A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems

TL;DR

This paper introduces a comprehensive formulation of the Fokker-Planck-Kolmogorov equation for stochastic hybrid systems, unifying previous models and aiding practitioners in deriving the probability evolution for complex systems.

Contribution

It presents a unified derivation of the FPK equation for GSHS using mean jump intensity, encompassing spontaneous and forced jumps, and unifies existing formulations.

Findings

Unified FPK equation for GSHS derived

Applicable to systems with spontaneous and forced jumps

Provides a practical tool for probability law evolution

Abstract

A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.