Tempered stable Ornstein– Uhlenbeck processes: A practical view

ABSTRACT

We study the one-dimensional Ornstein–Uhlenbeck (OU) processes with marginal law given by tempered stable and tempered infinitely divisible distributions. We investigate the transition law between consecutive observations of these processes and evaluate the characteristic function of integrated tempered OU processes with a view toward practical applications. We then analyze how to draw a random sample from this class of processes by considering both the classical inverse transform algorithm and an acceptance–rejection method based on simulating a stable random sample. Using a maximum likelihood estimation method based on the fast Fourier transform, we empirically assess the simulation algorithm performance.

Keywords:

• Acceptance–rejection sampling
• Maximum likelihood estimation
• Ornstein–Uhlenbeck processes
• Tempered infinitely divisible distributions
• Tempered stable distributions

MATHEMATICS SUBJECT CLASSIFICATION:

• C02
• C46
• 33C05
• 33C15
• 60E07
• 60G51
• 91G60

Acknowledgments

The authors are grateful to Piotr Jelonek and Massimo Sbracia, participants at the 6th International conference on Computational and Financial Econometrics, and two anonymous referees for their comments and suggestions.

Statement of interest

Michele Leonardo Bianchi acknowledges that the views expressed in the article are those of the author and do not involve the responsibility of the Bank of Italy.