Deep Neural Operator Learning for Probabilistic Models
This provides a theoretical foundation for applying neural operators to financial modeling, though it appears incremental as an extension of existing operator learning methods to probabilistic contexts.
The authors developed a deep neural-operator framework for approximating probabilistic models, proving a universal approximation theorem with explicit network-size bounds under Lipschitz conditions. They demonstrated its application to option-pricing problems, showing the learned model could generate optimal stopping boundaries for new strike prices without retraining.
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.