State dependent jump processes: It\^o-Stratonovich interpretations, potential, and transient solutions

TL;DR

This paper develops a comprehensive framework for analyzing state-dependent jump processes, deriving transition probabilities for Itô and Stratonovich interpretations, and providing new transient and steady-state solutions applicable to physical systems with abrupt changes.

Contribution

It introduces novel transition probabilities and solutions for state-dependent jump processes, extending the understanding beyond traditional Brownian motion limits.

Findings

Derived transition probabilities for Itô and Stratonovich jump interpretations.

Presented a new class of transient solutions for exponential jump inputs.

Provided a generic steady state solution using a potential function.

Abstract

The abrupt changes that are ubiquitous in physical and natural systems are often well characterized by shot noise with a state dependent recurrence frequency and jump amplitude. For such state dependent behavior, we derive the transition probability for both the It\^o and Stratonovich jump interpretations, and subsequently use the transition probability to pose a master equation for the jump process. For exponentially distributed inputs, we present a novel class of transient solutions, as well as a generic steady state solution in terms of a potential function and the Pope-Ching formula. These new results allow us to describe state dependent jumps in a double well potential for steady state particle dynamics, as well as transient salinity dynamics forced by state dependent jumps. Both examples showcase a stochastic description that is more general than the limiting case of Brownian motion to which the jump process defaults in the limit of infinitely frequent and small jumps. Accordingly, our analysis may be used to explore a continuum of stochastic behavior from infrequent, large jumps to frequent, small jumps approaching a diffusion process.