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Efficient simulation of Lévy-driven point processes

Part of: Distribution theory - Probability Markov processes Stochastic processes Distribution theory Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  15 November 2019

Yan Qu ,
Angelos Dassios  and
Hongbiao Zhao Open the ORCID record for Hongbiao Zhao [Opens in a new window]
Yan Qu*
Affiliation:
London School of Economics and Political Science
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Hongbiao Zhao*
Affiliation:
Shanghai University of Finance and Economics
*
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK.
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK.
**Postal address: School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China; Shanghai Institute of International Finance and Economics, 777 Guoding Road, Shanghai 200433, China.
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Abstract

In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.

Keywords

Contagion riskportfolio risk managementMonte Carlo simulationexact simulationexact decompositionself-exciting jump process with non-Gaussian Ornstein–Uhlenbeck intensitypoint processbranching processstochastic intensity modelnon-Gaussian Ornstein–Uhlenbeck processgamma processinverse Gaussian subordinatortempered stable subordinator

MSC classification

Primary: 60G55: Point processes 62E15: Exact distribution theory 65C05: Monte Carlo methods
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; L'evy processes 60J75: Jump processes
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 927 - 966
Copyright
© Applied Probability Trust 2019 

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