Ruin by dynamic contagion claims

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Abstract

In this paper, we consider a risk process with the arrival of claims modelled by a dynamic contagion process, a generalisation of the Cox process and Hawkes process introduced by Dassios and Zhao (2011). We derive results for the infinite horizon model that are generalisations of the Cramér–Lundberg approximation, Lundberg’s fundamental equation, some asymptotics as well as bounds for the probability of ruin. Special attention is given to the case of exponential jumps and a numerical example is provided.

Highlights

► Modelling of claims by a dynamic contagion process. ► Process generalises the Cox process and the Hawkes process. ► Asymptotics of ruin probabilities are discussed. ► A very efficient simulation procedure is implemented.

Introduction

In the classical Cramér–Lundberg risk model, the arrival of claims is modelled by a Poisson process. As substantially discussed in the literature, this model is often not realistic in practice and hence a variety of extensions have been studied. Many researchers, such as Björk and Grandell (1988) and Embrechts et al. (1993) had already suggested using the Cox process to model the arrival of claims, (see also the book by Grandel (1991)). Schmidli (1996) investigated the case for a Cox process with a piecewise constant intensity. More recently, Albrecher and Asmussen (2006) discussed a Cox process with shot noise intensity. On the other hand, only a few researchers have proposed risk models using self-excited processes, due to the observation of the clustering arrival of claims in reality, a similar pattern in the credit risk from the financial market, particularly during the current economic crisis. Stabile and Torrisi (2010) looked at the ruin problem in a model using the Hawkes process, a self-excited point process introduced by Hawkes (1971).

To capture the clustering phenomenon as well as some common external factors involved for the arrival of claims within one single consistent framework, in this paper, we extend further to use the dynamic contagion process introduced by Dassios and Zhao (2011), a generalisation of the externally excited Cox process with shot noise intensity (with exponential decay) and the self-excited Hawkes process (with exponential decay). It could be particularly useful for modelling the dependence structure of the underlying arriving events with dynamic contagion impact from both endogenous and exogenous factors. In this paper, we try to generalise results obtained for the classical model.

We organise our paper as follows. Section 2 provides distributional results we will use, mainly developed in Dassios and Zhao (2011). Section 3 formulates the problem. It also provides a numerical example and some asymptotics that are based on simulations. In Section 4, we use the martingale method and generalise Lundberg’s fundamental equation. We derive bounds for the ruin probability in Section 5. In Section 6, we derive all results via a change of measure. This makes simulations more efficient as ruin is certain under the new measure. Section 7 concentrates on exponentially distributed claims. Our results are illustrated by a numerical example.

Section snippets

Dynamic contagion process

The dynamic contagion process includes both the self-excited jumps (which are distributed according to the branching structure of a Hawkes process with exponential fertility rate) and the externally excited jumps (which are distributed according to a particular shot noise Cox process). We directly use the definition of the dynamic contagion process from Dassios and Zhao (2011).

Definition 2.1 Dynamic Contagion Process

The dynamic contagion process is a cluster point process D on R+: The number of points in the time interval (0,t] is

Ruin problem

We consider a company with its surplus process Xt in continuous time on a probability space (Ω,F,P), Xt=x+cti=1NtZi,t0, where

  • X0=x0 is the initial reserve at time t=0;

  • c>0 is the constant rate of premium payment per time unit;

  • Nt is a point process (N0=0) counting the number of cumulative arrived claims in the time interval (0,t], driven by a dynamic contagion process with its stochastic intensity process λt and the initial intensity λ0=λ>0;

  • {Zi}i=1,2, is a sequence of independent identical

Exponential martingales and generalised Lundberg’s fundamental equation

In this section, we find some useful exponential martingales which link to the generalised Lundberg’s fundamental equation. More importantly, they are crucial for deriving some key results of the ruin problem in the later sections.

Theorem 4.1

Assume δ>μ1G and the net profit condition (6), we have a martingaleevrXteηrλtert,r0,where constants r,vr and ηr satisfy a generalised Lundberg’s fundamental equation{δηr+zˆ(vr)gˆ(ηr)1=0ρ(hˆ(ηr)1)r+aδηrcvr=0(c>μ1Hρ+aδδμ1Gμ1Z,δ>μ1G).If 0r<r, then(7) has a

Ruin probability via original measure

Theorem 5.1

The ruin probability conditional on λ0 and X0 is given byP{τ<|X0=x,λ0=λ}=ev0+xeη0+λE[ev0+Xτeη0+λτ|τ<;X0=x,λ0=λ].

Proof

By the optional stopping theorem, a bounded martingale stopped at a stopping time is still a martingale. Now we consider the martingale found by Theorem 4.1 stopped at the ruin time, i.e. evr+X(τt)eηr+λ(τt)er(τt),0r<r. By the martingale property, we have E[evr+X(τt)eηr+λ(τt)er(τt)]=E[evr+X(τt)eηr+λ(τt)er(τt)|X0=x,λ0=λ]=evr+xeηr+λ, and E[evr+Xτeηr+λτ

Ruin probability via change of measure

In this section, we investigate the ruin probability and asymptotics by change of measure via the martingale derived by Theorem 4.1. We will find that under this new measure the ruin becomes certain, and this makes the simulation more efficient than under the original measure where the ruin is not certain and even rare. Similar ideas of improving simulation of rare events by change of measure can also be found in Asmussen (1985) and more recently Asmussen and Glynn (2007).

Example: jumps with exponential distributions

To represent the previous results in explicit forms, in this section, we further assume the externally excited and self-excited jumps in the intensity process λt and the claim sizes all follow exponential distributions, i.e. HExp(α),GExp(β) and ZExp(γ), with the density functions h(y)=αeαy,g(y)=βeβy,z(z)=γeγz,y,z;α,β,γ>0, and the Laplace transforms hˆ(u)=αα+u,gˆ(u)=ββ+u,zˆ(u)=γγ+u.

Acknowledgment

We would like to thank an anonymous referee for various helpful suggestions.

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