Linear prediction of point process times and marks

In this paper, we are interested in linear prediction of a particular kind of stochastic process, namely a marked temporal point process. The observations are event times recorded on the real line, with marks attached to each event. We show that in this case, linear prediction generalises straightforwardly from the theory of prediction for stationary stochastic processes. We propose two recursive methods to solve the linear prediction problem and show that these are computationally efficient. The first relies on a Wiener-Hopf integral equation and a corresponding set of differential equations. It is particularly well-adapted to autoregressive processes. In the second method, we develop an innovations algorithm tailored for moving-average processes. Both methods are assessed by an extensive simulation study and applied to a real-world dataset of polling data ahead of the 2022 French elections. In a particular case, we extend the "model independent origin" idea of Jaisson (2015) to the marked Hawkes process through its autoregressive representation. As a corollary, we also improve on existing non-recursive estimators such as that proposed by Bacry and Muzy (2016).