LAN property for sde s with additive fractional noise and continuous time observation

Transcription

1 LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday, Université du Maine, 7th October, 215 Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

2 The Ornstein-Uhlenbeck process dx t = θx t dt + db t, t [, τ], θ >. B t is a standard Brownian motion. Let ˆθ τ be the MLE of θ from the continuous observation of X in [, τ]. Then, it is well-known and that lim ˆθ τ = θ τ a.s. L(P θ ) lim τ(ˆθτ θ) = N (, 2θ). τ where P θ is the probability law of the solution in the space C(R +; R). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

3 The LAN propety for the Ornstein-Uhlenbeck process The parametric statistical model {P θ, θ Θ} satisfies the LAN property at θ Θ with rate τ since for any u R, as τ : log ( dp τ θ+ u τ dp τ θ ) ( L(P θ ) un, 1 ) u2 2θ 4θ, where P τ θ is probability law of the solution in the space C([, τ]; R)). The local log likelihood ratio is asymptotically normal, with a locally constant covariance matrix and a mean equal to minus one half the variance. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

4 Consequence of the LAN property Minimax Theorem : Let (ˆθ τ ) τ be a family of estimators of the parameter θ. Then [ τ(ˆθ τ θ ) 2] 2θ. lim δ lim inf τ sup E θ θ θ <δ In particular, the MLE is asymptotic minimax efficient. The LAN property is an important tool in order to quantify the identifiability of a system. Started by Le Cam 6. Parallel theory to Cramér-Rao bound. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

5 LAN property for ergodic diffusions Consider a non-linear d-dimensional ergodic diffusion dx t = b(x t; θ)dt + σ(x t)db t, t [, τ], θ Θ R q. Under regularity, ellipticity, and ergodic assumptions, for any θ Θ and u R q, as τ : ( dp τ ) θ+ u L(P τ θ ) log u T N (, Γ(θ)) 1 2 ut Γ(θ)u, dp τ θ where X is the ergodic limit of X, and Γ(θ) = E θ [ θ b(x; θ) T σ 1 (X) T σ 1 (X) θ b(x; θ)]. Proof : Girsanov s theorem, CLT for martingales and ergodicity. Consequence : Minimax theorem : lim δ lim inf τ sup E θ θ θ <δ [ τ(ˆθ τ θ ) 2] Γ(θ) 1. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

6 The fractional Ornstein-Uhlenbeck process X t = θ t Xs ds + Bt, t [, τ], θ >. B t fractional Brownian motion with Hurst parameter H > 1/2. P θ is the probability law of the solution in the space C λ (R +; R), for any λ < H. Let ˆθ τ be the MLE of θ from the continuous observation of X in [, τ]. Then, it is well-known and that lim ˆθ τ = θ τ a.s. L(P θ ) lim τ(ˆθτ θ) = N (, 2θ). τ This suggests that the LAN property holds with the same rate τ. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

7 Ergodic sde s with additive fractional noise t X t = x + b(x s; θ) ds + d σ j B j t, j=1 t [, τ]. θ Θ, where Θ is compactly embedded in R q. ergodicity condition : b(x; θ) b(y; θ), x y α x y 2. ˆb is the Jacobian matrix θ b. assumptions : xb, xxb, x ˆb, xx ˆb bounded, b, ˆb linear growth, ˆb Lipschitz in θ and x, and σ invertible. The solution converges for t a.s. to a unique stationary process (X t, t ). P τ θ is the probability laws of the solution in the spaces C λ ([, τ]; R d ), for any λ < H. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

8 The LAN property Theorem : For any θ Θ and u R q, as τ, ( dp τ ) θ+ u L(P τ θ ) log u T N (, Γ(θ)) 1 2 ut Γ(θ)u, dp τ θ where the matrix Γ(θ) is defined by Γ(θ) = R 2 + E θ [(ˆb(X ; θ) ˆb(X r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(X ; θ) ˆb(X r2 ; θ))] dr r 1/2+H 1 r 1/2+H 1 dr 2. 2 Remark : The efficiency of the MLE in the fractional Ornstein-Uhlenbeck case remains open... Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

9 Steps of the proof of the LAN property Use the representation of the fbm given in Hairer 5 (introduced by Mandelbrot and Van Ness 68) which is suitable to get the desired ergodic properties. Apply Girsanov s theorem for the fbm following Moret and Nualart 2. Handle the singularities popping out the fractional derivatives in the Girsanov exponent. Get ergodic results in Hölder type norms for our process X. In order to apply a CLT for Brownian martingales, we use Malliavin calculus techniques : derive concentration properties for the Girsanov exponents by means of a Poincaré type inequality (Üstunel 95), which needs to conviniently upper bound some Malliavin derivatives. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

10 Mendelbrot and Van Ness representation of fbm Let W be a two sided Wiener process, then the following defines a two-sided fbm : for any t R : B t = c H R ( r) H 1/2 [dw t+r dw r ] = c H { [ ( (r t)) H 1/2 ( r) H 1/2] dw r t ( (r t)) H 1/2 dw r }. Abstract Wiener space : (B, H, P), where B = { f C(R; R d ); } f t 1 + t <, P is the law of our fbm, and h is an element of the Cameron-Martin space H iff there exists an element X h in the first chaos such that h t = E[B t X h ], and h H = X h L 2 (Ω). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

11 Properties of the SDE Proposition : There exists a unique continuous pathwise solution on any arbitrary interval [, τ] such that : The map X : (x, B) R d C([, τ]; R d ) C([, τ]; R d ) is locally Lipschitz continuous. For any θ Θ, p 1, and s, t, E [ X t p] c p, and E [ δx st p] k p t s ph, where δ denotes the increment. For all ε (, H) there exists a random variable Z ε p 1 L p (Ω) such that a.s. X t Z ε (1 + t) 2ε, and δx st Z ε (1 + t) 2ε t s H ε, uniformly for s t. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

12 Ergodic properties of the SDE Garrido-Atienza, Kloeden and Neuenkirch 9 : Shift operators θ t : Ω Ω : θ tω( ) = ω( + t) ω(t), t R, ω Ω. The shifted process (B s(θ t )) s R is still a d-dimensional fractional Brownian motion and for any integrable random variable F : Ω R for P-almost all ω Ω. 1 lim τ τ τ F(θ t(ω)) dt = E[F], Theorem : There exists a random variable X : Ω R d such that lim t Xt(ω) X(θtω) = for P-almost all ω Ω. Moreover, we have E[ X p ] < for all p 1. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

13 Ergodic properties of the SDE Theorem : For any θ Θ and any f C 1 (R d ; R) such that ( f (x) + xf (x) c 1 + x N), x R d, we have 1 τ lim f (X τ t) dt = E[f (X)], τ P-a.s. Proposition : Let α (, H). There exists a random variable Z admitting moments of any order such that for all s t [ ] X t X t Z e cs and δ X X Z e cs (t s) α. st Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

14 Operator that transforms W into B Proposition : For w C c (R) and H (, 1), set [K H w] t = c H R ( r) H 1/2 [ẇ t+r ẇ r ] dr. Then : (i) There exists a constant c H such that ( ) c H [I H 1/2 + w] t [I H 1/2 + w], for H > 1 2 [K H w] t = ( ) c H [D 1/2 H + w] t [D 1/2 H + w], for H < 1, 2 where [D α +ϕ] t = c α ϕ t ϕ t r dr, and [I+ϕ] α R + r 1+α t = c α R + ϕ t r r α 1 dr. (ii) For H > 1/2, K H can be extended as an isometry from L 2 (R) to I H 1/2 + (L 2 (R)). (iii) There exists a constant c H such that K 1 H = c H K 1 H. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

15 Girsanov s transformation Proposition : For a given θ Θ, onsider the d-dimensional process Q t = t σ 1 b(x s; θ) ds + B t. Then Q is a d-dimensional fractional Brownian motion under the probability P θ defined by d P θ dp θ [,τ] = e L, with L = τ σ 1 [D H 1/2 + b(x; θ)] u, dw u + 1 τ σ 1 [D H 1/2 + b(x; θ)] u 2 du. 2 Proof : show that D H 1/2 + b(x; θ) is well defined on [, τ], and Novikov s condition : there exists λ > such that sup E θ [exp t [,τ] ( λ t σ 1 [D H 1/2 + b(x; θ)] s 2 ds )] <. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

16 Proof of the LAN property Step 1 : Apply Girsanov s theorem. Fix θ Θ, and set θ τ = θ + τ 1/2 u. Then ( dp τ ) τ θτ log = σ 1 ([D H 1/2 dp τ + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t), dw t θ 1 2 τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t) 2 dt. Step 2 : Linearize this relation : add and substract the d-dimensional vector [D H 1/2 + ˆb(X; θ)] t(θ τ θ) = 1 [D H 1/2 τ + ˆb(X; θ)] t, where ˆb = θ b. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

17 Step 2 : linearization where I 1 = 1 τ τ I 2 = I 3 = τ τ ( dp τ ) θτ log = I dp τ 1 I 2 1 θ 2 I 3 I 4, σ 1 [D H 1/2 + ˆb(X; θ)] tu, dw t 1 τ 2τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 dt + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)), dw t + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)) 2 dt I 4 = τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)), σ 1 [D H 1/2 + ˆb(X; θ)] t(θ τ θ) dt. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

18 Remaining steps of the proof Step 3 : Main contribution to our log-likelihood : we show that as τ 1 τ τ σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 P dt θ u T Γ(θ)u. Together with multivariate central limit theorem for Brownian martingales implies that as τ I 1 L(P θ ) u T N (, Γ(θ)) 1 2 ut Γ(θ)u. Step 4 : Negligible contributions : We show that the terms I 2, I 3 and I 4 converge to zero in P θ -probability as τ. For I 3 apply Taylor s expansion and some computations, I 3 is the quadratic variation of the martingale I 2, and by Cauchy-Schwarz inequality, I 4 is bounded by I 3. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

19 Proof of Step 3 Step 3a : We have that 1 τ τ σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 dt 1 τ Jτ (X) = 1 τ where (ˆb(X t; θ) N t(x) = ˆb(X t r ; θ))u dr = N R + r H+1/2 1,t (X) + N 2,t (X) = t (ˆb(X t; θ) ˆb(X t r ; θ))u r H+1/2 dr + where we have set X t = x for all t. t τ σ 1 N t(x) 2 dt, (ˆb(X t; θ) ˆb(x ; θ))u r H+1/2 We denote by J τ (X), N t(x), N 1,t (X), N 2,t (X) the same quantities with X replaced by X. dr Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

20 Proof of Step 3 Step 3b : We show that N t(x) N t(x) is of order t η Z with η > and Z p 1 L p (Ω). Hence J τ (X) J τ (X) is of order τ 1 2η, which is a negligible term on the scale τ. This allows us to consider the limiting behavior of J τ (X) instead of J τ (X). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

21 Proof of Step 3 Step 3c : This step is devoted to reduce our computations to an evaluation for the expected value. We show that 1 lim τ τ E θ [ J τ (X) E θ [J τ (X)] ] =, Poincaré type inequality : Let F : B R be a functional in D 1,2. Then, This reduces to show that E [ F E[F] ] π E [ DF H] 2 [ E θ DJτ (X) lim H τ τ ] =. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

22 Proof of Step 3 Proposition :For all t >, X t belongs to D 1,2, and the Malliavin derivative satisfies that ( DX t H c exp α t ), 2 uniformly in t R +. Moreover, for u v, ( D(δX uv) H c 1 exp α u ) (v u) H/2, 2 uniformly in u and v. Idea of proof : Derive contraction properties of the map h X h, h H, where X h is the solution to our SDE driven by B + h : ( Xt h X t c exp αt ) h H, 2 uniformly in t R +. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

23 Proof of Step 3 Step 3d : We are now reduced to the analysis of the quantity E θ [J τ (X)]. This is equal to u T Ψu where Ψ equals the matrix τ E θ [(ˆb(Y t; θ) dt ˆb(Y t r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(Y t; θ) ˆb(Y t r2 ; θ))] dr R 2 + r 1/2+H 1 r 1/2+H 1 dr 2 2 By stationarity of Y, the expected value does not depend on t and ( ) ( ) E θ [(ˆb(Y ; θ) ˆb(Y r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(Y ; θ) ˆb(Y r2 ; θ))] r1 H 1 r2 H 1 We obtain that Ψ is a convergent integral and E θ [J τ (Y )] = τu T Γ(θ)u. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24

24 BON ANNIVERSAIRE VLAD!!!! Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24