Governing equations for Probability densities of stochastic differential equations with discrete time delays

摘要

The time evolution of probability densities for solutions to stochasticdifferential equations (SDEs) without delay is usually described byFokker-Planck equations, which require the adjoint of the infinitesimalgenerator for the solutions. However, Fokker-Planck equations do not exist forstochastic delay differential equations (SDDEs) because the solutions to SDDEsare not Markov processes and have no corresponding infinitesimal generators. Inthis paper, we address the open question of finding the governing equations forprobability densities of SDDEs with discrete time delays. The governingequation is given in a simple form that facilitates theoretical analysis andnumerical computation. An illustrative example is presented to verify theproposed governing equations.