# Numerical weak approximation of stochastic differential equations driven by Levy processes

Zhang, Changyong.

Numerical weak approximation of stochastic differential equations driven by Levy processes.

Degree: PhD, Applied Mathematics, 2010, University of Southern California

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▼ Levy processes are the simplest generic class of processes having a.s. continuous paths interspersed with jumps of arbitrary sizes occurring at random times, which makes them useful tools in a variety of fields including mathematics, physics, engineering, and finance.; In stochastic analysis, it is frequently necessary to evaluate functionals of the process modeling the system of interest. In general, the law of the process is unknown and a closed-form solution is unrealistic. An alternative possibility is to numerically approximate the functionals by discrete time Monte-Carlo simulation, which is widely applied in practice. The simplest scheme for Monte-Carlo simulation is the weak Euler approximation. In such numerical treatment of stochastic differential equations, it is of theoretical and practical importance to estimate the rate of convergence of the discrete time approximation.; In this dissertation, the weak Euler approximation for stochastic differential equations driven by Levy processes is studied. The model under consideration is in a more general form but with weaker assumptions than those in existence. Hence, it is applicable to a broader range of processes arising from various fields. In order to investigate the convergence of the weak Euler approximation to the process considered, the existence of a unique solution to the corresponding integro-differential equation in Holder space is first proved. It is then identified that the Euler scheme yields a positive weak order of convergence, provided that the coefficients of the stochastic differential equation are Holder-continuous and the test function is continuously differentiable to some positive order. In particular, if the coefficients are slightly more than twice differentiable and the test function has up to the fourth order derivative, then first weak order convergence is guaranteed. Advisors/Committee Members: Mikulevicius, Remigijus(Committee Chair), Lototsky, Sergey (Committee Member), Zhang, Jianfeng (Committee Member), Ghanem, Roger(Committee Member).

Subjects/Keywords: weak Euler approximation; rate of convergence; stochastic differential equations; Levy processes;Nondegenerate; Holder continuity