We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.

The most popular ways to test for independence of two ordinal random variables are by means of Kendall's tau and Spearman's rho. However, such tests are not consistent, only having power for alternatives with "monotonic" association. In this paper, we introduce a natural extension of Kendall's tau, called τ * , which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, τ * is the simplest, similarly to Kendall's tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average p-values.

No abstract

In [lo] M.H.A. Davis introduced a class of non-diffusionmodels, called piecewise-deterministic Markov processes. As was pointed out by Embrechts [14] in the discussion to Davis's paper, these processes should provide a standard theory for studying applications in insurance risk theory. It is our aim to explain this in more detail by unifying the analysis of stochastic insurance models. Some new results will also be provided.In Section 1 we introduce the mathematics of the basic model together with a formulation of the classical risk processes as piecewise deterministic Markov (PD) processes. We distinguish between two different approaches. The first approach is used in Section 2 to find expressions for the probability of ruin in the classical Andersen model.Some new results are obtained for varying claim arrival rate, e.g. under periodicity assumptions. A general model including service payments is also analysed. Finally in Section 3 we discuss the second approach to establish exact results for Gamma claim sizes and investment policies, including borrowing. It should be stressed early on that the piecewise deterministic structure is essentially a vehicle for obtaining 182 DASSIOS AND EMBRECHTS interesting martingales, which can then be used to calculate relevant functionals of the process.If the Davis theory is to be used for the analysis of more realistic insurance models, then it is important to understand in detail its scope and versatility. The present paper stresses the request made by E. Arjas [2] : I t . . . I hope to see in the future many worked out examples of how the PD theory can be used as an aid in solving practical OR problems". PD PROCESSES AND INSURANCE MODELSIn general risk theory, one is interested in the so-called surplus process. It represents the surplus (liquid funds, accumulated capital) of a company and consists essentially of three different processes.(i) The income process (premiums etc.) (Ut : t 2 0 ) . The variable t will always denote time; it can either be discrete or continuous or indeed be operational time. For + reasons of uniformity, we assume t E IR in this paper.(ii)The counting process of. claim arrivals (N . t 2 0). t .(iii) An infinite sequence of random variables (Y.) the claim 1 i' sizes.The aggregate claim size process is defined by Yt = 1 Yi. So = Ut -Yt.i-1 that the surplus process equals Z t Whenever we shall speak about the classical model, we shall assume the following extra assumptions. Al.Ut is deterministic, i.e. u -u + ct for t r 0 , where t c > 0 is interpreted as the constant premium income rate and u is the initial capital.A2.(Nt : t > 0) is a homogeneous Poisson process with parameter A , say. A 3 .The claim sizes Y1,Y2,. . . are i.i.d. with distribution function G(y). A4. The processes (N . t > 0 ) and (Yi)iare independent. t . A5. Net-profit condition : 3 t > 0, Vt 2 t : E(Zt) -ZO > 0 1 1 Downloaded by [New York University] at 20:37 18 June 2015 MARTINGALES AND INSURANCE RISK 183 A6. Assume G(y) is supported by [0 ,.ojand its Laplace-Stieltjes ca UX trans...

No abstract