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Fokker-Planck-Kolmogorov equation for stochastic differential equations with boundary hitting resets

Research paper by Julien Bect, Hana Baili, Gilles Fleury

Indexed on: 28 Apr '05Published on: 28 Apr '05Published in: Mathematics - Probability



Abstract

We consider a Markov process on a Riemannian manifold, which solves a stochastic differential equation in the interior of the manifold and jumps according to a deterministic reset map when it reaches the boundary. We derive a partial differential equation for the probability density function, involving a non-local boundary condition which accounts for the jumping behaviour of the process. This is a generalisation of the usual Fokker-Planck-Kolmogorov equation for diffusion processes. The result is illustrated with an example in the field of stochastic hybrid systems.