论文标题
多级PICARD近似值，用于高维向前靠背随机微分方程
Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations
论文作者
论文摘要
向后的随机微分方程（BSDE）出现在数值应用中。经典的近似方法受维度的诅咒，基于深度学习的近似方法尚不融合到BSDE解决方案。最近，Hutzenthaler等。 （ARXIV：2108.10602）为BSDE引入了一种新的近似方法，其正向扩散是布朗运动，并证明该方法基本上是最佳速率，而不会受到维数的诅咒。本文的中心对象是将此结果扩展到一般的正向扩散。主要的挑战是我们需要在时间空间Hölder规范中建立收敛性，因为通常不能准确地采样正向扩散。
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE solution. Recently, Hutzenthaler et al. (arXiv:2108.10602) introduced a new approximation method for BSDEs whose forward diffusion is Brownian motion and proved that this method converges with essentially optimal rate without suffering from the curse of dimensionality. The central object of this article is to extend this result to general forward diffusions. The main challenge is that we need to establish convergence in temporal-spatial Hölder norms since the forward diffusion cannot be sampled exactly in general.
