  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.
