Optimal Estimation of Generic Dynamics by Path-Dependent Neural Jump ODEs

TL;DR

This paper extends the Neural Jump ODE framework to handle complex, non-Markovian stochastic processes with incomplete data, providing theoretical guarantees and demonstrating superior empirical performance on real-world non-Markovian and filtering tasks.

Contribution

It generalizes Neural Jump ODEs to non-Markovian and discontinuous processes using signature transforms, with proven convergence and improved empirical results.

Findings

Path-dependent NJ-ODE outperforms original NJ-ODE on non-Markovian data.

Theoretical guarantees are extended to generic stochastic processes.

Successful application to stochastic filtering and limit order book data.

Abstract

This paper studies the problem of forecasting general stochastic processes using a path-dependent extension of the Neural Jump ODE (NJ-ODE) framework \citep{herrera2021neural}. While NJ-ODE was the first framework to establish convergence guarantees for the prediction of irregularly observed time series, these results were limited to data stemming from It\^o-diffusions with complete observations, in particular Markov processes, where all coordinates are observed simultaneously. In this work, we generalise these results to generic, possibly non-Markovian or discontinuous, stochastic processes with incomplete observations, by utilising the reconstruction properties of the signature transform. These theoretical results are supported by empirical studies, where it is shown that the path-dependent NJ-ODE outperforms the original NJ-ODE framework in the case of non-Markovian data. Moreover, we show that PD-NJ-ODE can be applied successfully to classical stochastic filtering problems and to limit order book (LOB) data.