  We propose an extension to Hawkes processes by treating the levels of
self-excitation as a stochastic differential equation. Our new point process
allows better approximation in application domains where events and intensities
accelerate each other with correlated levels of contagion. We generalize a
recent algorithm for simulating draws from Hawkes processes whose levels of
excitation are stochastic processes, and propose a hybrid Markov chain Monte
Carlo approach for model fitting. Our sampling procedure scales linearly with
the number of required events and does not require stationarity of the point
process. A modular inference procedure consisting of a combination between
Gibbs and Metropolis Hastings steps is put forward. We recover expectation
maximization as a special case. Our general approach is illustrated for
contagion following geometric Brownian motion and exponential Langevin
dynamics.
