Mean field limits for Hawkes processes in a diffusive regime Xavier - PowerPoint PPT Presentation

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Mean field limits for Hawkes processes in a diffusive regime Xavier Erny 1 ocherbach 2 and Dasha Loukianova 1 with Eva L¨ 1 Universit´ e d’Evry Val d’Essonne (LaMME) 2 Universit´ e Paris 1 Panth´ eon-Sorbonne (SAMM) Les Probabilit´ es de Demain, 14 juin 2019 Xavier ERNY Diffusive limit for Hawkes processes 1 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Hawkes processes Point process = Jump process Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Hawkes processes Point process = Jump process = (random) Set of the jump times Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Hawkes processes Point process = Jump process = (random) Set of the jump times = (random) Point measure on R + Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Hawkes processes Point process = Jump process = (random) Set of the jump times = (random) Point measure on R + Hawkes processes = Interacting point processes on R + Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Hawkes processes Point process = Jump process = (random) Set of the jump times = (random) Point measure on R + Hawkes processes = Interacting point processes on R + Example : 2 processes Z 1 and Z 2 Z 1 inhibits Z 2 Z 2 self-excitation Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Modeling in neurosciences Neural activity = Set of spike times Xavier ERNY Diffusive limit for Hawkes processes 3 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Modeling in neurosciences Neural activity = Set of spike times = Point process Xavier ERNY Diffusive limit for Hawkes processes 3 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Modeling in neurosciences Neural activity = Set of spike times = Point process Spike rate depends on the potential of the neuron Xavier ERNY Diffusive limit for Hawkes processes 3 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Modeling in neurosciences Neural activity = Set of spike times = Point process Spike rate depends on the potential of the neuron Each spike modifies the potential of the neurons Xavier ERNY Diffusive limit for Hawkes processes 3 / 17

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Contents Introduction 1 Hawkes Processes 2 Stochastic Intensity Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime 3 Model Convergence Xavier ERNY Diffusive limit for Hawkes processes 4 / 17

Introduction Stochastic Intensity Hawkes Processes Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Z point process on R + λ : R + → R + stochastic process Xavier ERNY Diffusive limit for Hawkes processes 5 / 17

Introduction Stochastic Intensity Hawkes Processes Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Z point process on R + λ : R + → R + stochastic process λ stochastic intensity of Z if : �� b � � � � ∀ 0 ≤ a < b , E [ Z ([ a , b ]) |F a ] = E λ ( t ) dt � F a a Xavier ERNY Diffusive limit for Hawkes processes 5 / 17

Introduction Stochastic Intensity Hawkes Processes Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Definition : Hawkes processes ( Z 1 , . . . , Z N ) system of Hawkes processes : λ i stochastic intensity of Z i Xavier ERNY Diffusive limit for Hawkes processes 6 / 17

Introduction Stochastic Intensity Hawkes Processes Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Definition : Hawkes processes ( Z 1 , . . . , Z N ) system of Hawkes processes : λ i stochastic intensity of Z i � � � � N h ji ( t − s ) dZ j ( s ) λ i ( t ) = f i [0 , t [ j =1 Xavier ERNY Diffusive limit for Hawkes processes 6 / 17

Introduction Stochastic Intensity Hawkes Processes Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Definition : Hawkes processes ( Z 1 , . . . , Z N ) system of Hawkes processes : λ i stochastic intensity of Z i � � � � N h ji ( t − s ) dZ j ( s ) λ i ( t ) = f i [0 , t [ j =1 X N , i t Z i ([0 , t ]) = number of spikes of neuron i in [0 , t ] X N , i = potential of neuron i at time t t f i = spike rate function h ji = leakage function Xavier ERNY Diffusive limit for Hawkes processes 6 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N , i stochastic intensity of Z N , i � � � � N λ N , i ( t ) = f i dZ N , j ( s ) h ji ( t − s ) j =1 [0 , t [ Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N , i stochastic intensity of Z N , i � � � � N λ N , i ( t ) = f i dZ N , j ( s ) h ji ( t − s ) j =1 [0 , t [ Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f dZ N , j ( s ) h ( t − s ) j =1 [0 , t [ Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f 1 dZ N , j ( s ) h ( t − s ) √ N j =1 [0 , t [ Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f 1 h ( t − s ) U j ( s ) dZ N , j ( s ) √ N j =1 [0 , t [ U j ( s ) iid with mean 0 and variance 1 Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f 1 h ( t − s ) U j ( s ) dZ N , j ( s ) √ N j =1 [0 , t [ X N t U j ( s ) iid with mean 0 and variance 1 Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f 1 h ( t − s ) U j ( s ) dZ N , j ( s ) √ N j =1 [0 , t [ X N t U j ( s ) iid with mean 0 and variance 1 Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Hawkes processes in diffusive mean field � Z N , 1 , . . . , Z N , N � For each N ∈ N ∗ , we consider : λ N stochastic intensity of Z N , i � � � � N λ N ( t ) = f e − α ( t − s ) U j ( s ) dZ N , j ( s ) 1 √ N j =1 [0 , t [ X N t U j ( s ) iid with mean 0 and variance 1 Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Dynamique de X N � N � 1 X N e − α ( t − s ) U j ( s ) dZ N , j ( s ) t := √ N [0 , t ] j =1 Xavier ERNY Diffusive limit for Hawkes processes 8 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Dynamique de X N � N � 1 X N e − α ( t − s ) U j ( s ) dZ N , j ( s ) t := √ N [0 , t ] j =1 � if none of the Z N , j charge [ s , t ] X N t = X N s e − α ( t − s ) t − + U j ( t ) if Z N , j charges t X N t = X N √ N 2 1 0 − 1 0 2 4 6 8 10 Xavier ERNY Diffusive limit for Hawkes processes 8 / 17

Introduction Model Hawkes Processes Convergence Limit of Hawkes Processes in a Diffusive Regime Markov Process ( X t ) t ≥ 0 Markov process Xavier ERNY Diffusive limit for Hawkes processes 9 / 17