# Deep Learning for Constrained Utility Maximisation

arXiv: https://arxiv.org/abs/2008.11757 · HTML: https://mlsanity.com/paper/2008.11757/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Aug 2020.

## Built on by (most cited first, 4 of 4)

- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?](https://mlsanity.com/paper/2206.02016/) (2022)
- [Convergence of Proximal Policy Gradient Method for Problems with Control Dependent Diffusion Coefficients](https://mlsanity.com/paper/2505.18379/) (2025)
- [Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point](https://mlsanity.com/paper/2410.05524/) (2024)

## Abstract (arXiv)

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.