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

Abstract

Abstract. Exact yet simple simulation algorithms are developed for a wide class of Ornstein-Uhlenbeck processes with tempered stable stationary distribution of finite variation with the help of their exact transition probability between consecutive time points. Random elements involved can be divided into independent tempered stable and compound Poisson distributions, each of which can be simulated in the exact sense through acceptance-rejection sampling, respectively, with stable and gamma proposal distributions. We discuss various alternative simulation methods within our algorithms on the basis of acceptance rate in acceptance-rejection sampling for both high-and low-frequency sampling. Numerical results illustrate their advantage relative to the existing approximative simulation method based on infinite shot noise series representation.

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

“…(In principle, the stepsize does not need to be equidistant and can be set different positive values for different steps.) The difference from the finite variation setting [18,9] lies in the integrability of Lévy density of the transition law around the origin. As a consequence, the Lévy density has to be decomposed twice to extract all infinite activity part, while only once in the finite variation case.…”

“…For more details about related acceptance-rejection sampling methods, see Baeumer and Meerschaert [1], Devroye [6] and Kawai and Masuda [9].…”

“…It is known that exact, yet simple, simulation methods are available for both random elements. The particular setting of inverse-Gaussian OU processes was studied in Zhang and Zhang [17], while the general setting in [9]. Also, it was shown in Zhang and Zhang [18] that the transition law is selfdecomposable when the stability index is no less than 1/2.…”

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.…”