This paper studies the problem of forecasting general stochastic processes using a path-dependent extension of the Neural Jump ODE (NJ-ODE) framework. 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：本文研究利用神经跳跃ODE（NJ-ODE）框架的路径依赖扩展对一般随机过程进行预测的问题。虽然NJ-ODE是首个为不规则观测时间序列预测建立收敛保证的框架，但这些结果仅适用于源自伊藤扩散（具有完全观测数据）的情形，特别是所有坐标同时被观测的马尔可夫过程。本研究通过利用签名变换的重构性质，将这些结果推广至具有不完全观测的通用随机过程（可能为非马尔可夫或非连续过程）。理论结果得到实验研究的支持，实验表明在非马尔可夫数据场景下，路径依赖NJ-ODE的性能优于原始NJ-ODE框架。此外，我们证明PD-NJ-ODE可成功应用于经典随机滤波问题及限价订单簿（LOB）数据。