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

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  • Ashley Davey
  • Harry Zheng

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.

Suggested Citation

  • Ashley Davey & Harry Zheng, 2024. "Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point," Papers 2410.05524, arXiv.org.
    • Handle: RePEc:arx:papers:2410.05524
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    References listed on IDEAS

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    1. Jennifer N. Carpenter, 2000. "Does Option Compensation Increase Managerial Risk Appetite?," Journal of Finance, American Finance Association, vol. 55(5), pages 2311-2331, October.
    2. Ashley Davey & Harry Zheng, 2022. "Deep Learning for Constrained Utility Maximisation," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 661-692, June.
    3. Dong, Yinghui & Zheng, Harry, 2020. "Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan," European Journal of Operational Research, Elsevier, vol. 281(2), pages 341-356.
    4. Alex S. L. Tse & Harry Zheng, 2023. "Speculative trading, prospect theory and transaction costs," Finance and Stochastics, Springer, vol. 27(1), pages 49-96, January.
    5. (**), Hui Wang & Jaksa Cvitanic & (*), Walter Schachermayer, 2001. "Utility maximization in incomplete markets with random endowment," Finance and Stochastics, Springer, vol. 5(2), pages 259-272.
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