Thomas Norton with Wojbor A. Woyczyński
Stochastic Differential Equations Driven by Compound Poisson and Lévy Processes
Stochastic differential equations (SDEs) provide a rigorous setting for investigating the dynamics of systems subject to “noise,” by which we mean source terms whose behavior is not deterministic, but governed by a probability distribution. The noise can have quite irregular behavior (such as continuous but nowhere differentiable paths), giving rise to a rich theory of stochastic integration that allows for multiple interpretations of the same SDE. Most commonly, the noise is assumed to follow the same probability distribution as Brownian Motion, a pure continuous-time random walk, but the theory is further enriched by considering sources of noise whose amplitude is governed by alternative distributions, such as a compound Poisson process or, more generally, a Lévy process. The student’s summer research focused on exploring the consequences of differing interpretations of SDEs driven by Brownian Motion noise, with applications to biological networks. This research compared the two most common interpretations of this class of SDEs, the Itô and Stratonovich interpretations. In this project, the student will explore the theory of SDEs driven by Lévy processes, with particular attention to stochastic processes on graphs (the dynamics of a population of particles moving on a lattice would be an example) and with the goal of investigating (at least numerically) an analogue of the Stratonovich SDE interpretation for SDEs driven by Lévy processes.