Stochastic calculus and sample path estimation for jump processes

TL;DR

This paper develops stochastic calculus tools for jump processes, including Markov chains, providing sample path estimates and applications such as ODE approximation and stochastic logistic models.

Contribution

It introduces a framework for stochastic calculus in jump processes, extending Itô calculus to processes with jumps and deriving sample path estimates.

Findings

Sample path estimates for jump processes

Applications to stochastic logistic models

Framework applicable to Markov and non-Markov processes

Abstract

We describe stochastic calculus in the context of processes that are driven by an adapted point process of locally finite intensity and are differentiable between jumps. This includes Markov chains as well as non-Markov processes. By analogy with It\^o processes we define the drift and diffusivity, which we then use to describe a general sample path estimate. We then give several examples, including ODE approximation, processes with linear drift, first passage times, and an application to the stochastic logistic model.