Drift estimation for differential equations driven by fractional - PowerPoint PPT Presentation

Drift estimation for differential equations driven by fractional Brownian motions Samy Tindel Purdue University University of Tennessee - Probability Seminar 2015 Ongoing joint works with Fabien Panloup and Eulalia Nualart Samy T. (Purdue) Drift estimation UTN 2015 1 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 2 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 3 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 4 / 25

Definition of fBm Definition 1. A 1-d fBm is a continuous process B = { B t ; t ∈ R } such that B 0 = 0 and for H ∈ (0 , 1): B is a centered Gaussian process 2 ( | s | 2 H + | t | 2 H − | t − s | 2 H ) E [ B t B s ] = 1 d -dimensional fBm: B = ( B 1 , . . . , B d ), with B i independent 1-d fBm Variance of increments: E [ | δ B j st | 2 ] ≡ E [ | B j t − B j s | 2 ] = | t − s | 2 H Samy T. (Purdue) Drift estimation UTN 2015 5 / 25

Examples of fBm paths H = 0 . 3 H = 0 . 5 H = 0 . 7 Samy T. (Purdue) Drift estimation UTN 2015 6 / 25

System under consideration Equation: � t d � σ j B j Y t = y 0 + 0 b ( Y s ; θ ) ds + t , t ∈ [0 , τ ] . (1) j =1 Coefficients: y 0 ∈ R d fixed. θ ∈ Θ, where Θ compact set of R q ( x , θ ) ∈ R d × Θ �→ b ( x ; θ ) ∈ R d smooth enough � b ( x ; θ ) − b ( y ; θ ) , x − y � ≤ − α | x − y | 2 ( B 1 , . . . , B d ) collection of d -dimensional fBms σ = ( σ 1 , . . . , σ d ) ∈ R d × d invertible Samy T. (Purdue) Drift estimation UTN 2015 7 / 25

Basic aim Objective: Estimate parameter θ with one observation of path for Y Example: Two Ornstein-Uhlenbeck type processes 0.8 4 3 0.4 2 1 0.0 0 −0.4 −1 0 2 4 6 8 10 0 2 4 6 8 10 Figure: H = 0 . 7, d = 1, b ( x ) = − 3 x Figure: H = 0 . 7, d = 1, b ( x ) = − 0 . 1 x Samy T. (Purdue) Drift estimation UTN 2015 8 / 25

A motivation from biophysics Source: Series of papers by S. Kou Anomalous fluctuations: New observations at molecule scale Fluctuations, end of a protein ֒ → changes in shape of protein Subdiffusive behavior for fluctuations. Mathematical model: � t −∞ K H ( t − u ) Y u du + (2 ζ k B T ) 1 / 2 dB t m dY t = − ζ Friction coefficient ζ to be estimated from observation Samy T. (Purdue) Drift estimation UTN 2015 9 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 10 / 25

Estimation of σ and H Notation: On a finite interval [0 , τ ] we set δ Y st = Y t − Y s i = i τ For n ≥ 1, take t n n Estimator: ˆ H n consistent and asymptotically normal with � 2 � � 2 n δ Y t 2 n H n = 1 2 − ln( R n ) k − 1 t 2 n k =1 ˆ R n = k � 2 , and 2 ln(2) � � n δ Y t n k − 1 t n k =1 k Extensions: Joint estimation of ( σ, H ) Use of filters (weights on increments δ Y t n k ) k − 1 t n Contributors: León-Berzin, Kubilius-Mishura, Brouste-Iacus Samy T. (Purdue) Drift estimation UTN 2015 11 / 25

Drift estimation for H > 1 / 2 known Fractional Ornstein-Uhlenbeck: For θ ∈ R , set � t Y t = θ 0 Y s ds + B t A fractional kernel: Define k H ( t , s ) = c H s 1 / 2 − H ( t − s ) 1 / 2 − H Fundamental semi-martingale and tilted drift: Set � t � t � � t 2 H − 1 Z t + 0 r 2 H − 1 dZ r Z t = 0 k H ( t , s ) dY s , and Q t = c H Estimator: � t 0 Q s dZ s ˆ θ t = c H � t s s 1 − 2 H ds 0 Q 2 Samy T. (Purdue) Drift estimation UTN 2015 12 / 25

Drift estimation for H > 1 / 2 known (2) FOU case: ˆ θ t is consistent ֒ → Kleptsyna - Le Breton Extension: If drift is θ b ( x ), consistent estimator ֒ → Tudor - Viens, Mishura - Schevshenko Problem: Numerically, estimators perform poorly ֒ → Due to singularity of kernel k H Estimator without weights: for FOU, � 2 � − 1 � n k =1 Y 2 2 H ˆ k n θ n = c H σ 2 is consistent and asymptotically Gaussian (Hu-Nualart). Samy T. (Purdue) Drift estimation UTN 2015 13 / 25

Drift estimation, nonlinear cases Case of interest: general drift b ( x ; θ ) Contribution 1: Ladroue-Papavasiliou Polynomial coefficients Based on rough paths analysis Method of moments Contribution 2: Neuenkirch-T Coercive drift � b ( x ; θ ) − b ( y ; θ ) , x − y � ≤ − α | x − y | 2 Based on ergodic properties Least square estimator Samy T. (Purdue) Drift estimation UTN 2015 14 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 15 / 25

Setting � t j =1 σ j B j � d Equation: Y t = y 0 + 0 b ( Y s ; θ 0 ) ds + t Observation: { Y k τ n α ; k ≤ n } with α < 1 and unknown θ 0 Main assumption 1: � b ( x ; θ ) − b ( y ; θ ) , x − y � ≤ − α | x − y | 2 (2) Ergodic behavior: Under Hypothesis (2), Unique invariant measure ν θ for L ( Y t ) Ergodic convergence towards stationary solution ¯ Y = ¯ Y ( θ ) Main assumption 2: Identifiability, For all θ ∈ Θ , ν θ = ν θ 0 ⇐ ⇒ θ = θ 0 Samy T. (Purdue) Drift estimation UTN 2015 16 / 25

Least square procedure (with F. Panloup) Numerical approximation of ¯ Y : for a small γ > 0, define X γ,θ → Numerical approx. of ¯ ֒ Y obtained with Euler scheme, step γ Theorem 2. Define ˆ θ n as:     n M n  1 1 ˆ    ; θ ∈ Θ � � θ n = argmin δ Y k τ n α , δ X γ,θ  d TV n M n k γ  k =1 k =1 Under previous assumptions we have a . s − lim n →∞ ,γ → 0 ˆ θ n = θ 0 . Remark: In fact we use a discretized or smoothed version of d TV Samy T. (Purdue) Drift estimation UTN 2015 17 / 25

Numerical experiments Quadratic test function: n = M n = 1000 τ Step: T = γ = n α = 1 FOU with θ 0 = 1 k =1 ( Y kT ) 2 − ( X γ,θ � kT ) 2 � � � n � � � Smoothed TV distance: n = M n τ Step: T = γ = n α FOU with θ 0 = 1 L 1 distance for densities Samy T. (Purdue) Drift estimation UTN 2015 18 / 25

Numerical experiments (2) Equation: dY t = − (1 + cos( θ Y t )) dt + dB t Parameters: θ 0 = 1 and H = 2 3 � �� n k =1 ( Y kT ) 2 − ( X γ,θ kT ) 2 � Figure: With Figure: TV type distance � Open questions: Convexity, gradient descent Samy T. (Purdue) Drift estimation UTN 2015 19 / 25

Outline Introduction 1 Setting of the problem Estimation for systems driven by fBm: brief review Generalized least squares estimator 2 LAN property 3 Samy T. (Purdue) Drift estimation UTN 2015 20 / 25

Setting � t j =1 σ j B j 0 b ( Y s ; θ ) ds + � d Equation: Y t = y 0 + t Observation: { Y t ; t ≥ 0 } with unknown θ Main assumption 1: � b ( x ; θ ) − b ( y ; θ ) , x − y � ≤ − α | x − y | 2 Ergodic behavior: Under Hypothesis (2), Unique invariant measure ν θ for L ( Y t ) Ergodic convergence towards stationary solution ¯ Y = ¯ Y ( θ ) Samy T. (Purdue) Drift estimation UTN 2015 21 / 25

LAN property: definition Definition 3. LAN property for { P τ θ , θ ∈ Θ , τ > 0 } satisfied if there exists: ϕ τ ∈ R such that lim τ →∞ ϕ τ = 0 Γ( θ ) ∈ R q , q positive definite matrix such that for any u ∈ R q , as τ → ∞ : � d P τ � → u T N (0 , Γ( θ )) − 1 L ( P θ ) θ + ϕ τ u 2 u T Γ( θ ) u log − − − d P τ θ Interpretation: Statistical model behaves locally like a Gaussian i.i.d model Samy T. (Purdue) Drift estimation UTN 2015 22 / 25

Cramer-Rao type bound Theorem 4. Suppose: LAN property is satisfied We have a family of estimators (ˆ θ τ ) τ ≥ 0 Then: 2   ˆ � � θ τ − θ � �  ≥ Tr (Γ( θ )) lim inf τ →∞ E θ � �  ϕ τ � � � � Samy T. (Purdue) Drift estimation UTN 2015 23 / 25

LAN for fBm systems (with E. Nualart) Theorem 5. Consider: Our system with coercive hypothesis (2) Then as τ → ∞ we have: d P τ θ + u → u T N (0 , Γ( θ )) − 1 L ( P θ ) √ τ 2 u T Γ( θ ) u , log − − − d P τ θ where the quantity Γ( θ ) is defined by ( ¯ Y ergodic limit of Y ): � dr 1 dr 2 r − (1 / 2+ H ) r − (1 / 2+ H ) 1 2 R 2 + � σ − 1 (ˆ b ( ¯ Y 0 ; θ ) − ˆ b ( ¯ Y r 1 ; θ ))(ˆ b ( ¯ Y 0 ; θ ) − ˆ b ( ¯ Y r 2 ; θ )) T ( σ − 1 ) T � E θ Samy T. (Purdue) Drift estimation UTN 2015 24 / 25

Perspective: rate of convergence A consequence of LAN: θ τ of order τ − 1 / 2 (does not depend on H ) Best convergence rate for ˆ Case of fractional Ornstein-Uhlenbeck: Rate τ − 1 / 2 achieved Other cases: No rate of convergence! Samy T. (Purdue) Drift estimation UTN 2015 25 / 25