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In this paper, we are interested in linear prediction of a particular kind ofstochastic process, namely a marked temporal point process. The observationsare event times recorded on the real line, with marks attached to each event.We show that in this case, linear prediction extends straightforwardly from thetheory of prediction for stationary stochastic processes. Following classicallines, we derive a Wiener-Hopf-type integral equation to characterise thelinear predictor, extending the "model independent origin" of the Hawkesprocess (Jaisson, 2015) as a corollary. We propose two recursive methods tosolve the linear prediction problem and show that these are computationallyefficient in known cases. The first solves the Wiener-Hopf equation via a setof differential equations. It is particularly well-adapted to autoregressiveprocesses. In the second method, we develop an innovations algorithm tailoredfor moving-average processes. A small simulation study on two typical examplesshows the application of numerical schemes for estimation of a Hawkes processintensity.

Linear prediction of point process times and marks