Deep Neural Operator Learning for Probabilistic Models

TL;DR

This paper introduces a deep neural-operator framework capable of approximating a wide range of probabilistic models, with theoretical guarantees and practical applications in option pricing.

Contribution

It develops a universal approximation theorem for neural operators applied to probabilistic models, including FBSDEs and PDEs, with explicit network size bounds and real-world financial applications.

Findings

The framework accurately models European and American options.

It produces optimal stopping boundaries without retraining.

The approach is validated through a numerical example.

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.