The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from issues related to statistical bias and sampling efficiency. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.

Feynman-Kac 公式提供了一种根据连续时间马尔可夫过程的期望来理解椭圆偏微分方程解的方法。这种联系允许基于这些马尔可夫过程的样本创建解决方案的数值方案，在某些情况下，这些方案比传统数值方法具有优势。然而，朴素的数值实现存在与统计偏差和采样效率相关的问题。我们提出了离散化出现在 Feynman-Kac 公式中的随机过程的方法，以减少数值方案的偏差。我们还建议使用时间差异学习以比传统蒙特卡罗方法更有效的方式从随机样本中组合信息。