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：本文研究了利用路径依赖扩展的神经跳跃常微分方程（NJ-ODE）框架 \citep{herrera2021neural} 预测通用随机过程的问题。虽然 NJ-ODE 是首个为不规则观测时间序列预测建立收敛性保证的框架，但这些结果仅限于源于具有完全观测的 Itô 扩散的数据，特别是马尔可夫过程，其中所有坐标被同时观测。在本工作中，我们通过利用签名变换的重构特性，将这些结果推广至具有不完全观测的、可能为非马尔可夫或非连续的通用随机过程。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形下，路径依赖 NJ-ODE 的表现优于原始 NJ-ODE 框架。此外，我们证明了 PD-NJ-ODE 可以成功应用于经典随机滤波问题以及限价订单簿（LOB）数据。