标题： 全历史递归多级Picard逼近对于随机控制问题的哈密顿-雅可比-贝尔曼方程受到维数灾难的影响 显示英文标题

标题： Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem

摘要： 全历史递归多级Picard（MLP）近似已被证明能够克服在半线性热方程数值逼近中的维数灾难，其中非线性项在最大范数下是全局Lipschitz连续的。 在随机控制理论中的Hamilton-Jacobi-Bellman方程中的非线性项，然而，通常在标准欧几里得范数下是（局部）Lipschitz连续的。 在本文中，我们证明了一个令人惊讶的事实，即对于这样一个示例方程的MLP近似会受到维数灾难的影响。 显示英文摘要

摘要： Full history recursive multilevel Picard (MLP) approximations have been proved to overcome the curse of dimensionality in the numerical approximation of semilinear heat equations with nonlinearities which are globally Lipschitz continuous with respect to the maximum-norm. Nonlinearities in Hamilton-Jacobi-Bellman equations in stochastic control theory, however, are often (locally) Lipschitz continuous with respect to the standard Euclidean norm. In this paper we prove the surprising fact that MLP approximations for one such example equation suffer from the curse of dimensionality.