A Neural RDE approach for continuous-time non-Markovian stochastic control problems

TL;DR

This paper introduces a Neural RDE-based framework for solving continuous-time non-Markovian stochastic control problems, enabling efficient simulation and improved accuracy over traditional methods.

Contribution

It develops a novel Neural RDE approach that models control processes with historical dependence, providing theoretical guarantees and superior numerical performance.

Findings

Time-resolution-invariance demonstrated in experiments

Higher accuracy compared to RNN-based methods

Enhanced stability with irregular sampling

Abstract

We propose a novel framework for solving continuous-time non-Markovian stochastic control problems by means of neural rough differential equations (Neural RDEs) introduced in Morrill et al. (2021). Non-Markovianity naturally arises in control problems due to the time delay effects in the system coefficients or the driving noises, which leads to optimal control strategies depending explicitly on the historical trajectories of the system state. By modelling the control process as the solution of a Neural RDE driven by the state process, we show that the control-state joint dynamics are governed by an uncontrolled, augmented Neural RDE, allowing for fast Monte-Carlo estimation of the value function via trajectories simulation and memory-efficient backpropagation. We provide theoretical underpinnings for the proposed algorithmic framework by demonstrating that Neural RDEs serve as universal approximators for functions of random rough paths. Exhaustive numerical experiments on non-Markovian stochastic control problems are presented, which reveal that the proposed framework is time-resolution-invariant and achieves higher accuracy and better stability in irregular sampling compared to existing RNN-based approaches.