Markov processes with spatial delay: path space characterization, occupation time and properties

TL;DR

This paper characterizes one-dimensional Markov processes with spatial delay using pathwise methods, showing they are semi-martingales and deriving explicit formulas for occupation times and local times.

Contribution

It provides a novel pathwise SDE characterization for Markov processes with delay, extending stochastic calculus tools to these processes.

Findings

Processes with delay are semi-martingales.

Explicit occupation time and local time formulas are derived.

Processes with delay can be analyzed using standard stochastic calculus.

Abstract

In this paper, we study one dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator's domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this article we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob-Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including It\^{o} formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.