Stochastic differential equations - A study on Levy processes
Master thesis
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http://hdl.handle.net/11250/2616013Utgivelsesdato
2015Metadata
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Sammendrag
The main goal of this thesis is to understand, describe, and implement some of the techniques and methods used in a research article by Kohatsu-Higa, Tankov, and Ortiz-Latorre on the subject of optimal simulation schemes for Levy driven stochastic differential equations.
Since the author did not have a background on Levy driven stochastic differential equations, a significant part of the thesis work was to get familiar with the fundamental aspects of the theory. After an introduction to the existing theory, we start describing the research article by introducing the necessary notation. Then an error bound is proved, is used for the optimisation problem later. Later we use the optimisation problem to find optimal approximating Levy measures.
Next, we develop a novel approach for approximating Levy measures, based on approximation by atomic measures, using techniques from linear programming. We call this the LP procedure. Later we show how to make the approximation measures obtained by the LP procedure symmetric when the original measures are symmetric. Later, inspired by the article by Kohatsu-Higa et. al. we demonstrate how to use the LP procedure for local moment matching.
We conclude the text with some numerical analysis, where we discuss sampling methods and do some numerical experiments.