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ASYMPTOTIC EXPANSION FOR THE TRANSITION DENSITIES OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE GAMMA PROCESSES

Published online by Cambridge University Press:  17 September 2020

Fan Jiang ,
Xin Zang Open the ORCID record for Xin Zang [Opens in a new window]  and
Jingping Yang
Fan Jiang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn
Xin Zang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn
Jingping Yang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn
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Abstract

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

Keywords

asymptotic expansiongamma processstochastic differential equationtransition density
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 401 - 432
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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