GOVERNING EQUATIONS FOR PROBABILITY DENSITIES OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCRETE TIME DELAYS

摘要

The time evolution of probability densities for solutions to stochastic differential equations (SDEs) without delay is usually described by Fokker-Planck equations, which require the adjoint of the infinitesimal generator for the solutions. However, Fokker-Planck equations do not exist for stochastic delay differential equations (SDDEs) since the solutions to SDDEs are not Markov processes and have no corresponding infinitesimal generators. In this paper, we address the open question of finding governing equations for probability densities of SDDEs with discrete time delays. In the governing equation, densities for SDDEs with discrete time delays are expressed in terms of those for SDEs without delay. The latter have been well studied and can be obtained by solving the corresponding Fokker-Planck equations. The governing equation is given in a simple form that facilitates theoretical analysis and numerical computation. Some example are presented to illustrate the proposed governing equations.