Lévy driven CARMA generalized processes and stochastic partial differential equations

Optimal simulation schemes for Lévy driven stochastic differential equations

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic...

An Euler-Poisson scheme for Lévy driven stochastic differential equations

We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of FerreiroCastilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that th...

Stochastic Partial Differential Equations Driven by Lévy Space - Time White Noise

In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d ≤ 3, then this s...

Asymptotic Problems for Stochastic Processes and Corresponding Partial Differential Equations

Title of dissertation: ASYMPTOTIC PROBLEMS FOR STOCHASTIC PROCESSES AND CORRESPONDING PARTIAL DIFFERENTIAL EQUATIONS Lucas Tcheuko, Doctor of Philosophy, 2014 Dissertation directed by: Professor Mark Freidlin & Professor Leonid Koralov Department of Mathematics We consider asymptotic problems for diffusion processes that rely on large deviations. In Chapter 2, we study the long time behavior (a...

Stochastic Partial Differential Equations