Stability, convergence to equilibrium and simulation of non-linear\n Hawkes Processes with memory kernels given by the sum of Erlang kernels

2016/10/11 by Aline Duarte, Eva Löcherbach, Duarte, Aline +3 · 5 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1610.03300

openalex publication_date 2016/10/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Non-linear Hawkes processes with memory kernels given by the sum of Erlang\nkernels are considered. It is shown that their stability properties can be\nstudied in terms of an associated class of piecewise deterministic Markov\nprocesses, called Markovian cascades of successive memory terms. Explicit\nconditions implying the positive Harris recurrence of these processes are\npresented. The proof is based on integration by parts with respect to the jump\ntimes. A crucial property is the non-degeneracy of the transition semigroup\nwhich is obtained thanks to the invertibility of an associated Vandermonde\nmatrix. For Lipschitz continuous rate functions we also show that these\nMarkovian cascades converge to equilibrium exponentially fast with respect to\nthe Wasserstein distance. Finally, an extension of the classical thinning\nalgorithm is proposed to simulate such Markovian cascades.\n

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