Numerical Methods for Stochastic Delay Differential Equations Via the Wong--Zakai Approximation

Abstract

We use the Wong-Zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations. By approximating the Brownian motion with its truncated spectral expansion and then using different discretizations in time, we present three schemes: a predictor-corrector scheme, a midpoint scheme, and a Milstein-like scheme. We prove that the predictor-corrector scheme converges with order half in the mean-square sense while the Milstein-like scheme converges with order one. Numerical tests confirm the theoretical prediction and demonstrate that the midpoint scheme is of half-order convergence. Numerical results also show that the predictor-corrector and midpoint schemes can be of first-order convergence under commutative noises when there is no delay in the diffusion coefficients. Introduction.Numerical solution of stochastic delay differential equations (SDDEs) has attracted increasing interest recently, as memory effects in the presence of noise are modeled with SDDEs in engineering and finance, e.g., [10,13,34,37,43]. Most numerical methods for SDDEs have focused on the convergence and stability of time-discretization schemes since the early works [38,39]. Currently, several timediscretization schemes have been well studied: the Euler-type schemes (the forward Euler scheme [1,21] and the drift-implicit Euler scheme [16,23,48]), the Milstein schemes [3,14,15,20], the split-step schemes [11,44,49], and also some multistep schemes [4,5,6,7].Although SDDEs can be thought as a special class of stochastic differential equations (SDEs), the extension of numerical methods for SDEs to SDDEs is nontrivial especially since the delay may induce instabilities in the underlying SDDEs while the corresponding SDEs are stable; see, e.g., [16,26]. Also, the formulation of appropriate numerical methods requires a somewhat different calculus because of the delay nature of SDDEs (see the Ito-Taylor expansion, e.g., [20,35]), or anticipative calculus (see, e.g., [15]). Further, the presence of time delay affects the convergence order and computational complexity of numerical methods, as will be shown in section 3.

Mitigation of tipping point transitions by time-delay feedback control

“…Numerical integration of SDDEs in Section VI are carried out by the predictor-corrector scheme developed by Cao et al 37 . For completeness, we present their method here.…”

“…where f : R n → R n , W (t) ∈ R m is a standard multidimensional Wiener process, and σ(X) ∈ R n×m is the diffusion matrix. For the smoothness requirements of the maps f and σ refer to Section 2 of Cao et al 37 . In the context of our paper, we have…”

“…If the maps f and σ are smooth enough, the above scheme has a global mean-square convergence rate of O(h) (see Theorem 2.2 in Ref. 37). The numerical integrations in Section VI are carried out using the scheme (C4) with h = 0.01.…”

The parallel waveform relaxation stochastic Runge–Kutta method for stochastic differential equations

“…Its theory has been widely used in scientific research fields such as stochastic control, electronic engineering, financial economics, stochastic neural network, population dynamics [17] , etc. Many scholars are concerned about the aspect of the numerical solution of SDEs (see, e.g., [4,7,8,15,22,23]). The reason for this is that most SDEs with application background are nonlinear because of their complexity and cannot be solved explicitly, especially some of them are large systems and, consequently, traditional numerical methods take a lot of computer memory and can be highly timeconsuming.…”

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

“…In this section, we verify the convergence rate of the EM method given in (11) for some SFIDEs with weakly singular kernels. In a similar way as [6], we use sample average to approximate the expectation. More precisely, we measure the mean square error of numerical solutions by h,T = 1 5000 We can verify that the functions f i (i = 0, 1, 2) satisfy the hypotheses of Theorem 4.6.…”