (Cox and Isham, 1980, 1986 and Klüppelberg and Mikosch, 1995). The shot noise process is particularly useful within the claim arrival process as it

3 measures the frequency, magnitude and time period needed to determine the effect of primary events. As time passes, the shot noise process decreases as more and more claims are settled. This decrease continues until another event occurs which will result in a positive jump in the shot noise process. Therefore the shot noise process can be used as the parameter of doubly stochastic Poisson process to measure the number of claims due to primary events, i.e. we will use it as a claim intensity function to generate the Cox process. We will adopt the shot noise process used by Cox and Isham (1980):

Mt −δ t −−δ ()tSi λλti=+0eY∑ e i=1 where

• λ 0 is initial value of λt . • Y is a sequence of independent and identically distributed random { i }i=1,2,"

variables with distribution function Gy() (y > 0), where EY()i = µ1 . • S is the sequence representing the event times of a Poisson process { i }i=1,2,"

M t with constant intensity ρ . • δ is rate of exponential decay. We assume that the Poisson process M and the sequences Y are independent t { i }i=1,2," of each other.

The Cox process w ith shot noise intensity

40 s

t 30

. o

N 20

0 50 100 150 200 250 300 350 t

4 The generator of the process (Λtt,λ ,t) acting on a function f (Λ,λ,t) belonging to its domain is given by ∂∂ff∂f∞ ΑΛ f (,λλ,tf)= + −δλ+ρ[∫ (Λ,λ+y,t)dG(y)−f(Λ,λ,t)]. (2.6) ∂∂t Λ ∂λ0 For f (Λ,λ,t) to belong to the domain of the generator Α, it is sufficient that f (,Λ λ,t) is differentiable w.r.t. Λ , λ , t for all Λ , λ , t and that ∞ ∫ fy(,⋅+λλ,⋅)dG(y)−f(,⋅,⋅) <∞. 0

Let us find a suitable martingale in order to derive the p.g.f. (probability generating function) of Nt at time t (Dassios 1987).

* Theorem 2.2 Let us assume that Λt and λt evolve up to a fixed time t . −δt* Considering constants k1 and k2 are such that k1 ≥ 0 and k2≥−k1e , ⎡ t ^ ⎤ exp −Λkδλexp −(k+keδδts) exp ρ{1−g(k+ke )}ds (2.7) ()11tt{}2⎢ ∫ 12⎥ ⎣ 0 ⎦ is a martingale.

Proof δt Define Wt=Λδ t+λt and Zt = λt e, then the generator of the process (,Wt Zt ,t) acting on a function f (,wzt,) is given by ∂ f ∞ Α= f (,wz,t) +ρ[∫ f(w+y,z+yeδ t ,t)dG(y)−f(wz,,t)]. (2.8) ∂ t 0 and f (,wzt,) has to satisfy Α f = 0 for f (,Wt Zt ,t) to be a martingale. Setting ee−−kw12kzh(t) we get the equation ^ δt ht'( )−−ρ[1g(k12+ke)]h(t) =0. (2.9)

−−kw12kz eeh(t) belongs to the domain of the generator because of our choice of k1, k2 ; the function is bounded for all t≤ t* and our process evolves up to time t * only. Solving (2.9) t ^ δ s ρ ∫{1−+g ( kk12e)}ds ht()=Κe 0 (2.10) where Κ is an arbitrary constant. Therefore t ^ δ s ρ ∫{1−+g ( kk12e)}ds ee−−kW12ttk Z e0 is a martingale and hence the result follows.

5 Corollary 2.3 Let ν12≥≥00,, ν ν ≥0, 0 ≤ θ ≤ 1 and t1, t2 be fixed times. Then −Λνν()−Λ −λ Ee{,12tt221 e tΛ λ } t1t1 tt− 21 ∧ νν ⎡ νν ⎤ =−exp ⎡⎤{ 11+(νλ−)e−−δ ()tt21} exp ⎢−ρ[1−g{ 11+(ν−)e−δ s}]ds⎥ (2.11) ⎣⎦δδ22t1 ∫ δδ ⎣⎢ 0 ⎦⎥ and ()NN−−νλ Ee{,θ tt221 tNλ } t1t1 tt− ⎡ 21 ∧ ⎤ =exp ⎡⎤−{11−−θθ+−(νλ)e−−δ ()tt21} exp ⎢−ρ[1−g{11−θθ+−(ν−)e−δ s}]ds⎥. (2.12) ⎣⎦δδ t1 ∫ δδ ⎣⎢ 0 ⎦⎥

Proof ν ν We set kk==1 ,( ν −1 )e−δt2 , t* ≥t in Theorem 2.2 and (2.11) follows 1 δ 22δ 2 immediately. (2.12) follows from (2.11) and (2.5).

Now we can easily derive the p.g.f. (probability generating function) of Nt and the

Laplace transform of λt using Corollary 2.3.

Corollary 2.4 The probability generating function of Nt is given by tt− 21 ∧ ()NN− ⎡ ⎤ Ee{}θλtt221 =−exp⎡⎤11−−θθ{1−−−δ ()tt1}λexp⎢−ρ[1−g{(1−e−δ s}]ds⎥ ,(2.13) tt11⎣⎦δδ∫ ⎣⎢ 0 ⎦⎥

the Laplace transform of the distribution of λt is given by t ^ −νλ −δt ⎡ −δs ⎤ E e t λ = exp −νλ e exp − ρ {1− g(νe )}ds (2.14) {}0 ()0 ⎢ ∫ ⎥ ⎣ 0 ⎦ and if λt is asymptotic (stationary), it is given by ∞ ^ −νλt ⎡ −δ s ⎤ Ee()=−exp⎢ ρν∫{1−g(e )}ds⎥ (2.15) ⎣ 0 ⎦ which can also be written as ν ∧ −νλ ⎧ ρ ⎫ Ee()t =−exp⎨ ∫G(ud)u⎬ (2.16) ⎩⎭δ 0 ∧ ∧ 1− gu() where G()u = . u

Proof

If we set ν = 0 in (2.12) then (2.13) follows. (2.14) follows if we either set ν1 = 0 in (2.11) or set θ =1 in (2.12). Let t → ∞ in (2.14) and the result follows immediately.

If we differentiate (2.14) and (2.16) with respect to θ and put θ =1, we can easily obtain the first moments of λt , i.e. µ ρ µ ρ E()λ λ = 1 + (λ − 1 )e −δt (2.17) t 0 δ 0 δ and µ ρ E()λ = 1 . (2.18) t δ The higher moments can be obtained by differentiating them further, i.e. µ ρ Var()λ λ = (1− e −2δt ) 2 (2.19) t 0 2δ and µ ρ Var()λ = 2 (2.20) t 2δ ∞ where µ ==EY()22ydG(y). 2 ∫ 0

3. The distribution of the interval of the Cox process with shot noise intensity and its moment

Let us examine the Laplace transform of the distribution of the shot noise intensity at th claim times. To do so, let us denote the time of the n claim of Nt by τn and denote the value of λ , when N takes the value n for the first time by λ . Since a claim t t τn occurs at time τ , this implies that the intensity at claim times, λ τ , should be higher than the intensity at any times λt . Therefore the distribution of λ τ should not be the same as the distribution of λt .

Let us start with the following lemma in order to obtain the Laplace transform of the distribution of the shot noise intensity at claim times. We assume that the claims and jumps (or primary events) in shot noise intensity do not occur at the same time.

Lemma 3.1 Suppose that f (λ) is a function belonging to its domain and furthermore that it satisfies t lim Ef{ (λλts)exp(−∫ ds) λ0} = 0 . (3.1) t→∞ 0 If h(λ) is such that λ{(h λ)− f (λ)}+ Α f (λ)= 0 (3.2) then Eh{(λ )λ }= f(λ ). (3.3) τ1 00

Proof From (3.2) t f (λ ) + [λ {h(λ ) − f (λ )}]ds t ∫ s s s 0 is a martingale and since τ1 ∧ t is a stopping time, where Pr()τ1 ≤ s =Pr()N s >0 and

N s is λ s-measurable, we have τ1 ∧t E{ f (λ λ ) + E[ [λ {h(λ ) − f (λ )}]ds λ ] = f (λ ) . (3.4) τ1 ∧t 0 ∫ s s s 0 0 0

If we now place a condition on the realisation λ v ; 0 ≤ v≤ t, then the first term of the left-hand side in (3.4) is Ef{(λλ)}=≤Ef{(λ)λ;0 v≤t}⋅dP(λ;0≤v≤t) (3.5) ττ11∧∧tt0 ∫ vv Ω and the second term of the left-hand side in (3.4) is

ττ1 ∧∧tt1 E[[λλ{h()−=f (λ)}]ds λ]E[λλ{h()−f (λ)}ds λ;0]≤v ≤t ⋅dP(λ;0)≤v ≤t ∫∫ss s 0 ∫ss s v v 00Ω (3.6) where dP(λv ;0 ≤≤v t) is the probability differential of a particular realisation in Ω, the set of all possible realisations.

r Since τ1 ∧ t is distributed with density, λr exp(−∫ λs ds) on (,0 t) and a mass, 0 t exp(−∫ λs ds) at t, conditionally on λr ;0≤ r≤ t, where Nt is the Cox process, we have 0 t rt E{ f (λλ) ;0 ≤≤νt} ={ f (λ)λexp(−∫∫λds)}dr +f (λ)exp(−λds) (3.7) τ1 ∧tv ∫ rr s t s 0 00 and

τ1 ∧t Eh[{λλ()−≤f(λ)};dsλ0ν≤t] ∫ ss s v 0 tr t = [ λ {h(λ ) − f (λ )}ds Pr(τ = r)]dr + λ {h(λ ) − f (λ )}ds Pr(τ > t) ∫∫ s s s 1 ∫ s s s 1 00 0 tr rtt =−[ λλ{h( ) f (λ)}ds ⋅λexp(−λds)]dr +λ{h(λ) −f (λ)}ds exp(−λds) ∫∫ ss s r ∫∫s ∫ ss s s 00 000

tt r t t = [− − λ exp(− λ ds)dr]⋅ λ {h(λ ) − f (λ )}ds + λ {h(λ ) − f (λ )}ds exp(− λ ds) ∫∫ r ∫ s s s s ∫ s s s ∫ s 0 s 0 0 0

8 t s t t t = {exp(− λ ds) − exp(− λ ds)}⋅ λ {h(λ ) − f (λ )}ds + λ {h(λ ) − f (λ )}ds exp(− λ ds) ∫ ∫ s ∫ s s s s ∫ s s s ∫ s 0 0 0 0 0

t s = λ {h(λ ) − f (λ )}exp(− λ ds)ds . (3.8) ∫ s s s ∫ s 0 0 Put s = r in (3.8), then we have τ ∧t 1 t r E[ λ {h(λ ) − f (λ )}ds λ ,0 ≤ν ≤ t] = λ {h(λ ) − f (λ )}exp(− λ ds)dr . (3.9) ∫ s s s v ∫ r r r ∫ s 0 0 0

Therefore (3.4) becomes

τ1 ∧t E{ f (λ λ ) + E[ [λ {h(λ ) − f (λ )}]ds λ ] τ1 ∧t 0 ∫ s s s 0 0 t rt =−[(fdλλ)exp(λ s)dr+f(λ)exp(−λds)]dP(λ;0≤v≤t) ∫∫ rr ∫∫s t s v Ω 0 00 t r +−[λλ{h() f ()λ}exp(−λds)dr]dP(λ;0≤v ≤t) ∫∫ rr r ∫ s v Ω 0 0 trt = f (λλ)exp(− ds)dP(λ;0 ≤≤v t) +[ λh(λ)exp(−λds)dr]dP(λ;0 ≤≤v t) ∫∫ts∫∫v ∫rrs v ΩΩ000 trt =−Ef{ (λλ)exp( ds) λ}+[ h(λ)λexp(−λds)dr]dP(λ;0 ≤v≤t) (3.10) ts∫0 ∫∫ rr∫s v 00Ω 0

= f ()λ 0 .

Letting t →∞ in (3.10), then from (3.1), the first term in the left-hand side tends to 0 r and the second term to Eh{(λ )λ }, as λexp(− λds) is a density. Therefore we τ1 0 r∫ s 0 have Eh{(λ )λ }= f(λ ). τ1 00

Assuming that the shot noise process λt is stationary, let us derive the Laplace transform of the distribution of the shot noise process at claim times, λ τ .

Theorem 3.2 If the shot noise process λt is stationary, the Laplace transform of the distribution of the shot noise process at claim times is given by ∧ ν ∧ −νλ G()νρ⎧ ⎫ Ee()τi =⋅exp ⎨−∫G(u)du⎬ . (3.11) µδ1 ⎩⎭0

Proof From Lemma 3.1, which implies that if f (λ) and h(λ) are such that

9 ∞ λλ{(hf)−−(λ)} δλf'(λ)+ρ{∫ f(λ+y)dG(y)−f(λ)}=0 (3.12) 0 and (3.1) is satisfied, we have Eh()λ λ= f(λ) (3.13) { τii+1 τ} τi ⎧⎫∧ ⎪g '(ν ) ⎪−νλ by starting the process from τi . Employing f ()λλ=−⎨∧ ⎬e , the function ⎩⎭⎪⎪1(− g ν ) f (λ) clearly satisfies (3.1) and substituting into (3.12), then we have

⎧⎫∧∧⎧⎫ ⎪⎪−−νλνgg'(νν) λ⎪'( ) ⎪−νλ−νλ λλ⎨⎬he()−+λ ∧∧e+δνλ⎨λ− ⎬e−δλe ⎩⎭⎪⎪1(−−ggνν) ⎩⎪1()⎭⎪ ∧∧ ⎡⎤⎧⎫∧∧ ∧ ⎧⎫ −νλ ⎪⎪gg'(νν) ⎪'( ) ⎪−νλ =−ρλ⎢⎥egg⎨⎬'(ν)+ '()ν− g()ν−⎨λ− ⎬e ⎢⎥∧∧ ⎣⎦⎩⎭⎪⎪1(−−ggνν) ⎩⎪1()⎭⎪ ∧ =−ρλeg−νλ { ()ν −1}. Divide by λ and simplify then we have ∧ ∧ −−νλ νλ g'(ν) −−νλ νλ he(λ)=−λ ()11δν +δe−()−δν ∧ ee+ρ{1−g(ν)}. (3.14) 1− g()ν From (3.13), it is given that Eh()λ =E⎡Eh()λλ⎤=E{f(λ)}. (3.15) { ττii++11} ⎣{ τi}⎦τi So put (3.14) into (3.15), then ∧ ∧ g '(ν ) E[λττexp(−−νλ )(1 δν ) +δ exp(−νλτ) −(1−δν ) ∧ exp(−νλτ) +ρ exp(−νλτ){1−g(ν )}] ii++11 i+1 1(−g ν ) i+1 i+1 ∧ g '(ν ) =− E{λνexp( λ) −exp(−νλ)}. (3.16) ττii++11∧ τi 1(− g ν )

When the process λ is stationary, λ and λ have the same distribution whose t τi+1 τi −νλ Laplace transform we denote by HE()ν = (eτi ) . Therefore from (3.16), we have ∧ g '(ν ) ∧ −−(1 δνν)HH'()−(1−δν) ∧ (ν)+[δ+ρ{1−g()ν}]H()ν 1(− g ν ) ∧ g '(ν ) =−H'(ν ) − ∧ H(ν ). (3.17) 1(− g ν )

Divide both sides of (3.17) by δν, then we have

10 ∧ ∧ g'(ν) 11ρ − g()ν H'(ν)+ ∧ H()ν ++{ }(H ν)= 0. (3.18) 1− g()ν ν δ ν

Solving (3.18), subject to H()0= 1 (3.19) then the Laplace transform of a distribution of the shot noise process at claim times is given by ⎛⎞∧ 1(− g νρ) ⎧ ν ∧ ⎫ HK()ν =⎜⎟exp⎨−∫G(u)du⎬ ⎜⎟νδ⎩⎭0 ⎝⎠ 1 where K is a constant. Therefore from (3.19), K = and µ1 ∧ ∧ 1 1− g(ν ) ⎧ ρ ν ∧ ⎫ G(ν ) ⎧ ρ ν ∧ ⎫ H (ν ) = ⋅ exp⎨− ∫G(u)du⎬ = ⋅ exp⎨− ∫G(u)du⎬ . (3.20) µ1 ν ⎩ δ 0 ⎭ µ1 ⎩ δ 0 ⎭

Equation (3.20) provides us with a very interesting result that this is the distribution of the sum of two random variables; one having the stationary distribution of λt (see G(y) Corollary 2.4) and the other, having density , where G(yG) =−1 (y) . µ1

Comparing it with the distribution of the shot noise process, λt at any times, we can easily find that ∧ G()νρ⎧⎫νν∧∧⎧⎫ρ ⋅−exp ⎨⎬∫∫GG(ud) u > exp ⎨⎬−(u)du . µδ1 ⎩⎭00⎩⎭δ

It is therefore the case that λ τ is stochastically larger than λt . In other words, the intensity at claim times is higher than the intensity at any times.

Now let us derive the distribution of the interval of the Cox process with shot noise intensity for insurance claims using Theorem 3.2.

Corollary 3.3 Assume that 0 is the time at which a claim of Nt has occurred and the stationary of λt has been achieved. Then the tail of the distribution of the interval of the Cox process with shot noise intensity is given by ∧ G()11− e−δ t ⎪⎧ ρ t ∧ ⎪⎫ Pr(τ >=te) δδ exp − G( 11− −δ s )ds. (3.21) ⎨ ∫ δδ ⎬ µδ1 ⎩⎭⎪ 0 ⎪

11 Proof

From (2.13), the probability generating function of Nt is given by ⎡ t ∧ ⎤ EeθλNt =−exp 1−θ(1−−δt)λexp −ρ[1−g{1−θ(1−e−δs)}]ds. (3.22) ()00{}δ⎢ ∫ δ⎥ ⎣ 0 ⎦ Set θ = 0 in (3.22) and take expectation, then the tail of the distribution of τ is given by t ∧ δ s ⎡⎤1− e −δt Pr(τ >=tg) exp −ρλ{1−( )}dsE⎡exp −(1−e ) ⎤ . (3.23) ⎢∫ ⎥⎣⎢ {δ 0}⎦⎥ ⎣⎦0 δ Substitute (3.11) into (3.23), then the result follows immediately as 0 is the time at which a claim has occurred and λt is stationary.

Corollary 3.4 The expectation and variance of the interval between claims are given by ∞ δ E()ττ=>∫ Pr( t)dt= (3.24) 0 µ1ρ and ∧ 2 ∞ G()11− e−δ u ρδu ∧ ⎛⎞ Var(τ ) =⋅2 [u δδ exp{−G( 11−e−δ s )ds}]du − . (3.25) ∫∫δδ ⎜⎟ 00µ11δµ⎝⎠ρ

Proof Integrate (3.21), then (3.24) follows. (3.25) is obtained from ∧ ∞∞G()11− e−δ u ρ u ∧ E(τ 22) ==t f (t)dt 2 [u ⋅δδ exp{−G( 11−e−δ s )ds}]du . ∫∫ ∫δδ 00µδ1 0

An interesting result we can find from (3.24) and (2.18) is that the expected interval between claims is the inverse of the expected number of claims, where the number of claims follows the Cox process with shot noise intensity, which is also the case for a Poisson process.

4. Conclusion

We started with deriving the probability generating function of the Cox process with shot noise intensity, employing piecewise deterministic Markov processes theory. It was necessary to obtain the distribution of the shot noise process at claim times as it is not the same as the distribution of the shot noise process at any times, i.e. the intensity

12 of claim times are higher than the intensity at any times. Assuming that the shot noise process is stationary, we derived the distribution of the interval of the Cox process with shot noise intensity for insurance claims and its moments from its probability generating function. The result of this paper can be used or easily modified in computer science/telecommunications modeling, electrical engineering and queueing theory as an alternative counting process of a Poisson process.

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