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We propose an extension to Hawkes processes by treating the levels ofself-excitation as a stochastic differential equation. Our new point processallows better approximation in application domains where events and intensitiesaccelerate each other with correlated levels of contagion. We generalize arecent algorithm for simulating draws from Hawkes processes whose levels ofexcitation are stochastic processes, and propose a hybrid Markov chain MonteCarlo approach for model fitting. Our sampling procedure scales linearly withthe number of required events and does not require stationarity of the pointprocess. A modular inference procedure consisting of a combination betweenGibbs and Metropolis Hastings steps is put forward. We recover expectationmaximization as a special case. Our general approach is illustrated forcontagion following geometric Brownian motion and exponential Langevindynamics.

Hawkes Processes with Stochastic Excitations