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

Autor(es)

Duarte, Aline ^[1] ; Locherbach, Eva ^[2] ; Ost, Guilherme ^[3]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Sao Paulo, SP - Brazil

^[2] Univ Paris 1 Pantheon Sorbonne, Paris - France

^[3] Univ Fed Rio de Janeiro, Rio De Janeiro - Brazil

• Tipo de documento: Artigo Científico
• Fonte: ESAIM-PROBABILITY AND STATISTICS; v. 23, p. 770-796, DEC 20 2019.
• Citações Web of Science: 0

Resumo

Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades. (AU)

• Processo FAPESP: 16/17791-9 - Processos interagentes com memória de alcance variável em modelos neurobiológicos
• Beneficiário: Aline Duarte de Oliveira
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado
• Processo FAPESP: 13/07699-0 - Centro de Pesquisa, Inovação e Difusão em Neuromatemática - NeuroMat
• Beneficiário: Oswaldo Baffa Filho
• Modalidade de apoio: Auxílio à Pesquisa - Centros de Pesquisa, Inovação e Difusão - CEPIDs
• Processo FAPESP: 16/17789-4 - Estabilidade, seleção estatística de modelos e limite hidrodinâmico para sistemas de cadeias interagentes com memória de alcance variável
• Beneficiário: Guilherme Ost de Aguiar
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado