Convergence and Stability of the Split-Step<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi mathvariant="bold">θ</mml:mi></mml:mrow></mml:math>-Milstein Method for Stochastic Delay Hopfield Neural Networks

Guo, Qian; Xie, Wenwen; Mitsui, Toshio

doi:10.1155/2013/169214

Cited by 2 publications

(2 citation statements)

References 23 publications

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Section: Introductionmentioning

confidence: 99%

“…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]).…”

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

Cao¹,

Zhang²,

Karniadakis³

2015

SIAM J. Sci. Comput.

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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.

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