# The Distribution of the Interval between Events of a Cox Process with Shot Noise Intensity

Authors: Angelos Dassios; Jiwook Jang
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

Applying piecewise deterministic Markov processes theory, the probability generating function of a 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 a Cox process with shot noise intensity, we obtain the distribution of the interval of a Cox process with shot noise intensity for insurance claims and its moments, that is, mean and variance.


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## Body

## 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 [1–6]. However there has been a significant volume of literature that
questions the suitability of the Poisson process in insurance modeling [7, 8]. 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 a Cox
process or a doubly stochastic Poisson process [9–15]. An important book
on Cox processes is the book by Bening and Korolev [16], where the
applications in both insurance and finance are discussed. A Cox process
provides us with the flexibility to allow the intensity not only to depend on
time but also to be a stochastic process.
Dassios and Jang [17] demonstrated how a 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 a 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 a doubly stochastic (with a
shot-noise intensity) point process of claim arrivals. Then, we derive the
probability generating function of a Cox process with shot noise intensity
using piecewise deterministic Markov processes (PDMPs) theory, for which see
the appendix. The piecewise deterministic Markov processes theory is a powerful
mathematical tool for examining nondiffusion models. For details, we refer the
reader to [17–25]. 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 a Cox process with shot
noise intensity, we derive the distribution between events of a Cox process
with shot noise  intensity. These can be insurance
claims for examples. We also derive the
first two moments of this distribution. Section 4 contains some concluding remarks.


## 2. A Cox Process and The Shot Noise Process

A Cox process
(or a doubly stochastic Poisson process) can be viewed as a two-step
randomisation procedure. A processλt 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 [15].Definition 2.1.
Let(Ω,F,P) be a probability space with information
structure given by F={ℑt,t∈[0,T]}. Let Nt be a point process adapted to F. Let λt be a nonnegative process adapted to F such that  (2.1)∫0tλsds<∞almost surely (no explosions). If for all 0≤t1≤t2 and u∈ℜ(2.2)E{eiu(Nt2−Nt1)∣ℑt2λ}=exp⁡{(eiu−1)∫t1t2λs⁢ds} then Nt is called a ℑt-doubly stochastic Poisson process with
intensity, λt where ℑtλ  is the σ-algebra generated by λ up to time t, that is, ℑtλ=σ{λs;s≤t}.Equation (2.2)
gives us (2.3)Pr⁡{Nt2−Nt1=k∣λs;t1≤s≤t2}=exp⁡(−∫t1t2λs⁢ds)(∫t1t2λs⁢ds)kk!,(2.4)Pr⁡{τ2>t∣λs;t1≤s≤t2}=Pr⁡{Nt2−Nt1=0∣λs;t1≤s≤t2}=exp⁡(−∫t1t2λs⁢ds), where τk=inf{t>0:Nt=k}. Therefore from
(2.4), we can easily find
that (2.5)Pr⁡(τ2≤t)=E{λt2exp⁡(−∫t1t2λs⁢ds)}.If we consider
the process Λt=∫0tλsds (the
aggregated process), then from
(2.3) we can also easily find that (2.6)E(θNt2−Nt1)=E{e−(1−θ)(Λt2−Λt1)},where θ is a constant between 0 and 1. Equation (2.6)
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 probability generating function (p.g.f.) of  Nt to retrieve the moment generating function (m.g.f.)
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 [26–28].
The shot noise process is particularly useful within the claim arrival process
as it   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, that is, 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 [26]: (2.7)λt=λ0e−δt+∑i=1MtYie−δ(t−Si), where (i)
λ0 is initial value of λt;(ii)
{Yi}i=1,2,… is a sequence of independent and
identically distributed random variables with distribution function G(y)
(y>0),
where E(Yi)=μ1;(iii)
{Si}i=1,2,… is the
sequence representing the event times of a Poisson process Mt with constant intensity ρ;(iv)
δ is rate of exponential decay. We assume that
the Poisson process Mt and the sequences {Yi}i=1,2,… are independent  of each other. Figure 1 is the graph illustrating shot
noise process.
Figure 2
is the
graph illustrating a Cox process with shot noise intensity.The generator of
the  process(Λt,λt,t) acting on a function f(Λ,λ,t) belonging to its domain is given by (2.8)Af(Λ,λ,t)=∂f∂t+λ∂f∂Λ−δλ∂f∂λ+ρ[∫0∞f(Λ,λ+y,t)dG(y)−f(Λ,λ,t)].For f(Λ,λ,t) to belong to the domain of the generator
A, it is sufficient that f(Λ,λ,t) is differentiable with respect to Λ, λ, t for  all Λ, λ, t and that |∫0∞f(⋅,λ+y,⋅)dG(y)−f(⋅,λ,⋅)|<∞.Let us find a
suitable martingale in order to derive the probability generating function (p.g.f.)
ofNt at time t.Theorem 2.2.
Let us assume thatΛt and λt evolve up to a fixed time t*. Considering constants k1 and k2 are such that k1≥0 and k2≥−k1e−δt*, (2.9)exp⁡(−k1δΛt)exp⁡{−(k1+k2eδt)λt}exp⁡[ρ∫0t{1−g^(k1+k2eδs)}ds] is a martingale,
where g^(u)=∫0te−uydG(y)  and t>0.Proof.
DefineWt=δΛt+λt and Zt=λteδt,
then the generator of the process (Wt,Zt,t) acting on a function f(w,z,t) is given by (2.10)Af(w,z,t)=∂f∂t+ρ[∫0∞f(w+y,z+yeδt,t)dG(y)−f(w,z,t)],and f(w,z,t) has to satisfy Af=0 for f(Wt,Zt,t) to be a martingale. We try a solution of the form e−k1we−k2zh(t),
where h(t) is a differentiable function. Then we get the following equation: (2.11)h′(t)−ρ[1−g^(k1+k2eδt)]h(t)=0.e−k1we−k2zh(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 (2.11) (2.12)h(t)=Keρ∫0t{1−g^(k1+k2eδs)}ds,where K is an arbitrary constant. Therefore (2.13)e−k1Wte−k2Zteρ∫0t{1−g^(k1+k2eδs)}ds is a martingale
and hence the result follows.Corollary 2.3.
Letν1≥0,ν2≥0,ν≥0,0≤θ≤1,
and t1,t2 be fixed times. Then  (2.14)E{e−ν1(Λt2−Λt1)e−ν2λt2∣Λt1,λt1}=exp⁡[−{ν1δ+(ν2−ν1δ)e−δ(t2−t1)}λt1]×exp⁡[−ρ∫0t2−t1[1−g^{ν1δ+(ν2−ν1δ)e−δs}]ds],(2.15)E{θ(Nt2−Nt1)e−νλt2∣Nt1,λt1}=exp⁡[−{1−θδ+(ν−1−θδ)e−δ(t2−t1)}λt1]×exp⁡[−ρ∫0t2−t1[1−g^{1−θδ+(ν−1−θδ)e−δs}]ds].Proof.
We setk1=ν1/δ,k2=(ν2−ν1/δ)e−δt2,t*≥t2 in Theorem 2.2 and
(2.14) follows
immediately. Equation (2.15) follows
from (2.14) and
(2.6).Now we can easily derive the probability generating
function (p.g.f.) ofNt and the Laplace transform of λt using
Corollary 2.3.Corollary 2.4.
The  probability
generating function ofNt is given by  (2.16)E{θ(Nt2−Nt1)∣λt1}=exp⁡[−1−θδ{1−e−δ(t2−t1)}λt1]×exp⁡[−ρ∫0t2−t1[1−g^{1−θδ(1−e−δs)}]ds],  the Laplace  transform of the distribution of λt is given by (2.17)E{e−νλt∣λ0}=exp⁡(−νλ0e−δt)exp⁡[−ρ∫0t{1−g^(νe−δs)}ds]
and if λt is asymptotic (stationary), it is given by (2.18)E(e−νλt)=exp⁡[−ρ∫0∞{1−g^(νe−δs)}ds]  which can also be written as (2.19)E(e−νλt)=exp⁡{−ρδ∫0νG^(u)du},where G^(u)=(1−g^(u))/u.Proof.
If we setν=0 in (2.15) then
(2.16) follows. Equation
(2.17) follows if we either set ν1=0 in (2.14) or set θ=1 in (2.15). Let t→∞ in (2.17) and the result follows immediately.Theorem2.2,
Corollaries 2.3 and 2.4 can be found in [17, 19], but they have been included here for completeness and for
comparison purposes.If we differentiate (2.17) and
(2.19) with respect to ν and put ν=0,
we can easily obtain the first moments of λt,
that is, (2.20)E(λt∣λ0)=μ1ρδ+(λ0−μ1ρδ)e−δt,(2.21)E(λt)=μ1ρδ.
The higher
moments can be obtained by differentiating them further, that is, (2.22)Var(λt∣λ0)=(1−e−2δt)μ2ρ2δ,Var(λt)=μ2ρ2δ, where μ2=E(Y2)=∫0∞y2dG(y).


## 3. The Distribution of The Interval between Events of A Cox Process with Shot Noise Intensity and Its Moment

Let us examine
the Laplace transform of the distribution of
the shot noise intensity at claim times. To do so, let us denote the time of thenth claim of Nt by τn and denote the value of λt,
when Nt takes the value n for the first time by λτn. Since a claim occurs at time τ,
this implies that the intensity at claim times, λτ,
should be higher than the intensity at any times λt. Therefore the distribution of λτ should not be the same as the distribution of λt,
which will be clear from Theorem 3.2.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.
LetNt be a Cox process with shot noise intensity λt. Let A be the generator of the process λt  and  suppose that f(λ) is a function belonging to its domain and
furthermore that it satisfies (3.1)lim⁡t→∞E{f(λt)exp⁡(−∫0tλsds)∣λ0}=0. If h(λ) is such that (3.2)λ{h(λ)−f(λ)}+Af(λ)=0 then (3.3)E{h(λτ1)∣λ0}=f(λ0).Proof.
From (3.2) (3.4)f(λt)+∫0t[λs{h(λs)−f(λs)}]ds is a martingale
and since τ1^t is a stopping time, where Pr(τ1≤s)=Pr(Ns>0) and Ns is λs-measurable, we have (3.5)Ef{(λτ1^t∣λ0)}+E[∫0τ1^t[λs{h(λs)−f(λs)}]ds∣λ0]=f(λ0). Conditioning on
the realisation λv, 0≤v≤t, τ1^t is distributed with density (3.6)λrexp⁡(−∫0rλu⁢du) on (0,t) and a mass exp(−∫0tλudu) at t. Hence, (3.7)E{f(λτ1^t)∣λv,0≤v≤t}=∫0t{f(λr)λrexp⁡(−∫0rλu⁢du)}dr+f(λt)exp⁡(−∫0tλu⁢du),(3.8)E[∫0τ1^tλs{h(λs)−f(λs)}ds∣λv,0≤v≤t]=∫0t∫0rλs{h(λs)−f(λs)}ds⁢λrexp⁡(−∫0rλu⁢du)dr+∫0tλs{h(λs)−f(λs)}dsexp⁡(−∫0tλu⁢du). Changing the
order of integration on the first term of this, it becomes (3.9)=∫0t∫stλrexp⁡(−∫0rλu⁢du)drλs{h(λs)−f(λs)}ds+∫0tλs{h(λs)−f(λs)}dsexp⁡(−∫0tλu⁢du)=∫0t{exp⁡(−∫0sλu⁢du)−exp⁡(−∫0tλu⁢du)}λs{h(λs)−f(λs)}ds+∫0tλs{h(λs)−f(λs)}dsexp⁡(−∫0tλu⁢du)=∫0texp⁡(−∫0sλu⁢du)λs{h(λs)−f(λs)}ds. Adding (3.7) and
(3.9), we notice that more terms cancel and we get (3.10)E{f(λτ1^t)+∫0τ1^tλs{h(λs)−f(λs)}ds∣λv,0≤v≤t}=∫0texp⁡(−∫0sλu⁢du)λsh(λs)ds+f(λt)exp⁡(−∫0tλu⁢du)=E{h(λτ1)1{τ1≤t}∣λv,0≤v≤t}+f(λt)exp⁡(−∫0tλu⁢du), and hence (3.11)E{f(λτ1^t)+∫0τ1^tλs{h(λs)−f(λs)}ds∣λ0}=E{(h(λτ1)1{τ1≤t})+f(λt)exp⁡(−∫0tλu⁢du)∣λ0}.
From (3.5), we
then have (3.12)E{(h(λτ1)1{τ1≤t})+f(λt)exp⁡(−∫0tλu⁢du)∣λ0}=f(λ0) and setting t→∞,
we get (3.3).Assuming that
the shot noise processλt is stationary, let us derive the Laplace transform of the distribution of the shot noise
process at claim times, λτ.Theorem 3.2.
If the shot noise processλt is stationary, the Laplace transform of the
distribution of the shot noise process at claim times is given by (3.13)E(e−νλτi)=G^(ν)μ1⋅exp⁡{−ρδ∫0νG^(u)du},where G^(u)=(1−g^(u))/u and g^(u)=∫0te−uydG(y).Proof.
From Lemma3.1,
which implies that if f(λ) and h(λ) are such that (3.14)λ{h(λ)−f(λ)}−δλf′(λ)+ρ{∫0∞f(λ+y)dG(y)−f(λ)}=0 and
(3.1) is
satisfied, we have (3.15)E{h(λτi+1)∣λτi}=f(λτi) by starting the process from τi. Employing f(λ)={λ−g′^(ν)/(1−g^(ν))}e−νλ,
the function f(λ) clearly satisfies (3.1) and substituting into
(3.14), then we have (3.16)λ{h(λ)−λe−νλ+g′^(ν)1−g^(ν)e−νλ}+δνλ{λ−g′^(ν)1−g^(ν)}e−νλ−δλe−νλ=−ρλe−νλ{g^(ν)−1}. Divide by λ and simplify then we have (3.17)h(λ)=λe−νλ(1−δν)+δe−νλ−(1−δν)g′^(ν)1−g^(ν)e−νλ+ρe−νλ{1−g^(ν)}. From
(3.15), it
is given that (3.18)E{h(λτi+1)}=E[E{h(λτi+1)∣λτi}]=E{f(λτi)}. So put
(3.17)
into (3.18), then (3.19)E[λτi+1exp⁡(−νλτi+1)(1−δν)+δexp⁡(−νλτi+1)−(1−δν)g′^(ν)1−g^(ν)×exp⁡(−νλτi+1)+ρexp⁡(−νλτi+1){1−g^(ν)}]=E{λτiexp⁡(−νλτi+1)−g′^(ν)1−g^(ν)exp⁡(−νλτi+1)}.
When the process λt is stationary, λτi+1,
and λτi have the same distribution whose Laplace transform we denote by H(ν)=E(e−νλτi).
Therefore from (3.19), we have (3.20)−(1−δν)H′(ν)−(1−δν)g′^(ν)1−g^(ν)H(ν)+[δ+ρ{1−g^(ν)}]H(ν)=−H′(ν)−g′^(ν)1−g^(ν)H(ν).
Divide both
sides of (3.20) by δν,
then we have (3.21)H′(ν)+g′^(ν)1−g^(ν)H(ν)+{1ν+ρδ1−g^(ν)ν}H(ν)=0.
Solving (3.21),
subject to (3.22)H(0)=1 then the Laplace transform of a distribution of the shot noise
process at claim times is given by (3.23)H(ν)=K(1−g^(ν)ν)exp⁡{−ρδ∫0νG^(u)du},where
K
is a constant. Therefore from (3.22), K=1/μ1 and (3.24)H(ν)=1μ11−g^(ν)ν⋅exp⁡{−ρδ∫0νG^(u)du}=G^(ν)μ1⋅exp⁡{−ρδ∫0νG^(u)du}.Equation (3.24)
provides us with an interesting result. The distribution defined by the Laplace
transform (3.24) (and (3.13)) is the same as the distribution of two random
variables; one having the stationary distribution of λt (see Corollary 2.4) and the other having
density G(y)/μ1,
where G(y)=1−G(y).
Comparing it with the distribution of the shot noise process, λt at any times, we can easily find that (3.25)G^(ν)μ1⋅exp⁡{−ρδ∫0νG^(u)du}>exp⁡{−ρδ∫0νG^(u)du}.It is therefore
the case that λτ 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 a Cox
process with shot noise intensity for insurance claims using Theorem3.2.Corollary 3.3.
Assume that 0 is the
time at which a claim ofNt has occurred and the stationary of λt has been achieved. Then the tail of the
distribution of the interval of a Cox process with shot noise intensity is
given by (3.26)Pr⁡(τ>t)=G^(1/δ−(1/δ)e−δt)μ1exp⁡{−ρδ∫0tG^(1δ−1δe−δs)ds}.Proof.
From (2.16), the
probability generating function of Nt is given by (3.27)E(θNt∣λ0)=exp⁡{−1−θδ(1−e−δt)λ0}exp⁡[−ρ∫0t[1−g^{1−θδ(1−e−δs)}]ds]. Set θ=0 in (3.27) and take expectation, then the tail
of the distribution of τ is given by (3.28)Pr⁡(τ>t)=exp⁡[−ρ∫0t{1−g^(1−eδsδ)}ds]E[exp⁡{−(1−e−δt)δλ0}].Substitute
(3.13)
into (3.28), 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(3.29)E(τ)=∫0∞Pr⁡(τ>t)dt=δμ1ρ,(3.30)Var(τ)=2∫0∞[uG^(1/δ−(1/δ)e−δu)μ1exp⁡{−ρδ∫0uG^(1δ−1δe−δs)ds}]du−(δμ1ρ)2.Proof.
Integrate (3.26), then
(3.29)
follows. (3.30) is obtained from (3.31)E(τ2)=∫0∞t2f(t)dt=2∫0∞[uG^(1/δ−(1/δ)e−δu)μ1exp⁡{−ρδ∫0uG^(1δ−1δe−δs)ds}]du.An interesting
result we can find from (3.29) and (2.21) is that the expected interval between
claims is the inverse of the expected number of claims, where the number of
claims follows a 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 a 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. Assuming that the shot noise process is
stationary, we derived the distribution of the interval of a 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 to a Poisson process.Figure 1
Graph illustrating shot
noise process.Figure 2
Graph illustrating a Cox
process with shot noise process.



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