Notice
A Palm Space Approach to Non-Linear Hawkes Processes
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Descriptif
A Hawkes process on the real line is a point process whose intensity function at time is a functional of its past activity before time. It is defined by its activation and memory functions. By using the classical correspondence between a stationary point process and its Palm measure, for stationary Hawkes point processes we establish a characterization of the Palm measure in terms of an invariant distribution of a Markovian kernel. We prove that if the activation function is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded activation function is therefore relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, plays also an important role. The classical linear case of Hawkes and Oakes is revisited at this occasion. Limit results for some explosive Hawkes point processes, i.e. when there does not exist a stationary version, are also obtained.
Joint work with Gaëtan Vignoud (INRIA and Dioxycle).
Thème
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