Abstract In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.

中文翻译：

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随机微分方程无偏模拟的二阶概率参数法

摘要 在本文中，遵循偏差-方差权衡哲学的范式，我们基于具有随机分区的 Euler-Maruyama 方案推导出二阶参数展开，目的是构建多维随机微分方程的无偏模拟方法. 这些公式导致了可以轻松并行化的蒙特卡罗模拟方法。与一阶方法相比，这里提出的二阶方法需要系数的进一步规律性，但即使在对分区使用泊松采样时也能实现有限矩，与 Andersson 和 Kohatsu-Higa（2017）相反。而且，使用指数缩放技术实现了一种无偏模拟方法，它类似于空间重要性采样技术，显着提高了所提出方法的效率。还提供了如何导出更高阶展开的提示。