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.

翻译：全历史递归多层Picard（MLP）近似方法已被证明能够克服具有最大范数全局Lipschitz连续非线性项的半线性热方程数值近似中的维度灾难。然而，随机控制理论中Hamilton-Jacobi-Bellman方程的非线性项通常（局部）关于标准欧几里得范数是Lipschitz连续的。本文证明了一个令人惊讶的事实：针对此类方程示例的MLP近似方法仍然遭受维度灾难的困扰。