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Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed.

An insurance company insures occurring claims in exchange for a regular premium. Numerous works study the determination of an optimal premium: for example, the premium should be high enough such that the ruin probability of the insurance company is sufficiently small. The claim sizes themselves are often considered to be independent and identically distributed, with arrival times being jump times from a Poisson process. A by now classical extension of this model considers renewal times, where the inter-arrival times are no longer exponential.

In this paper, we extend this set-up further and study arrival times with a random arrival rate. In particular, we will consider arrival rates having shot-noise features. This could, for example, be used to model the claims arrivals after a catastrophe in a dynamic way: many claims will be reported right after the catastrophe, such that the arrival rate in the beginning is high. Further claims will be announced later and later corresponding to a decreasing arrival rate. Shot-noise arrival rates directly model such an effect. An alternative application appears when considering claims caused by a flood or hail: they typically admit spatial patterns with a centre where the majority of the claims are located and a decreasing number of claims with increasing distance from the centre. In a life insurance context, a natural disaster, such as a tsunami also leads to a similar patterns.

The main idea we follow here is to give a new view on insurance claims processes inspired by recent results in credit risk. In particular, we propose a model with multiple claim arrivals, i.e., claims can occur at the same time. This is an important issue for catastrophe modeling and for pricing Catastrophe Bonds (CAT bonds). The size of CAT bond markets has been increasing tremendously over the last decade. Currently, it reaches an all-time high: the outstanding volume hit $19 billion dollars in October 2013 (sources: [

Shot-noise processes are a well-known and well-studied object. Inspired by physical effects as in [

More precisely, consider a Poisson process

It turns out that

The structure of the paper is as follows: in

We study models allowing for factor-driven dynamics by borrowing heavily from current developments in credit risk, in particular reduced form modeling, see [

Consider a probability space

From a general viewpoint, non-life insurance can be described as follows: insurance claims are reported at the

We start by revisiting some well-known facts for marked point processes. A detailed exposition of the theory of point processes and marked point processes may be found in [

By

First, for a point process

The cumulated intensity measure determines the compensator in the Doob-Meyer decomposition, such that

In this section we study a general class of shot-noise processes driven by time-inhomogeneous Poisson processes. In Section this class will build the cornerstone for our modeling of the cumulated intensity process

Consider an inhomogeneous Poisson process

If

This representation shows that in general,

For the following result, we denote by

In the exponential case, i.e., when

For applications it is important to have a repertory of parametric families which can be used to estimate the shot-noise process from data. We give some specifications in the following example which lead to highly tractable models. These examples will partly be taken up in Example 10 in an integrated form.

Regime switching:

Exponential structure:

We close this section with an example where claims are discounted with respect to deterministic, but non-constant interest rates.

Illustration of a shot-noise process (

For the description of the statistical properties of the model, the Fourier transform of the shot-noise process is a central quantity which is given in the following result. For convenience of the reader we give a proof of this classical result in our general set-up. We denote by

Central to the proof is the following lemma which gives a relation of the jump times of the Poisson process to order statistics of i.i.d., uniformly distributed random variables. The order statistic of

For a proof of Lemma 4 we refer to p.502 in [

Now we utilize the representation of an inhomogeneous Poisson process as time-transformation of a standard Poisson process: the process

Related results may be found in [

Now we are in the position to put our ingredients together for the modeling of insurance claims. Let

As before, claims arrive at times

As indicated in the above example we will consider integrals over shot-noise processes as cumulated intensity processes. In view of classical applications this class of processes is quite unusual as the noise function is increasing. We distinguish these two cases in our notation by always using

For concrete implementations it is important to have a repertory of non-decreasing shot-noise processes which can be used to estimate the shot-noise process from data. We give some specifications in the following example which lead to highly tractable models.

Linear structure:

Exponential structure:

Rational structure:

An illustration of the last example may be found in

Illustration of the cumulated shot-noise intensity

Catastrophe bonds (CAT bonds) are risk-linked securities which allow to transfer insurance risks to investors. While the valuation of car insurance can effectively be done using the law of large numbers, catastrophe risks pose a large challenge due to highly dependent claim arrivals. Our shot-noise approach sets a framework which is ideally suited to model such risks.

The size of CAT bond markets has been increasing continuously over the last decade and has reached an outstanding volume of $19 billion dollars in October 2013.

We consider the following stylized version: a CAT bond offers a coupon payment

As an example we consider as trigger event if the claims process

For the pricing of the CAT bond we need to choose a risk-neutral measure

First, we assume that

If interest rates are deterministic, we obtain that

For more information on CAT bonds we refer to [

Following the results in [

The basic driver of the shot-noise process

We consider an initial filtration

Consider a

For tractability reasons one often considers shot-noise processes driven by a marked point process which has independent increments. If the increments are moreover stationary, then Φ is a Lévy process. We cover both cases in this section.

In [

We take a more general approach here and only assume that certain properties of the shot-noise process hold under

According to Theorem 6 we assume a simple structure of Φ under

This assumption will be satisfied under an Esscher change of measure, which is an important class for insurance applications. If we have a deterministic interest rate,

The key to efficient pricing methodologies is to obtain the Fourier transform of the claims arrival process. In this regard, we consider the set-up as in

Of course, if

In the doubly-stochastic case as in Example 1 we have the following, important result: recall that this setting can be viewed as a stochastic time change:

In Example 8 the

Now the way to pricing of the CAT-bond is clear: one can either invert the Fourier transform by Fast-Fourier methods or, alternatively compute

The estimation of shot-noise processes is an important part in the application of these models. A possible approach in this direction uses filtering methods and has been started in [

The key assumption in the approach of [

We will consider the following case: observations consist of data of

We assume that

Choosing a parametric approach, we follow Equation (

The first step towards the estimation is the introduction of the aggregated point process and the aggregated compensator:

The second step is to define a suitable distance. For the finite measure

The following weak identifiability assumption will be needed for consistency. By

The following result, given in Theorem 1 in [

The proof of the theorem may be found in [

This estimation procedure seems a very promising approach compared to existing methodologies and will be taken up in a future article for an estimation on insurance catastrophe data.

Efficient simulation algorithms are often the key to widespread application of a model. In particular, when closed-form results are expensive or not at hand, Monte-Carlo simulation always provides an alternative which is nowadays often feasible due to available computer power. Similar to [

Consider a fixed time horizon

We shortly recall our model:

Draw the number of jumps

Simulate

Simulate

Compute the path

Simulate the claim arrival times by taking i.i.d. exponential(1)-random variables

Simulate the claim sizes

Simulation of a claims process driven by a shot-noise process with rational structure. The graph shows the intensity process

The author declares no conflicts of interest.