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Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering

Author

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  • Calypso Herrera
  • Florian Krach
  • Josef Teichmann

Abstract

Combinations of neural ODEs with recurrent neural networks (RNN), like GRU-ODE-Bayes or ODE-RNN are well suited to model irregularly observed time series. While those models outperform existing discrete-time approaches, no theoretical guarantees for their predictive capabilities are available. Assuming that the irregularly-sampled time series data originates from a continuous stochastic process, the $L^2$-optimal online prediction is the conditional expectation given the currently available information. We introduce the Neural Jump ODE (NJ-ODE) that provides a data-driven approach to learn, continuously in time, the conditional expectation of a stochastic process. Our approach models the conditional expectation between two observations with a neural ODE and jumps whenever a new observation is made. We define a novel training framework, which allows us to prove theoretical guarantees for the first time. In particular, we show that the output of our model converges to the $L^2$-optimal prediction. This can be interpreted as solution to a special filtering problem. We provide experiments showing that the theoretical results also hold empirically. Moreover, we experimentally show that our model outperforms the baselines in more complex learning tasks and give comparisons on real-world datasets.

Suggested Citation

  • Calypso Herrera & Florian Krach & Josef Teichmann, 2020. "Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering," Papers 2006.04727, arXiv.org, revised Apr 2021.
  • Handle: RePEc:arx:papers:2006.04727
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Patrick Cheridito & John Ery & Mario V. Wüthrich, 2020. "Assessing Asset-Liability Risk with Neural Networks," Risks, MDPI, vol. 8(1), pages 1-17, February.
    3. Calypso Herrera & Florian Krach & Josef Teichmann, 2020. "Local Lipschitz Bounds of Deep Neural Networks," Papers 2004.13135, arXiv.org, revised Feb 2023.
    4. Bernard Lapeyre & Jérôme Lelong, 2020. "Neural network regression for Bermudan option pricing," Working Papers hal-02183587, HAL.
    5. Bernard Lapeyre & J'er^ome Lelong, 2019. "Neural network regression for Bermudan option pricing," Papers 1907.06474, arXiv.org, revised Dec 2020.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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    Cited by:

    1. Luca Galimberti & Anastasis Kratsios & Giulia Livieri, 2022. "Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis," Papers 2210.13300, arXiv.org, revised May 2023.
    2. Kohei Hayashi & Kei Nakagawa, 2022. "Fractional SDE-Net: Generation of Time Series Data with Long-term Memory," Papers 2201.05974, arXiv.org, revised Aug 2022.

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