Numerical Approximation of Stochastic Differential Equations Driven by Levy Motion with Infinitely Many Jumps

Jum, Ernest.

Numerical Approximation of Stochastic Differential Equations Driven by Levy Motion with Infinitely Many Jumps.

Degree: 2015, University of Tennessee – Knoxville

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▼ In this dissertation, we consider the problem of simulation of stochastic differential equations driven by pure jump Levyprocesses with infinite jump activity. Examples include, the class of stochastic differential equations driven by stable and tempered stable Levy processes, which are suited for modeling of a wide range of heavy tail phenomena. We replace the small jump part of the driving Levy process by a suitable Brownian motion, as proposed by Asmussen and Rosinski, which results in a jump-diffusion equation. We obtain Lp [the space of measurable functions with a finite p-norm], for p greater than or equal to 2, and weak error estimates for the error resulting from this step. Combining this with numerical schemes for jump diffusion equations, we provide a good approximation method for the original stochastic differential equation that can also be implemented numerically. We complement these results with concrete error estimates and simulation.

Subjects/Keywords: stochastic differential equation; numerical approximation; Levy motion; infinitely many jumps;Probability