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.
Highlights
The class of non-Gaussian Ornstein-Uhlenbeck (OU, in short) processes is closely related to the selfdecomposability of the infinitely divisible distribution
In terms of mathematical tractability, the transition law between consecutive observations can be written in the exact sense as a convolution of one compound Poisson and one tempered stable distributions
Due to its distributional flexibility and the positivity of sample paths, they have been used in financial economics and mathematical finance
Summary
Masuda [10] and references therein.) due to the growing practical interest, many authors have proposed statistical inference methods for non-Gaussian OU processes. (See, for example, Brockwell et al [4], Jongbloed et al [8] and Sun and Zhang [15].). We study the class of TS-OU processes of infinite variation, that is, OU processes with a tempered stable stationary distribution with stability index in (1, 2). The bilateral framework can produce more distributional flexibility through combinations of positive and negative jump components in terms of, for example, stability index and even finite and infinite variations. They may widen the applicability of OU processes in a variety of fields. We provide numerical results to illustrate the effectiveness of our exact transition law and proposed simulation techniques in sample paths generation, relative to the existing approximative method with infinite shot noise series representation of tempered stable.
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