Deep Learning for Constrained Utility Maximisation

TL;DR

This paper introduces two deep learning algorithms for solving stochastic control problems with utility maximisation, effectively handling both Markovian and non-Markovian cases with high accuracy and low computational cost.

Contribution

It presents novel deep learning algorithms that leverage duality and BSDE formulations to solve complex stochastic control problems, including non-Markovian cases.

Findings

High accuracy results in utility maximisation problems

Effective handling of non-Markovian stochastic control

Low computational cost demonstrated in numerical experiments

Abstract

This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.