Paper: arXiv 2511.07235

Authors: Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang

Abstract

We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical theory with universal approximation theorems, Lipschitz conditions, and explicit network-size bounds, indicating high mathematical density; while it includes a numerical example for American options, it lacks full backtest-ready code or extensive datasets, placing it in the high math/high rigor quadrant.

Research Flowchart

  flowchart TD
  A["Research Goal"] -->|Develop universal approximator for probability models| B["Methodology: Neural Operator Framework"]
  B --> C["Input: FBSDE System"]
  C --> D["Process: Train with Lipschitz & Integrability Constraints"]
  D --> E["Output: Learned Operator"]
  E --> F["Findings: Universal Approximation & Generalization"]
  F -->|Example: American Basket Options| G["Outcome: Optimal Stopping Boundaries for New Strikes"]