Deep-Learning Solution to Portfolio Selection with Serially-Dependent Returns

This paper investigates a deep-learning solution to high-dimensional multi-period portfolio optimization problems with bounding constraints on the control. We propose a deep neural network (DNN) architecture to describe the underlying control process. The DNN consists of $K$ subnetworks, where $K$ is the total number of decision steps. The feedback control function is determined solely by the network parameters. In this way, the multi-period portfolio optimization problem is linked to a training problem of the DNN, that can be efficiently computed by the standard optimization techniques for network training. We offer a sufficient condition for the algorithm to converge for a general utility function and general asset return dynamics including serially-dependent returns. Specifically, under the condition that the global minimum of the DNN training problem is attained, we prove that the algorithm converges with the quadratic utility function when the risky asset returns jointly follow multivariate AR(1) models and/or multivariate GARCH(1,1) models. Numerical examples demonstrate the superior performance of the DNN algorithm in various return dynamics for a high-dimensional portfolio (up to 100 dimensions)