Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but accurate estimates require huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree $D$. They prove that cubature formulae exist for arbitrary $D$, but explicit constructions are difficult and have only reached $D=7$, too small for practical use. We present ARCANE, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree. It reproduces the state of the art in seconds and reaches $\boldsymbol{D=19}$ within hours on modest hardware. In simulations across multiple different SDEs and error metrics, our cubature formulae robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.

翻译：蒙特卡洛采样是估计随机微分方程解的性质的标准方法，但精确估计需要巨大的样本量。Lyons和Victoir（2004）提出用“求积公式”——即匹配布朗运动“特征矩”至某阶数$D$的确定性加权路径集合——来替代独立采样的布朗驱动路径。他们证明了任意$D$的求积公式均存在，但显式构造困难且目前仅达到$D=7$，这在实际应用中阶数过低。我们提出了ARCANE算法，该算法能高效、自动地构造任意阶数的求积公式。它在数秒内复现了现有最佳结果，并在普通硬件上数小时内达到了$\boldsymbol{D=19}$。在多种不同随机微分方程和误差度量的模拟中，我们的求积公式在相同路径数下，稳健地实现了比蒙特卡洛方法低数个数量级的误差。