Statistics versus mean-field limit for Hawkes process. with Sylvain Delattre (P7)

Transcription

1 Saisics versus mean-field limi for Hawkes process wih Sylvain Delare (P7)

2 The model We have individuals. Z i, := number of acions of he i-h individual unil ime. Z i, jumps (is increased by 1) a rae λ i, = µ+ 1 θ ij j=1 0 ϕ( s)dz j, s, 1 i where µ (0, ), ϕ : [0, ) [0, ), (θ ij ) i,j=1,..., i.d.d. Bernoulli(p) Two ypes of acions: auonomous and by mimeism. Example : ϕ = 1 [0,K].

3 Goal We observe he aciviy of he individuals unil ime ha is Z i, s, i = 1,...,, s [0,] and we wan o esimae p (i.e. he main characerisic of he ineracion graph), in he asympoic and We consider µ and ϕ as nuisance parameers. Inuiively: no very easy... how o know if a jump is auonomous or is due o exciaion by anoher individual (and which one)? Hawkes 1971, Hawkes-Oakes Finance, eurons, Earhquake replicas, ec. Esimaion of (more general) µ and (nonparameric) ϕ ij a fixed as : Hansen, Reynaud, Rivoirard, Gaiffas, Hoffmann, Bacry, Muzzy, Rasmussen, ec.

4 Mean-field limi (, fixed) For each given k 1 and > 0, he process goes in law as o Z i, s, i = 1,...,k, s [0,] Y i s, i = 1,...,k, s [0,] a family of i.i.d. inhomogeneous Poisson processes wih inensiy (λ s ) s [0,] saisfying s λ s = µ+p ϕ(s u)λ u du, s [0,]. 0 The limi depends only on µ and pϕ hus i is no idenifiable.

5 Main resul Se Λ := 0 ϕ()d. Subcriical case: Λp < 1. Then roughly, Z (on an even where (θ ij ) behaves reasonably). We pu m = Supercriical case : Λp > 1. Then roughly, Z e α 0, wih α 0 defined by p 0 e α 0 ϕ()d = 1. We pu m = e α 0. Criical case: zoology, we do no rea. Theorem Under some (reasonnable) echnical assumpions on ϕ and if Λp 1, here exiss an (explici) esimaor ˆp such ha ( ) Pr ˆp p ε C ( 1 ) 1 + ε m where C depends only on p,µ,ϕ. The precision 1 + m seems o be opimal (Gaussian oy model). We need large if large because he MF limi is no idenifiable.

6 Subcriical case 1 Very pleasan poin: we will no have o esimae he non-parameric nuisance parameer ϕ (his would of course lead o a much less precise esimaion). We will build 3 esimaors E µ 1 Λp, V µ2 Λ 2 (1 Λp) 2p(1 p), W µ (1 Λp) 3. One hen easily find Φ such ha ˆp = Φ(E,V,W ) p (as well as esimaors of Λ and µ).

7 Subcriical case 2 I is easily seen ha E θ [Z i, ] = µ + 1 Assuming ha Z i, θ ij j=1 0 ϕ( s)e θ [Z j, s ]ds, E θ [Z i, ] γ (i) for large ( fixed). γ (i) µ+ 1 θ ij Λγ (j). j=1 Thus, wih A = ( 1 θ ij) 1 i,j and Q = (I ΛA ) 1. γ (i) = µ Q (i,j) = µ Λ k A k (i,j). j=1 j=1 k 0 Conclusion: Z (i) µl (i) wih l (i) = j=1 Q (i,j). Remark: Q exiss wih high probabiliy in he subcriical case.

8 Subcriical case 3 We sudy now l. True (and quanified) ha l (i) = l 0 since for l 1 Λ l A l (i,j) = 1 l j=1 j=1 A l (i,j) ( 1+ Λ 1 Λp 1 ) θ ij j=1 θ i,i1 θ i1,i 2...θ il 1,j j i 1,...,i l 1 = 1 l θ i,i1 θ i1,i 2...θ il 1,j p l 1 1 i 1 i 2,...,i l 1,j j }{{} l 1 p l 1 if l o relaed o known eigenvalues problems, no way o use momens, because all expecaions are infinie, because A does no exis on a small even. θ ij

9 Subcriical case 4 Thus Z (i) µ(1+ Λ 1 Λp L (i)) wih L (i) = 1 j=1 θ ij. Firs esimaor: E = Z µ(1+ Λ 1 Λp p) = µ 1 Λp.

10 Subcriical case 5 Thus Z (i) µ(1+ Λ 1 Λp L (i)) wih L (i) = 1 j=1 θ ij. Second esimaor: V := ( 1 Zi, 1 Z i=1 ) 2 Z 1 i=1 ( 1 Zi, 1 Z ) 2 ( 1 ) Var Z1, = Var ( E θ [ 1 Z1, ] ) + 1 2E[ Var θ (Z 1, ) ] and Var θ (Z 1, ) E θ [Z 1, ] since Z 1, Poisson. Thus V 2Var(E θ[z 1, ]) µ2 Λ 2 (1 Λp) 2p(1 p)

11 Subcriical case 6 Third esimaor: emporal empirical variance W, = / ( Z k Z (k 1) Z k=1 ) 2 where 1 (heory: = ) W, 1 Var θ( Z ) µ (1 Λp) 3 More complicaed. Really uses ha Z is no Poisson. Acually, Z resembles, very roughly, an auonomous (1D) Hawkes process wih parameers µ and pϕ.

12 Subcriical case 7 Remark: everyhing sars from Γ() := 0 sϕ( s)ds Λ. Wih e.g. ϕ = e, we have 0 sϕ( s)ds = Λ 1+e. So Γ(2) Γ() resembles Λ considerably much more precisely han Γ() (always rue when ϕ has a fas decay). We hus modify he 3 esimaors. For example, we use E = Z 2 Z insead of E = Z This is crucial o ge he nearly opimal (??) precision.

13 Supercriical case 1 We expec ha Z i, and ha E θ [Z i, ] γ (i)e α. Bu E θ [Z i, ] = µ + 1 θ ij So, wih A (i,j) = 1 θ ij, H E θ [Z i, ] for some r.v. H > 0; j=1 0 ϕ( s)e θ [Z j, s ]ds γ = A γ e αs ϕ(s)ds. 0 The vecor γ being posiive, i is a Perron-Frobenius eigenvecor of A, so ha ρ = ( 0 e αs ϕ(s)ds) 1 is is Perron-Frobenius eigenvalue. Since A (i,j) p, we conclude ha ρ p and hus α α 0. We consider he Perron-Frobenius eigenvecor V such ha i=1 (V (i)) 2 = and conclude ha (wih anoher r.v. K > 0) Z i, K V (i)e α 0

14 Supercriical case 2 Thus Z i, K V (i)e α 0. Single esimaor: U = 1 [ ( Z ) 2 1 (Z i, Z ) 2 Z ] 1 ( V ) 2 (V (i) V ) 2 1 Indeed, 1 ( i=1 Z i, Z ) 2 Var ( Z 1, ) = Var ( E θ [Z 1, ] ) +E [ Var θ (Z 1, ) ] and Var θ (Z 1, ) E θ [Z 1, ] since Z 1, Poisson. Thus U ( Z ) 2Var(E θ[z 1, ]) 1 ( V ) 2 (V (i) V ) 2 1

15 Supercriical case 3 U 1 ( V ) 2 1 (V (i) V ) 2. Bu V is almos colinear o L (wih L (i) = 1 A (i,j)): very roughly, he marix A 2 is almos consan (A2 (i,j) p), so ha is Perron Frobenius eigenvecor (V ) is almos consan (V (i) 1), so ha A V = ρ V gives V (i) ρ 1 j A (i,j) = ρ 1 L (i) (no very convincing). Since L is a vecor of i.i.d. Binomial(,p), we conclude ha U 1 ( L ) 2 1 (L (i) L ) 2 p(1 p) p 2 = 1 p 1.

16 Choice beween sub and super We hus have wo differen esimaors p 1 (,) p (if Λp < 1) and p 2 (,) p (if Λp > 1). If we do no know, we se p(,) = p 1 (,)1 { Z exp((log) 2 )} +p 2(,)1 { Z >exp((log) 2 )} (does no affec he precision).

17 Opimaliy? A Gaussian oy model Γ > 0 and p (0,1] unknown, (θ ij ) i,j=1,..., i.i.d. Ber(p) and he observaions are Z i, ( Poisson Γ θ ij ), i = 1,..., j=1 Then roughly, X i, = 1 Zi, ( ormal Γp, Γ2 p(1 p) + 1 ). Γp Assume ha Γp is known (his can only increase he precision). S = 1 1 (Xi, Γp) 2 is he B.E. of 1 Γ 2 p(1 p)+ 1 Γp. Thus T = (Γp) 2 (S 1 Γp) is he B.E. of (1/p 1). And VarT ( 1/ /2 ) 2. Thus opimal precision in 1/ /2.

18 Mersi pour vore aansion.