# Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis

## TL;DR

This paper introduces deep learning algorithms for high-dimensional stochastic control problems, providing theoretical convergence analysis and demonstrating their effectiveness through numerical experiments.

## Contribution

It develops neural network-based algorithms for stochastic control, with rigorous convergence analysis and error estimates, advancing the application of deep learning in control theory.

## Key findings

- Algorithms converge with rates depending on neural network approximation errors.
- Theoretical analysis confirms consistency of control and value function estimates.
- Numerical results show promising performance in high-dimensional problems.

## Abstract

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the dynamic programming recursion by performance or hybrid iteration, and regress now methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks, and of the statistical error when estimating network function, leaving aside the optimization error. Numerical results on various applications are presented in a companion paper (arxiv.org/abs/1812.05916) and illustrate the performance of the proposed algorithms.