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

Julien Bect (1)

1 E3S - Supélec Sciences des Systèmes

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.

Keywords: continuous-time Markov processes; jump processes; Fokker-Planck-Kolmogorov equation; generalized Fokker-Planck equation; general stochastic hybrid systems; Markov models

Domains: Automatic; Probability [math.PR]; Analysis of PDEs [math.AP]

Julien Bect. A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems. 17th IFAC World Congress, Jul 2008, Séoul, South Korea. pp. 15480-15485. ⟨hal-00215938v2⟩