Time evolution of probability density in stochastic dynamical systems with time delays: The governing equation and its numerical solution

While it is well known that the probability density for conventional stochastic differential equations can be obtained by numerically solving the corresponding Fokker–Planck equations, no feasible approaches have been reported to compute the probability density for stochastic differential equations with time delays. Lacking efficient algorithms to compute the probability density has greatly restricted the application of stochastic differential equations with time delays. A numerical method is developed in this paper to compute the probability density for stochastic differential equations with multiple time delays. It is shown that the proposed numerical method is of first order convergence with respect to time step. Numerical examples are presented to verify and illustrate the proposed method.