Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.

中文翻译：

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无限维中的迭代随机积分：近似和误差估计

不具有可交换噪声的随机偏微分方程的高阶数值格式需要对迭代随机积分进行仿真。在这项工作中，我们扩展了Kloeden等人的算法。（Stoch Anal Appl 10（4）：431–441，1992. https://doi.org/10.1080/07362999208809281）和Wiktorsson（Ann Appl Probab 11（2）：470-487，2001.https：// doi （org / 10.1214 / aoap / 1015345301）近似将涉及有限维随机常微分方程数值方案的两次迭代随机积分近似为无限维设置。这些方法为没有换向噪声的SPDE的新型近似方案扫清了道路。准确地讲，我们分析了两种算法以对无限维近似两次迭代积分Q -Wiener在跟踪类运营商的情况下处理Q给出的增量Q -Wiener过程。推导并讨论了两种方法的均方误差估计。与此相反的有限维设置，这被包含作为一个特殊的情况下，最佳近似算法不能被唯一地确定，而是依赖于协方差运算符Q。由于随机过程是无穷大的，所以会产生这种差异。