ARCANE：可扩展的高次立方公式用于无需蒙特卡洛误差的SDE模拟
ARCANE: Scalable high-degree cubature formulae for simulating SDEs without Monte Carlo error
AI总结：
ARCANE算法通过高效构造高次立方公式，实现了无需蒙特卡洛误差的SDE模拟，达到D=19的高精度表现。
摘要：
蒙特卡洛采样是估计随机微分方程（SDEs）解性质的标准方法，但准确估计需要巨大的样本量。Lyons和Victoir（2004）提出用“立方公式”代替独立采样的布朗路径，即确定性的加权路径集，能够匹配布朗路径的“特征矩”到某个程度D。他们证明了立方公式在任意D下都存在，但显式构造困难，目前仅达到D=7，不足以用于实际应用。我们提出了ARCANE算法，该算法能够高效自动构造任意程度的立方公式。它在数秒内达到当前最先进的水平，并在普通硬件上数小时内达到D=19。在多个不同SDE和误差度量的模拟中，我们的立方公式在相同路径数量下，误差比蒙特卡洛方法小多个数量级。
英文摘要：
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.
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