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A dynamic contagion process

Part of: Mathematical finance Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Angelos Dassios  and
Hongbiao Zhao
Angelos Dassios*
Affiliation:
London School of Economics
Hongbiao Zhao*
Affiliation:
London School of Economics
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK.
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Abstract

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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.

Keywords

Dynamic contagion processCox process with shot noise intensitypiecewise-deterministic Markov processcluster point processself-exciting point processHawkes process

MSC classification

Primary: 60J75: Jump processes
Secondary: 60G55: Point processes 60G44: Martingales with continuous parameter 91G40: Credit risk

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 814 - 846
Copyright
Copyright © Applied Probability Trust 2011 

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