This paper studies the problem of forecasting general stochastic processes
using an 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 limit
order book (LOB) data