Abstract
Various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation, in particular cases of very small scale parameter, which correspond to increments of a tempered stable Lévy process with a very short stepsize. Methods under consideration are based on acceptance–rejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. With a given computing budget, an approximative acceptance–rejection sampling technique Baeumer and Meerschaert (2009) [11] is both most efficient and handiest in the case of very small scale parameter and moreover, any desired level of accuracy may be attained with a small amount of additional computing effort.
Highlights
The class of tempered stable distributions was first proposed by Tweedie [21]
We have investigated various, existing and new, simulation methods of the tempered stable law with stability index greater than one, with primal interest in simulation of increments X(∆) over a very short stepsize ∆ > 0: a suitable setting for approximation of stochastic differential equations through the Euler scheme
The model-free acceptance-rejection sampling method of [8] provides an exact simulation method, in principle, but requires a lot of computing effort for computing density values. This method exhibits quite low acceptance rate when ∆ is small and the stability index α is close to 2, that is, when the target is close to Gaussian
Summary
The class of tempered stable distributions was first proposed by Tweedie [21]. Several featuring properties of tempered stable distributions and processes were revealed by Rosinski [18], such as a stable-like behavior over short intervals, the absolute continuity with respect to its short-range limiting stable process, an aggregational Gaussianity and an infinite shot noise series representation in closed form. To the best of our knowledge, on the other hand, there exist no practically exact simulation methods for tempered stable random variates with stability index greater than one. We will investigate various possible, existing and new, simulation techniques and discuss their advantages, limitations and trade-off between approximation error and computing effort, with a full view towards practical implementation. The infinite shot noise series only provides an exact simulation method for tempered stable Levy processes since it simulates complete information of sample paths, that is, size, direction and timing of every single jump. From a computational point of view, the form of infinite sum has raised important issues of finite truncation to be addressed. (See Imai and Kawai [10].)
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