Operator Neural Jump ODEs: $L^2$-optimal prediction in function spaces

2026-07-25

AI Summary

This work addresses the limitation of existing Neural Jump ODE (NJ-ODE) methods, which are confined to finite-dimensional processes and thus unsuitable for directly modeling continuous-time stochastic processes in function spaces—such as yield curves. The paper presents the first extension of NJ-ODE to infinite-dimensional $L^2$ function spaces by integrating neural operators with the NJ-ODE framework, yielding an end-to-end operator-valued neural differential equation model. This approach learns the conditional expectation of functional-valued processes directly from discrete, irregularly sampled, and potentially incomplete observations, circumventing information loss due to spatial discretization. Under weaker assumptions than prior work, the authors establish $L^2$-convergence of the model to the true conditional expectation, thereby unifying and generalizing finite-dimensional theory while offering a novel paradigm for modeling high-dimensional dynamic objects like financial surfaces.

Abstract

In this paper, we study the extension of Neural Jump ODEs to infinite-dimensional function spaces. In particular, the underlying process $X$ now takes values in $L^2(Ξ, \mathbb{R}^{d_X})$ instead of $\mathbb{R}^{d_X}$ and the Operator NJ-ODE approximates the optimal predictor of this process by producing a representative of the conditional expectation. The NJ-ODE model is a framework for online learning the optimal prediction of continuous-time stochastic processes, given discrete, possibly irregular and incomplete past observations. In a series of works, this model has been extended to deal with generic path-dependent processes, with observation noise and dependent observations, with long-term predictions, and with input-output systems. However, throughout all of these works, the underlying processes were restricted to be finite-dimensional. In particular, function-valued problems, like yield curve or volatility surface predictions, could only be handled through discretization, which inherently leads to a loss of information. In this work, we build on ideas from Neural Operator methods that allow us to extend the NJ-ODE framework to an infinite-dimensional output process. To prove convergence of the NJ-ODE to the optimal prediction process, we develop a new approximation strategy that also generalizes previous works in the finite-dimensional setting by considerably weakening the assumptions.

Problem

• Neural Jump ODEs
• function spaces
• L^2
• infinite-dimensional processes
• optimal prediction

Innovation

• Operator Neural Jump ODEs
• L² function spaces
• optimal prediction
• Neural Operators
• conditional expectation