On Bellman equations for continuous-time policy evaluation I: discretization and approximation

TL;DR

This paper introduces new algorithms for continuous-time policy evaluation that leverage discretization schemes, achieving high accuracy and bounded approximation errors even with infinite horizons.

Contribution

The authors develop a novel class of discretization-based algorithms for continuous-time RL, providing error guarantees and compatibility with function approximation.

Findings

High-order numerical accuracy achieved

Bounded approximation error independent of horizon

Compatible with discrete-time RL with function approximation

Abstract

We study the problem of computing the value function from a discretely-observed trajectory of a continuous-time diffusion process. We develop a new class of algorithms based on easily implementable numerical schemes that are compatible with discrete-time reinforcement learning (RL) with function approximation. We establish high-order numerical accuracy as well as the approximation error guarantees for the proposed approach. In contrast to discrete-time RL problems where the approximation factor depends on the effective horizon, we obtain a bounded approximation factor using the underlying elliptic structures, even if the effective horizon diverges to infinity.