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

arXiv: https://arxiv.org/abs/2410.05524 · HTML: https://mlsanity.com/paper/2410.05524/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 21 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 5 of 5)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Deep Learning for Constrained Utility Maximisation](https://mlsanity.com/paper/2008.11757/) (2020)
- [On the dual problem of utility maximization in incomplete markets](https://mlsanity.com/paper/1510.08323/) (2015)

## Abstract (arXiv)

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.