Beyond Bellman: High-Order Generator Regression for Continuous-Time Policy Evaluation

TL;DR

This paper introduces a high-order generator regression method for continuous-time policy evaluation, improving accuracy over traditional Bellman approaches by leveraging multi-step transition data.

Contribution

It develops a novel multi-step generator estimation technique that cancels lower-order errors, providing a more accurate and stable policy evaluation method in continuous time.

Findings

Second-order estimator outperforms Bellman baseline in various benchmarks.

The method remains stable within the theoretically predicted regime.

Higher-order gains are observable under specific decision-frequency regimes.

Abstract

We study finite-horizon continuous-time policy evaluation from discrete closed-loop trajectories under time-inhomogeneous dynamics. The target value surface solves a backward parabolic equation, but the Bellman baseline obtained from one-step recursion is only first-order in the grid width. We estimate the time-dependent generator from multi-step transitions using moment-matching coefficients that cancel lower-order truncation terms, and combine the resulting surrogate with backward regression. The main theory gives an end-to-end decomposition into generator misspecification, projection error, pooling bias, finite-sample error, and start-up error, together with a decision-frequency regime map explaining when higher-order gains should be visible. Across calibration studies, four-scale benchmarks, feature and start-up ablations, and gain-mismatch stress tests, the second-order estimator consistently improves on the Bellman baseline and remains stable in the regime where the theory predicts visible gains. These results position high-order generator regression as an interpretable continuous-time policy-evaluation method with a clear operating region.