A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems (extended version)

TL;DR

This paper introduces a comprehensive formulation of the Fokker-Planck-Kolmogorov equation within the framework of Generalized Stochastic Hybrid Systems, unifying previous approaches and aiding in modeling the evolution of hybrid system probabilities.

Contribution

It provides a unified derivation of the FPK equation for GSHS using mean jump intensity, covering all known cases and facilitating future applications.

Findings

Unified FPK equation for GSHS derived

Incorporates both spontaneous and forced jumps

Provides a practical tool for hybrid system analysis

Abstract

This paper has been withdrawn from the arXiv. It is now published by Elsevier in Nonlinear Analysis: Hybrid Systems, see http://dx.doi.org/10.1016/j.nahs.2009.07.008 . 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.