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.

翻译：本文研究了利用神经跳跃常微分方程（NJ-ODE）框架\citep{herrera2021neural}的路径依赖扩展来预测一般随机过程的问题。尽管NJ-ODE是首个为不规则观测时间序列预测建立收敛保证的框架，但这些结果仅限于来自完整观测伊藤扩散的数据，特别是马尔可夫过程，其中所有坐标同时被观测。在本工作中，我们利用签名变换的重构性质，将这些结果推广到具有不完全观测的通用、可能非马尔可夫或不连续的随机过程。这些理论结果得到了实证研究的支持，研究表明在非马尔可夫数据情况下，路径依赖的NJ-ODE优于原始NJ-ODE框架。此外，我们证明了PD-NJ-ODE可成功应用于经典随机滤波问题以及限价订单簿数据。