Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumps

Abstract

We develop a distributional decomposition approach for exactly simulating two types of Gamma-driven Ornstein–Uhlenbeck (OU) processes with time-varying marginal distributions: the Gamma-OU process and the OU-Gamma process. The former has finite-activity jumps, and its marginal distribution is asymptotically Gamma; the latter has infinite-activity jumps that are driven by a Gamma process. We prove that the transition distributions of the two processes at any given time can be exactly decomposed into simple elements: at any given time, the former is equal in distribution to the sum of one deterministic trend and one compound Poisson random variable (r.v.); the latter is equal in distribution to the sum of one deterministic trend, one compound Poisson r.v., and one Gamma r.v. The results immediately lead to very efficient algorithms for their exact simulations without numerical inversion. Extensive numerical experiments are reported to demonstrate the accuracy and efficiency of our algorithms.

Keywords:

• Monte Carlo simulation
• exact simulation
• non-Gaussian Ornstein–Uhlenbeck (OU) process
• Gamma-OU process
• OU-Gamma process
• shot-noise process
• finite activity
• infinite activity
• Gamma process

Acknowledgement

The authors would like to thank the reviewer for very helpful and constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 It is fundamentally built upon a Gamma distribution. As an important alternative to the normal distribution, the Gamma distribution is widely used as the building block for a variety of useful stochastic processes, such as the Gamma process and variance Gamma process, see Madan and Seneta (1990) and Madan, Carr, and Chang (1998).

2 The abbreviation “Γ-OU” is borrowed from Barndorff-Nielsen and Shephard (2003a).

3 When working with non-negative random variables, it is common for a lot of researchers to use Laplace transforms rather than characteristic functions, see Bertoin (1998, Chapter III), Bertoin (1999), Sato (1999, Chapter 6), and Norberg (2004). In fact, all of our calculations can be repeated using characteristic functions instead, and we avoid potential complications with complex arguments.

4 For details of the martingale approach, see Dassios and Embrechts (1989), Dassios and Jang (2003), and Dassios and Zhao (2011).

5 Our algorithms are also applicable to any irregularly spaced intervals.

Additional information

Funding

The corresponding author Hongbiao Zhao would like to acknowledge the financial support from the National Natural Science Foundation of China (#71401147) and the research fund provided by Shanghai University of Finance and Economics and Shanghai Institute of International Finance and Economics.