Computing the Density Function for a Nonlinear Stochastic Delay System

Abstract

We develop a numerical method to compute the density of a specific nonlinear stochastic delay system, with no sampling. This system arises as a switch-type control model for human balance. Numerical tests against the Euler-Maruyama method show that our method is capable of computing accurate solutions. In particular, the method captures the covariance of the solution at the present and delayed times. This is accomplished through the time-evolution of a Gaussian approximation of the joint density at the present and delayed times. Issues of circularity prevent the numerical solution of the Fokker-Planck equation for stochastic delay systems. Our method bypasses these issues and offers one of the first deterministic algorithms to compute the density of a nonlinear stochastic delay system.