Stochastic Properties of Student-Lévy Processes with Applications
The topic of this thesis goes back to two famous mathematicians. William S. Gosset, better known by his pseudonym Student, is the origin of the name of the Student t distribution. Over time, the Student t distribution proved to be extremely useful in many statistical fields, both in theoretical foundations and applications. Paul Lévy studied time-continuous stochastic processes with stationary and independent increments, a generalization of Brownian motion, which is based on the normal distribution. These processes, called Lévy processes, have become well known in stochastics as well as in many financial and physical applications. Any Lévy process has an underlying infinitely divisible distribution characterizing its behavior. Since the Student t distribution is infinitely divisible, there exist Lévy processes having Student t distributed increments, which we call Student-Lévy processes in this thesis. However, not every marginal of the Student-Lévy process is Student t distributed, in contrast to Brownian motion, where all marginals are normally distributed. There exists only one point in time where the Student-Lévy process is Student t distributed at t = 1. If t 6= 1, the distribution of the Student-Lévy process has no closed form. This may explain why the time-continuous Student-Lévy process has received little attention in the literature. Its complicated form in continuous time makes analytical derivations and numerical computations challenging. In this thesis, we contribute to the literature by developing new useful statistical techniques to make the Student-Lévy process accessible. The main goal is to work out an efficient estimation scheme for the t-increments with t 6= 1. The thesis consists of three parts. Due to the complicated nature of the Student- Lévy process standard simulation techniques are not applicable. Thus there is a need for an appropriate simulation routine to generate quasi-time-continuous paths. Series representations for Lévy processes play a prominent role in path generation. However, since the Lévy measure for the Student-Lévy process has a complicated form, series representations are not directly applicable. The first part of the thesis (Chapter 2) therefore proposes simulation algorithms based on series representations. Furthermore, we prove bounds for approximation errors. The second part deals with maximum likelihood parameter estimation. If a Student-Lévy path is observed in high frequency, it is of interest to establish how this data can be used to estimate the unknown parameters. Again, due to the complicated form of the Student-Lévy process, there is no straightforward closed-form maximum likelihood estimator. Chapter 3 thus develops a numerical maximum likelihood estimation procedure. We then study its asymptotic properties and prove that it is asymptotically normal and asymptotically efficient. The third part shows that the Student-Lévy process is of practical interest. In Chapter 4 we apply it to high-frequency financial data and observe that a model based on a Student-Lévy process is a reasonable alternative in finance if other models do not fit well.
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