Latin In this thesis, the tempered stable distributions and processes are introduced. Then, various simulation methods for tempered stable random variables with stability index greater than one and skewness parameter equals to one are investigated. Special attention will be to the case of very small scale parameters, which corresponds to increments of tempered stable Lévy processes with a very short stepsize. This was motivated by its application for approximating Lévy driven stochastic differential equations such as the Euler scheme, for which we have to have many small time independent increments of the Lévy process. Tempered stable random variables do not have density function in closed form. Due to this fact, their simulation is a very difficult task. However, these variables can be simulated by using acceptance-rejection sampling method by finding some relations between their density function and other density functions which can be simulated easily. According to this approach, two methods which are called “exact sampling using density function” and “approximative sampling with stable proposal distribution” are obtained. Another approach is a suitable decomposition of their Lévy measures into two Lévy measures. With this decomposition, a tempered stable random variable decomposes into a constant and two random variables which are named small and large jump components. Therefore, with simulating these two components, a tempered stable random variable will be simulated. Large jump component can simulate exactly with a compound Poisson random variable, while several approximations , such as Gaussian approximation, will be presented for simulating small jump component. Finally, numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. From a practical point of view, a suitable simulation method makes a good balance between computational load and approximation error, together with implementation ease. In the other hand, results can be summarized as follows: · Exact sampling using density function method provides an exact simulation method, but requires a lot of computing efforts for computing density values. This method exhibits quite low acceptance rate for the case of small time of the increments for tempered stable Lévy processes, when the stability index is close to two. · The acceptance–rejection sampling is very handy with both very small computing time and approximation error. Obtaining an optimal value of the tuning parameter, which appears in this method, is relatively straightforward. · Decomposition into small and large component provides a different method for generating approximation values. It is shown that in this framework, the approximation error can be made very small by either simulating more large jump component or simulating more mass of the small jump component as compound Poisson random variables, while an extraordinary large amount of computing efforts are additionally required for an improvement in the approximation error, which may be a drawback in practice when thousands of tempered stable variables with small scales are needed. For conclusion, the acceptance–rejection sampling seems to be most efficient and handiest among all methods which have been discussed in this thesis, based on numerical assessment of accuracy for simulation of increments in small time. MSC: 65C10, 68U20, 60E07, 60B10 Latin keywords: Stable distribution, Tempered stable distribution, Characteristic function, Infinitely divisible distribution, Acceptance–rejection sampling method, Compound Poisson distribution, Lévy process
