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

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

Jul 03, 2023 254 likes •880 views

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 dEvry Val dEssonne

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime

Mean field limits for Hawkes processes in a diffusive regime

Xavier Erny 1

with Eva L¨

  • cherbach 2 and Dasha Loukianova 1

1Universit´

e d’Evry Val d’Essonne (LaMME)

2Universit´

e Paris 1 Panth´ eon-Sorbonne (SAMM)

Les Probabilit´ es de Demain, 14 juin 2019

Xavier ERNY Diffusive limit for Hawkes processes 1 / 17

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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

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SLIDE 3

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

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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

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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

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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 Z1 and Z2 Z1 inhibits Z2 Z2 self-excitation

Xavier ERNY Diffusive limit for Hawkes processes 2 / 17

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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

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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

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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

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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

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime

Contents

1

Introduction

2

Hawkes Processes Stochastic Intensity Hawkes Processes

3

Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Xavier ERNY Diffusive limit for Hawkes processes 4 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Hawkes Processes

Stochastic Intensity

Z point process on R+ λ : R+ → R+ stochastic process

Xavier ERNY Diffusive limit for Hawkes processes 5 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Hawkes Processes

Stochastic Intensity

Z point process on R+ λ : R+ → R+ stochastic process λ stochastic intensity of Z if : ∀0 ≤ a < b, E [Z([a, b])|Fa] = E b

a

λ(t)dt

  • Fa
  • Xavier ERNY

Diffusive limit for Hawkes processes 5 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Hawkes Processes

Definition : Hawkes processes

(Z 1, . . . , Z N) system of Hawkes processes : λi stochastic intensity of Z i

Xavier ERNY Diffusive limit for Hawkes processes 6 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Hawkes Processes

Definition : Hawkes processes

(Z 1, . . . , Z N) system of Hawkes processes : λi stochastic intensity of Z i λi(t) = fi

  • N
  • j=1
  • [0,t[

hji(t − s)dZ j(s)

  • Xavier ERNY

Diffusive limit for Hawkes processes 6 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Stochastic Intensity Hawkes Processes

Definition : Hawkes processes

(Z 1, . . . , Z N) system of Hawkes processes : λi stochastic intensity of Z i λi(t) = fi

  • N
  • j=1
  • [0,t[

hji(t − s)dZ j(s)

  • X N,i

t

Z i([0, t]) = number of spikes of neuron i in [0, t] X N,i

t

= potential of neuron i at time t fi = spike rate function hji = leakage function

Xavier ERNY Diffusive limit for Hawkes processes 6 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN,i stochastic intensity of Z N,i λN,i(t) = fi

  • N
  • j=1
  • [0,t[

hji(t − s) dZ N,j(s)

  • Xavier ERNY

Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN,i stochastic intensity of Z N,i λN,i(t) = fi

  • N
  • j=1
  • [0,t[

hji(t − s) dZ N,j(s)

  • Xavier ERNY

Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • N
  • j=1
  • [0,t[

h (t − s) dZ N,j(s)

  • Xavier ERNY

Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • 1

√ N N

  • j=1
  • [0,t[

h (t − s) dZ N,j(s)

  • Xavier ERNY

Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • 1

√ N N

  • j=1
  • [0,t[

h (t − s)Uj(s)dZ N,j(s)

  • Uj(s) iid with mean 0 and variance 1

Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • 1

√ N N

  • j=1
  • [0,t[

h (t − s)Uj(s)dZ N,j(s)

  • X N

t

Uj(s) iid with mean 0 and variance 1

Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • 1

√ N N

  • j=1
  • [0,t[

h (t − s)Uj(s)dZ N,j(s)

  • X N

t

Uj(s) iid with mean 0 and variance 1

Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Hawkes processes in diffusive mean field

For each N ∈ N∗, we consider

  • Z N,1, . . . , Z N,N

: λN stochastic intensity of Z N,i λN (t) = f

  • 1

√ N N

  • j=1
  • [0,t[

e−α(t−s) Uj(s)dZ N,j(s)

  • X N

t

Uj(s) iid with mean 0 and variance 1

Xavier ERNY Diffusive limit for Hawkes processes 7 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Dynamique de X N

X N

t :=

1 √ N

N

  • j=1
  • [0,t]

e−α(t−s)Uj(s)dZ N,j(s)

Xavier ERNY Diffusive limit for Hawkes processes 8 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Dynamique de X N

X N

t :=

1 √ N

N

  • j=1
  • [0,t]

e−α(t−s)Uj(s)dZ N,j(s)

  • X N

t = X N s e−α(t−s)

if none of the Z N,j charge [s, t] X N

t = X N t− + Uj(t) √ N

if Z N,j charges t

2 4 6 8 10 −1 1 2

Xavier ERNY Diffusive limit for Hawkes processes 8 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Markov Process

(Xt)t≥0 Markov process

Xavier ERNY Diffusive limit for Hawkes processes 9 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Markov Process

(Xt)t≥0 Markov process (Pt)t≥0 semigroup of X : Ptg(x) := Ex [g(Xt)] := E [g(Xt)|X0 = x]

Xavier ERNY Diffusive limit for Hawkes processes 9 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Markov Process

(Xt)t≥0 Markov process (Pt)t≥0 semigroup of X : Ptg(x) := Ex [g(Xt)] := E [g(Xt)|X0 = x] A generator of X : Ag(x) := d dt (Ptg(x))|t=0

Xavier ERNY Diffusive limit for Hawkes processes 9 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t)

Xavier ERNY Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t) ANg(x) = −αxg′(x) + Nf (x)E

  • g
  • x +

U √ N

  • − g(x)
  • Xavier ERNY

Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t) ANg(x) = −αxg′(x) + Nf (x)E

  • g
  • x +

U √ N

  • − g(x)
  • U

√ N g′(x) + U2 2N g′′(x) + O(1/N

√ N)

Xavier ERNY Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t) ANg(x) = −αxg′(x) + Nf (x)E

  • g
  • x +

U √ N

  • − g(x)
  • U

√ N g′(x) + U2 2N g′′(x) + O(1/N

√ N)

Xavier ERNY Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t) ANg(x) = −αxg′(x) + Nf (x)E

  • g
  • x +

U √ N

  • − g(x)
  • U

√ N g′(x) + U2 2N g′′(x) + O(1/N

√ N) N − → +∞ : ¯ Ag(x) = −αxg′(x) + 1

2f (x)g′′(x)

Xavier ERNY Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the generators

dX N

t = −αX N t dt +

1 √ N

N

  • j=1

Uj(t)dZ N,j(t) ANg(x) = −αxg′(x) + Nf (x)E

  • g
  • x +

U √ N

  • − g(x)
  • U

√ N g′(x) + U2 2N g′′(x) + O(1/N

√ N) N − → +∞ : ¯ Ag(x) = −αxg′(x) + 1

2f (x)g′′(x)

d ¯ Xt = −α ¯ Xtdt +

  • f ( ¯

Xt)dBt

Xavier ERNY Diffusive limit for Hawkes processes 10 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds Sketch of proof : u(s) = PN

t−s ¯

Psg(x)

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds Sketch of proof : u(s) = PN

t−s ¯

Psg(x)

  • ¯

Pt − PN

t

  • g(x) =u(t) − u(0)

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds Sketch of proof : u(s) = PN

t−s ¯

Psg(x)

  • ¯

Pt − PN

t

  • g(x) =u(t) − u(0)

= t u′(s)ds

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds Sketch of proof : u(s) = PN

t−s ¯

Psg(x)

  • ¯

Pt − PN

t

  • g(x) =u(t) − u(0)

= t u′(s)ds = t

  • − d

du

  • PN

u ¯

Psg(x)

  • u=t−s + d

du

  • PN

t−s ¯

Pug(x)

  • u=s
  • ds

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (1)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds Sketch of proof : u(s) = PN

t−s ¯

Psg(x)

  • ¯

Pt − PN

t

  • g(x) =u(t) − u(0)

= t u′(s)ds = t

  • − d

du

  • PN

u ¯

Psg(x)

  • u=t−s + d

du

  • PN

t−s ¯

Pug(x)

  • u=s
  • ds

= t

  • −PN

t−sAN ¯

Psg(x) + PN

t−s ¯

A ¯ Psg(x)

  • ds

Xavier ERNY Diffusive limit for Hawkes processes 11 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (2)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds

Xavier ERNY Diffusive limit for Hawkes processes 12 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (2)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds

  • ¯

Pt − PN

t

  • g(x)
  • ≤

t

  • PN

t−s

  • ¯

A − AN ¯ Psg(x)

  • ds

Xavier ERNY Diffusive limit for Hawkes processes 12 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (2)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds

  • ¯

Pt − PN

t

  • g(x)
  • ≤

t

  • PN

t−s

  • ¯

A − AN ¯ Psg(x)

  • ds

≤ t Ex

  • ¯

A − AN ¯ Psg(X N

t−s)

  • ds

Xavier ERNY Diffusive limit for Hawkes processes 12 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the semigroups (2)

  • ¯

Pt − PN

t

  • g(x) =

t PN

t−s

  • ¯

A − AN ¯ Psg(x)ds

  • ¯

Pt − PN

t

  • g(x)
  • ≤

t

  • PN

t−s

  • ¯

A − AN ¯ Psg(x)

  • ds

≤ t Ex

  • ¯

A − AN ¯ Psg(X N

t−s)

  • ds

− → 0

Xavier ERNY Diffusive limit for Hawkes processes 12 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence in finite-dimensional distribution

Convergence of the semigroups : Ex

  • g
  • X N

t

  • −

→ Ex

  • g

¯ Xt

  • Xavier ERNY

Diffusive limit for Hawkes processes 13 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence in finite-dimensional distribution

Convergence of the semigroups : Ex

  • g
  • X N

t

  • −

→ Ex

  • g

¯ Xt

  • Induction + classical argument of Markov theory

= ⇒ Convergence in finite-dimensional distribution : Ex

  • g1
  • X N

t1

  • . . . gn
  • X N

tn

  • −

→ Ex

  • g1

¯ Xt1

  • . . . gn

¯ Xtn

  • Xavier ERNY

Diffusive limit for Hawkes processes 13 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the processes

X N converges in fidi distribution to ¯ X

  • X N : N ∈ N∗

tight on D(R+, R) (admited)

Xavier ERNY Diffusive limit for Hawkes processes 14 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the processes

X N converges in fidi distribution to ¯ X

  • X N : N ∈ N∗

tight on D(R+, R) (admited) = ⇒ X N converges to ¯ X in distribution in D(R+, R)

Xavier ERNY Diffusive limit for Hawkes processes 14 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of the processes

X N converges in fidi distribution to ¯ X

  • X N : N ∈ N∗

tight on D(R+, R) (admited) = ⇒ X N converges to ¯ X in distribution in D(R+, R)

2 4 6 8 10 10 20 2 4 6 8 10 −5 5 10

N = 10 N = 50

Xavier ERNY Diffusive limit for Hawkes processes 14 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z)

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z)

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t)

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as Skorohod’s Representation Theorem : (X N, πN

i )

( ¯ X, ¯ πi) L

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as Skorohod’s Representation Theorem : (X N, πN

i )

( ¯ X, ¯ πi) L ( X N, πN) ( X, π) L L as

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as Skorohod’s Representation Theorem : (X N, πN

i )

( ¯ X, ¯ πi) L ( X N, πN) ( X, π) L L as = ⇒ Φ( X N, πN) Φ( X, π) as

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

slide-58
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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as Skorohod’s Representation Theorem : (X N, πN

i )

( ¯ X, ¯ πi) L ( X N, πN) ( X, π) L L as = ⇒ Φ( X N, πN) Φ( X, π) as Φ(X N, πN

i )

Φ( ¯ X, ¯ πi) L L L

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

slide-59
SLIDE 59

Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime Model Convergence

Convergence of Z N,i

Z N,i

t

:=

  • ]0,t]×R+

1{z≤f (X N

s−)}dπN

i (s, z) = Φ(X N, πN i )(t)

¯ Z i

t :=

  • ]0,t]×R+

1{z≤f ( ¯

Xs−)}d¯

πi(s, z) = Φ( ¯ X, ¯ πi)(t) Φ continuous in ( ¯ X, ¯ πi) as Skorohod’s Representation Theorem : (X N, πN

i )

( ¯ X, ¯ πi) L ( X N, πN) ( X, π) L L as = ⇒ Φ( X N, πN) Φ( X, π) as Φ(X N, πN

i )

Φ( ¯ X, ¯ πi) L L L Result : (Z N,i)i≥1 converges to ( ¯ Z i)i≥1 in distribution in D(R+, R)N∗

Xavier ERNY Diffusive limit for Hawkes processes 15 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime

Bibliography

Erny, L¨

  • cherbach, Loukianova (2019). Mean field limits for

interacting Hawkes processes in a diffusive regime. HAL, arXiv. Br´ emaud, Massouli´ e (1996). Stability of nonlinear Hawkes

  • processes. Annals of probability.

Daley, Vere-Jones (2003). An Introduction to the Theory of Point Processes : Volume I : Elementary Theory and Methods. Springer.

Xavier ERNY Diffusive limit for Hawkes processes 16 / 17

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Introduction Hawkes Processes Limit of Hawkes Processes in a Diffusive Regime

Thank you for your attention ! Questions ?

Xavier ERNY Diffusive limit for Hawkes processes 17 / 17