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

Florian Krach, Marc Nübel, Josef Teichmann

TL;DR

This work extends neural Jump ODEs to a broad class of stochastic processes by introducing Path-Dependent NJ-ODE (PD-NJ-ODE) that utilizes the signature transform to capture path-dependence under irregular and incomplete observations. The authors prove convergence of the theoretical loss to the $L^2$-optimal conditional expectation and establish Monte Carlo consistency, enabling reliable online forecasting and uncertainty estimation. They demonstrate that PD-NJ-ODE can recover conditional moments and apply to stochastic filtering, with empirical gains on synthetic processes (jumps, non-Markovian paths) and real data (PhysioNet, limit order books). The results highlight PD-NJ-ODE’s ability to handle non-Markovian dynamics, partial observability, and Jump-diffusion effects, offering a practical tool for forecasting in finance, biology, and engineering. Overall, the paper combines rigorous theory with comprehensive experiments to show PD-NJ-ODE as a versatile framework for path-dependent stochastic forecasting and filtering.

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 Itô-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.

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

TL;DR

This work extends neural Jump ODEs to a broad class of stochastic processes by introducing Path-Dependent NJ-ODE (PD-NJ-ODE) that utilizes the signature transform to capture path-dependence under irregular and incomplete observations. The authors prove convergence of the theoretical loss to the -optimal conditional expectation and establish Monte Carlo consistency, enabling reliable online forecasting and uncertainty estimation. They demonstrate that PD-NJ-ODE can recover conditional moments and apply to stochastic filtering, with empirical gains on synthetic processes (jumps, non-Markovian paths) and real data (PhysioNet, limit order books). The results highlight PD-NJ-ODE’s ability to handle non-Markovian dynamics, partial observability, and Jump-diffusion effects, offering a practical tool for forecasting in finance, biology, and engineering. Overall, the paper combines rigorous theory with comprehensive experiments to show PD-NJ-ODE as a versatile framework for path-dependent stochastic forecasting and filtering.

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 Itô-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.
Paper Structure (68 sections, 16 theorems, 156 equations, 15 figures, 5 tables, 1 algorithm)

This paper contains 68 sections, 16 theorems, 156 equations, 15 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Outline of the Work
  4. Problem Setting
  5. Stochastic Process, Random Observation Times and Observation Mask
  6. Information sigma-algebra
  7. Notation and Assumptions on the Stochastic Process X
  8. Optimal Approximation
  9. Push-forward Measure of Known Information and Implied Metric
  10. Extension of the Neural Jump ODE Model
  11. Recall: Neural Jump ODE
  12. Signature Transform
  13. Path-Dependent Neural Jump ODE
  14. Objective Function
  15. Convergence Guarantees
  16. ...and 53 more sections

Key Result

Proposition 2.5

The optimal ($L^2$-norm minimizing) process in $L^2(\Omega \times \tilde{\Omega}, \mathbb{A} , \mathbb{P} \times \tilde{\mathbb{P}})$ approximating $(X_t)_{t \in [0,T]}$ is given byWhile we gave a pointwise definition, cohen2015stochastic allows to define $\hat{X}$ directly as the optional projectio

Figures (15)

  • Figure 1: Schematic representation of the interpolated observation path $\tilde{X}^{\leq t}$ of a $2$-dimensional process $X$ with discrete and incomplete observations (blue dots) at the observation times $t_i$.
  • Figure 2: Predicted and true conditional expectation on a test sample of the Poisson point process. All upward movements of the true path are jumps, the slope is only due to the discretization time grid.
  • Figure 3: Left: a test sample of a Brownian motion $X$ (top) and its square $X^2$ (bottom) together with the predicted and true conditional expectation. Right: the same test sample of the Brownian motion $X$ with a confidence interval given as $\hat{\mu}_t \pm \hat{\sigma}_t$.
  • Figure 4: NJ-ODE (top-left), NJ-ODE with signature input (top-right), NJ-ODE with recurrent jump network (bottom-left) and PD-NJ-ODE (bottom-right) on a test sample of a fractional Brownian motion with Hurst parameter $H=0.05$.
  • Figure 5: Predicted and true conditional expectation on a test sample of the 2-dimensional correlated Brownian motion (first coordinate plotted on top, second on bottom). The PD-NJ-ODE adjusts its prediction well for both coordinates when observing only one of them.
  • ...and 10 more figures

Theorems & Definitions (63)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • Definition 3.1
  • ...and 53 more