  The Path-Dependent Neural Jump Ordinary Differential Equation (PD-NJ-ODE) is
a model for predicting continuous-time stochastic processes with irregular and
incomplete observations. In particular, the method learns optimal forecasts
given irregularly sampled time series of incomplete past observations. So far
the process itself and the coordinate-wise observation times were assumed to be
independent and observations were assumed to be noiseless. In this work we
discuss two extensions to lift these restrictions and provide theoretical
guarantees as well as empirical examples for them. In particular, we can lift
the assumption of independence by extending the theory to much more realistic
settings of conditional independence without any need to change the algorithm.
Moreover, we introduce a new loss function, which allows us to deal with noisy
observations and explain why the previously used loss function did not lead to
a consistent estimator.
