Numerical Methods for Stochastic Delay Differential Equations Via the Wong-Zakai Approximation
Numerical Methods for Stochastic Delay Differential Equations Via the Wong-Zakai Approximation
复制标题
DOI：
10.1137/130942024
复制
发表时间：
2015-02
期刊：
影响因子：
--
通讯作者：
Wanrong Cao;Zhongqiang Zhang;G. Karniadakis
中科院分区：
文献类型：
--
作者：
Wanrong Cao;Zhongqiang Zhang;G. Karniadakis
我们使用Wong-Zakai近似作为中间步骤，推导出随机延迟微分方程的数值格式。通过近似的布朗运动的截断谱展开，然后使用不同的离散时间，我们提出了三个计划：预测校正计划，中点计划，和Milstein的计划。我们证明了预测校正格式在均方意义下以半阶收敛，而Milstein型格式以一阶收敛。数值试验证实了理论预言，并证明了中点格式具有半阶收敛性。数值结果还表明，当扩散系数不存在时滞时，预估校正格式和中点格式在交换噪声下均具有一阶收敛性。
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.