Deep Learning for Constrained Utility Maximisation

In this paper we propose algorithms that combine modern machine learning practises with theoretical stochastic control principles to solve a range of control problems with high accuracy, scalability with dimension, and low computational cost. There is a natural crossover between machine learning and stochastic control problems, as they both involve searching for features within a data set. For stochastic control the data set is a state process that is influenced by a controlling process that can be chosen with the aim to optimise some objective function. Often the optimal control is a function of the underlying state process, a so-called Markovian control. This is very similar to deep learning, where the control is chosen as the output of a neural network which takes the dataset as input, and some loss function is minimised. It is in this area of crossover that this paper resides. We propose deep learning methods to solve the utility maximisation problem, as an application of a more general stochastic control solver, in a wide range of markets with arbitrary convex control constraints. Dynamic portfolio optimisation has been extensively studied within the field of mathematical finance, see [19, 27] for exposition. A typical drawback of numerical methods for stochastic control problems is that their complexity increases dramatically with increases in the state and control dimensions.