| We develop a computational framework for the study of continuous-time stochastic processes with cadlag sample paths, then effectivize important results from the classical theory of Levy and Feller processes.;Probability theory (including stochastic processes) is based on measure. In Chapter 2, we review computable measure theory, and, as an application to probability, effectivize the Skorokhod representation theorem.;cadlag (right-continuous, left-limited) functions, representing possible sample paths of a stochastic process, form a metric space called Skorokhod space. In Chapter 3, we show that Skorokhod space is a computable metric space, and establish fundamental computable properties of this space.;In Chapter 4, we develop an effective theory of Levy processes. Levy processes are known to have cadlag modifications, and we show that such a modification is computable from a suitable representation of the process. We also show that the Levy-Ito decomposition is computable.;In Chapter 5, we extend the effective theory from Levy processes to the larger class of Feller processes. Feller processes, too, are known to admit cadlag modifications, and we show that such a modification is computable from a suitable representation of that type of process, which is quite different from how we represent a Levy process.;In Chapter 6, we outline two areas for future research: an effective theory of cadlag martingales, and algorithmic randomness for Levy processes. |
