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

TL;DR

This paper demonstrates that full history recursive multilevel Picard (MLP) approximations, previously effective for certain equations, also suffer from the curse of dimensionality when applied to Hamilton-Jacobi-Bellman equations in stochastic control.

Contribution

The paper shows that MLP methods do not overcome the curse of dimensionality for Hamilton-Jacobi-Bellman equations with locally Lipschitz nonlinearities, contrasting prior positive results.

Findings

MLP approximations suffer from the curse of dimensionality in this context

The effectiveness of MLP depends on the type of Lipschitz continuity of the nonlinearity

Theoretical proof provided for the limitations of MLP in high-dimensional stochastic control problems

Abstract

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.