Asymptotic Expansion for the Transition Densities of Stochastic Differential Equations Driven by the Gamma Processes

31 Pages Posted: 6 Apr 2020

See all articles by Fan Jiang

Fan Jiang

Peking University

Xin Zang

Peking University

Jingping Yang

Peking University - School of Mathematical Sciences

Date Written: March 11, 2020

Abstract

In this paper, enlightened by the asymptotic expansion methodology developed by Li (2013b) and Li and Chen (2016), 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 Levy 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 (OU) 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 suffciently small, which shows the efficiency of our approximated method.

Suggested Citation

Jiang, Fan and Zang, Xin and Yang, Jingping, Asymptotic Expansion for the Transition Densities of Stochastic Differential Equations Driven by the Gamma Processes (March 11, 2020). Available at SSRN: https://ssrn.com/abstract=3552353 or http://dx.doi.org/10.2139/ssrn.3552353

Fan Jiang

Peking University ( email )

No. 38 Xueyuan Road
Haidian District
Beijing, Beijing 100871
China

Xin Zang (Contact Author)

Peking University ( email )

School of Mathematical Sciences, Peking University
Beijing, Beijing 100871
China

Jingping Yang

Peking University - School of Mathematical Sciences ( email )

Peking
China

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