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Stochastic Dominance Constrained Optimization with S-shaped Utilities: Poor-Performance-Region Algorithm and Neural Network

Zeyun Hu, Yang Liu

TL;DR

The paper studies static portfolio optimization with non-concave S-shaped utilities under first- and second-order stochastic dominance constraints. It delivers an explicit FSD solution that replaces liquidation boundaries and introduces the Poor-Performance-Region Algorithm (PPRA) to obtain feasible suboptimal SSD solutions, complemented by an algorithm-guided piecewise neural-network framework that leverages PPRA structure to accelerate learning. Numerical experiments in a Black-Scholes market demonstrate the efficacy of PPRA and the NN approach across diverse utilities and benchmarks, including non-concave cases. The work provides practical risk-management insights and a scalable learning-based method for SSD problems, while noting suboptimality guarantees rather than universal optimality in the non-concave SSD setting.

Abstract

We investigate the static portfolio selection problem of S-shaped and non-concave utility maximization under first-order and second-order stochastic dominance (SD) constraints. In many S-shaped utility optimization problems, one should require a liquidation boundary to guarantee the existence of a finite concave envelope function. A first-order SD (FSD) constraint can replace this requirement and provide an alternative for risk management. We explicitly solve the optimal solution under a general S-shaped utility function with a first-order stochastic dominance constraint. However, the second-order SD (SSD) constrained problem under non-concave utilities is difficult to solve analytically due to the invalidity of Sion's maxmin theorem. For this sake, we propose a numerical algorithm to obtain a plausible and sub-optimal solution for general non-concave utilities. The key idea is to detect the poor performance region with respect to the SSD constraints, characterize its structure and modify the distribution on that region to obtain (sub-)optimality. A key financial insight is that the decision maker should follow the SD constraint on the poor performance scenario while conducting the unconstrained optimal strategy otherwise. We provide numerical experiments to show that our algorithm effectively finds a sub-optimal solution in many cases. Finally, we develop an algorithm-guided piecewise-neural-network framework to learn the solution of the SSD problem, which demonstrates accelerated convergence compared to standard neural network approaches.

Stochastic Dominance Constrained Optimization with S-shaped Utilities: Poor-Performance-Region Algorithm and Neural Network

TL;DR

The paper studies static portfolio optimization with non-concave S-shaped utilities under first- and second-order stochastic dominance constraints. It delivers an explicit FSD solution that replaces liquidation boundaries and introduces the Poor-Performance-Region Algorithm (PPRA) to obtain feasible suboptimal SSD solutions, complemented by an algorithm-guided piecewise neural-network framework that leverages PPRA structure to accelerate learning. Numerical experiments in a Black-Scholes market demonstrate the efficacy of PPRA and the NN approach across diverse utilities and benchmarks, including non-concave cases. The work provides practical risk-management insights and a scalable learning-based method for SSD problems, while noting suboptimality guarantees rather than universal optimality in the non-concave SSD setting.

Abstract

We investigate the static portfolio selection problem of S-shaped and non-concave utility maximization under first-order and second-order stochastic dominance (SD) constraints. In many S-shaped utility optimization problems, one should require a liquidation boundary to guarantee the existence of a finite concave envelope function. A first-order SD (FSD) constraint can replace this requirement and provide an alternative for risk management. We explicitly solve the optimal solution under a general S-shaped utility function with a first-order stochastic dominance constraint. However, the second-order SD (SSD) constrained problem under non-concave utilities is difficult to solve analytically due to the invalidity of Sion's maxmin theorem. For this sake, we propose a numerical algorithm to obtain a plausible and sub-optimal solution for general non-concave utilities. The key idea is to detect the poor performance region with respect to the SSD constraints, characterize its structure and modify the distribution on that region to obtain (sub-)optimality. A key financial insight is that the decision maker should follow the SD constraint on the poor performance scenario while conducting the unconstrained optimal strategy otherwise. We provide numerical experiments to show that our algorithm effectively finds a sub-optimal solution in many cases. Finally, we develop an algorithm-guided piecewise-neural-network framework to learn the solution of the SSD problem, which demonstrates accelerated convergence compared to standard neural network approaches.

Paper Structure

This paper contains 15 sections, 2 theorems, 54 equations, 7 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model Settings
  3. FSD Problem and Analytical Solution
  4. SSD Problem
  5. SSD Problem under Non-concavity: Analytical Difficulty
  6. SSD Problem: Numerical Algorithm
  7. Numerical Results
  8. FSD Problem: S-shaped Utility (Theorem \ref{['thm:FSD']})
  9. SSD Problem: Power Utility
  10. SSD Problem: S-shaped Utility
  11. SSD Problem: Various Utilities and Benchmark Quantiles
  12. Algorithm-Guided Piecewise-Neural-Network Framework
  13. Model Setting
  14. Experimental Results
  15. Conclusion

Key Result

Theorem 1

With a general S-shaped utility in Definition def:Sshaped, the optimal solution of Problem prob:main1 is given by where (i) the Lagrange multiplier $\lambda > 0$ is solved from the binding budget constraint $\mathbb{E}[\rho X_{\text{FSD}}^*] = \overline{x}$, and (ii) for any $Q_0\left(1 - F_\rho(\rho)\right) < B$, the (state-dependent) tangent point $C \in (B, \infty)$ is solved from

Figures (7)

  • Figure 1: First-order SD constraint: $\bar{x} = 5$
  • Figure 2: Impacts of $\mu_0$ and $\sigma_0$.
  • Figure 3: Impacts of $\mu_0$ and $\sigma_0$.
  • Figure 4: Setting (a)-(b)-(c).
  • Figure 5: Setting (d)-(e)-(f).
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1: Stochastic dominance
  • Definition 2: General S-shaped utility
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:FSD']}
  • Theorem 2: Theorem 5.10 of WX2021