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Exact simulation of Ornstein–Uhlenbeck tempered stable processes

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

Published online by Cambridge University Press:  23 June 2021

Yan Qu ,
Angelos Dassios  and
Hongbiao Zhao Open the ORCID record for Hongbiao Zhao [Opens in a new window]
Yan Qu*
Affiliation:
Peking University
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Hongbiao Zhao*
Affiliation:
Shanghai University of Finance and Economics
*
*Postal address: School of Mathematical Sciences, Peking University, Beijing 100871, China.
**Postal address: Department of Statistics, London School of Economics, Houghton Street, LondonWC2A 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. Email address: [email protected].
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Abstract

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

Keywords

Monte Carlo simulationexact simulationnon-Gaussian Ornstein–Uhlenbeck processtempered stable subordinatortempered stable OU processOU tempered stable process

MSC classification

Primary: 65C05: Monte Carlo methods 62E15: Exact distribution theory 60G51: Processes with independent increments; L'evy processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 68U20: Simulation 60H35: Computational methods for stochastic equations
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 347 - 371
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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