Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

Abstract

We investigate transition law between consecutive observations of Ornstein-Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.

Exact discrete sampling of finite variation tempered stable Ornstein–Uhlenbeck processes

“…It is of practical interest to extend to the infinite variation setting, in which no practical exact simulation method is known yet. Those topics are addressed in subsequent papers [15,16]. It would also be an interesting future research topic to improve Algorithm 5 to a uniformly fast algorithm, in a similar spirit to Algorithm 3.…”

Tempered stable Ornstein– Uhlenbeck processes: A practical view

“…By following Theorem 3.1 in Kawai and Masuda (2012), the Lévy measure of the random variable θΔ 0 e −θΔ+s dZ s is given by…”

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

“…However, focusing on univariate background driving processes, even when the background driving process is not a stable process, the law of the stochastic integral t 2 t 1 e a(t 2 −s) dL s (a is now a constant) may be identifiable and exactly (or almost exactly) simulatable for particular characteristics of the background driving process (see Barndorff-Nielsen and Shephard 2001;Imai and Kawai 2010;Imai and Kawai 2011;Imai and Kawai 2013;Kawai and Masuda 2011;Kawai and Masuda 2012;Samorodnitsky and Taqqu 1994;Zhang and Zhang 2008 and references therein). The approach for CARMA(1,0) however cannot be applied to higher order CARMA processes, with multivariate background driving processes.…”

“…Another special case is the class of Lévy-driven CARMA(1,0) processes, that is, Ornstein-Uhlenbeck processes (Barndorff-Nielsen and Shephard 2001;Sato 1999;Samorodnitsky and Taqqu 1994), in which all random elements are univariate. The univariate law of the stochastic integral may be fully characterized and exactly (or almost exactly) simulatable for some background driving Lévy processes, such as gamma, stable, tempered stable, and inverse Gaussian processes (Imai and Kawai 2010;Imai and Kawai 2011, Imai and Kawai 2013, Kawai and Masuda 2011, Kawai and Masuda 2012, Zhang and Zhang 2008. Apart from those special cases, however, it is difficult to construct exact simulation schemes for general higher order CARMA processes with multivariate background driving Lévy processes.…”