Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point

TL;DR

This paper introduces deep learning techniques to solve a complex portfolio optimization problem involving an S-shaped utility function and a random benchmark, addressing non-concavity in both complete and incomplete markets.

Contribution

It develops novel deep learning and duality methods to solve the Hamilton-Jacobi-Bellman and adjoint equations for non-concave utility maximization problems.

Findings

Deep learning methods accurately solve the non-concave utility maximization problem.

Comparison shows differences between solutions for original and concavified utilities.

Numerical results validate the effectiveness of the proposed algorithms.

Abstract

We consider the portfolio optimisation problem where the terminal function is an S-shaped utility applied at the difference between the wealth and a random benchmark process. We develop several numerical methods for solving the problem using deep learning and duality methods. We use deep learning methods to solve the associated Hamilton-Jacobi-Bellman equation for both the primal and dual problems, and the adjoint equation arising from the stochastic maximum principle. We compare the solution of this non-concave problem to that of concavified utility, a random function depending on the benchmark, in both complete and incomplete markets. We give some numerical results for power and log utilities to show the accuracy of the suggested algorithms.