# Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems

## TL;DR

This paper improves the theoretical error rate for approximating value functions in controlled diffusion processes using piecewise constant policies from 1/6 to 1/4, aligning with PDE literature standards.

## Contribution

The authors refine existing proofs to establish an improved 1/4 convergence rate, demonstrating optimality and enhancing error estimates for stochastic control approximations.

## Key findings

- Error rate improved from 1/6 to 1/4
- Aligns stochastic control approximation with PDE results
- Provides refined proof techniques for convergence analysis

## Abstract

In N.V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4(2), 1999, it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature.