  We consider the task of learning individual-specific intensities of counting
processes from a set of static variables and irregularly sampled time series.
We introduce a novel modelization approach in which the intensity is the
solution to a controlled differential equation. We first design a neural
estimator by building on neural controlled differential equations. In a second
time, we show that our model can be linearized in the signature space under
sufficient regularity conditions, yielding a signature-based estimator which we
call CoxSig. We provide theoretical learning guarantees for both estimators,
before showcasing the performance of our models on a vast array of simulated
and real-world datasets from finance, predictive maintenance and food supply
chain management.
