A semi-discretization method for delayed stochastic systems

Abstract

Consider ordinary stochastic differential equations with discrete time-delay governed by \[ dX(t) = [ A X(t) + A_\tau X(t-\tau) ] dt + \sigma(X,t) dW(t) \] where \(W(t), t\in R\) is a standard \(d\)-dimensional Wiener process \((W(0)=0\), \(E [W(t)]=0\), \(E [W(t)]^2=| t| \)), \(A\) and \(A_\tau\) are nonrandom real-valued matrices, and \(\tau > 0\) denotes a nonrandom time-delay. Suppose that one discretizes this type of equation with linear diffusion terms \(\sigma\) by Euler-type methods along nonrandom partitions of time-intervals \([0,T]\) with \(\tau=n \Delta t\), where \(\Delta t\) is the nonrandom mesh size and \(n \in N\) is the length of subintervals in each iteration. The authors discuss an example, and compare the results with known analytical solution with additive or multiplicative noises. The results of direct Itô integration and an exact drift-mapping method are compared with respect to the behavior of their second moments. An improvement of the accuracy by the semi-disretization method using the exact drift mapping is established. Their results are supported by some simulations. A related control problem is briefly discussed as well. This paper extends a deterministic semi-discretization method as presented in [\textit{T. Insperger} and \textit{G. Stepan}, Int. J. Numer. Methods Eng. 55, No. 5, 503--518 (2002; Zbl 1032.34071)] to the stochastic case without delay terms in the diffusion part.