Hawkes stochastic processes: theory and simulation

We assemble paper-following notes for linear Hawkes processes, emphasizing the bridge between (i) structural theory (compensators, cluster/branching representations, stationarity and convergence to equilibrium), and (ii) computation (Markovian reductions for exponential kernels and exact event-time simulation). Our starting point is the compensator- based framework of Bielecki et al [1]., which provides a broad kernel language for marked and multivariate Hawkes models. To obtain a constructive immigration–birth picture with general explicit generation-by-generation mechanics, we then follow Fierro et al. [6] and the subsequent analysis of Mehrdad–Zhu [5] for Hawkes processes with generation-dependent excitation functions. Under a subcriticality condition, an equilibrium (stationary) version exists and the forward-started process converges to equilibrium, yielding ergodic/LLN-type consequences for long-run event counts. We then present a well-studied special class of Hawkes process with (fixed) exponential (time-decay) kernels that leads to a finite-dimensional, piecewise- deterministic Markov state for the intensity (or memory variables), which admits an exact simulation mechanism (using clever technique to avoid numerical inverse by Dassios–Zhao [7] ) as a finite-horizon algorithm with no discretisation bias conditional on an initial state