Paper: arXiv 2603.06563

Authors: Chang Chen, Duy-Minh Dang

Abstract

We develop a neural-network framework for multi-period risk–reward stochastic control problems with constrained two-step feedback policies that may be discontinuous in the state. We allow a broad class of objectives built on a finite-dimensional performance vector, including terminal and path-dependent statistics, with risk functionals admitting auxiliary-variable optimization representations (e.g.\ Conditional Value-at-Risk and buffered probability of exceedance) and optional moment dependence. Our approach parametrizes the two-step policy using two coupled feedforward networks with constraint-enforcing output layers, reducing the constrained control problem to unconstrained training over network parameters. Under mild regularity conditions, we prove that the empirical optimum of the NN-parametrized objective converges in probability to the true optimal value as network capacity and training sample size increase. The proof is modular, separating policy approximation, propagation through the controlled recursion, and preservation under the scalarized risk–reward objective. Numerical experiments confirm the predicted convergence-in-probability behavior, show close agreement between learned and reference control heat maps, and demonstrate out-of-sample robustness on a large independent scenario set.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework and theoretical proof for the convergence of neural network policies in stochastic control problems with discontinuous feedback rules. While the theoretical machinery is highly sophisticated, the empirical evaluation is primary illustrative and lacks extensive multi-dataset validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimize Multi-Period Risk-Reward Stochastic Control via Neural Networks"] --> B["Inputs: Stochastic Dynamics, Path Statistics & Risk Functionals (e.g., CVaR, bPOE)"]
    B --> C["Methodology: Parametrize 2-Step Constrained Feedback Policies using Coupled NNs"]
    C --> D["Computational Process: Unconstrained Empirical Risk Minimization over NN Parameters"]
    D --> E["Theoretical Proof: Modular Proof of Convergence in Probability of Empirical Optimum"]
    D --> F["Numerical Validation: Synthetic Simulations & Large Out-of-Sample Scenario Sets"]
    E & F --> G["Outcomes: Verified Convergence, Heat Map Accuracy & Robust Control"]