Abstract
Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to Imai and Kawai [29] and Rosiński (2007) [3], in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis.
Highlights
The class of tempered stable law was first proposed by Tweedie [36]
Stochastic processes with heavy marginal probability tails and still with finite variance have been developed through various different direct truncations of the marginal density function of the stable law
The pioneering work of Mantegan and Stanley [28] is the constitution for the so-called truncated Levy flights in econophysics
Summary
The class of tempered stable law was first proposed by Tweedie [36]. Its associated Levy and OrnsteinUhlenbeck processes were studied in Barndorff-Nielsen and Shephard [4] and Rosinski [34]. Various other related models were studied in, for instance, Gupta and Campanha [15], Matsushita, Rathie and Da Silva [29], and Podobnik et al [32] Those truncated Levy flight models are adequately capable of describing a variety of stylized properties seen in time series of complex systems and have been appealing due to their intuitive approach, while they are no longer Levy processes and are not robust to the observation time scale. This fact ensure practical importance and sample path generation of tempered stable processes
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