Deep neural networks algorithms for stochastic control problems on finite horizon, Part 2: numerical applications

This paper presents several numerical applications of deep learning-based algorithms that have been analyzed in [11]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from [6] and on quadratic Backward Stochastic Differential equations as in [5]. We also provide numerical results for an option hedging problem in finance, and energy storage problems arising in the valuation of gas storage and in microgrid management.

Deep neural networks algorithms for stochastic control problems on finite horizon, part I: convergence analysis

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming (DP). Differently from the classical approximate DP approach, 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 DP recursion by performance or hybrid iteration, and regress now or later/quantization 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. Numerical results on various applications are presented in a companion paper [2] and illustrate the performance of our algorithms.