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

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Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

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Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

Samy T. (Purdue) Drift estimation UTN 2015 3 / 25

Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

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Definition of fBm

A 1-d fBm is a continuous process B = {Bt; t ∈ R} such that B0 = 0 and for H ∈ (0, 1): B is a centered Gaussian process E[BtBs] = 1

2(|s|2H + |t|2H − |t − s|2H)

Definition 1. d-dimensional fBm: B = (B1, . . . , Bd), with Bi independent 1-d fBm Variance of increments: E[|δBj

st|2] ≡ E[|Bj t − Bj s|2] = |t − s|2H

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Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7

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System under consideration

Equation: Yt = y0 +

t

0 b(Ys; θ) ds + d

• j=1

σjBj

t,

t ∈ [0, τ]. (1) Coefficients: y0 ∈ Rd fixed. θ ∈ Θ, where Θ compact set of Rq (x, θ) ∈ Rd × Θ → b(x; θ) ∈ Rd smooth enough b(x; θ) − b(y; θ), x − y ≤ −α |x − y|2 (B1, . . . , Bd) collection of d-dimensional fBms σ = (σ1, . . . , σd) ∈ Rd×d invertible

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Basic aim

Objective: Estimate parameter θ with one observation of path for Y Example: Two Ornstein-Uhlenbeck type processes

2 4 6 8 10 −0.4 0.0 0.4 0.8 Figure: H = 0.7, d = 1, b(x) = −3x 2 4 6 8 10 −1 1 2 3 4 Figure: H = 0.7, d = 1, b(x) = −0.1x

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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: m dYt = −ζ

t

−∞ KH(t − u) Yu du + (2ζ kB T)1/2 dBt

Friction coefficient ζ to be estimated from observation

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Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

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Estimation of σ and H

Notation: On a finite interval [0, τ] we set δYst = Yt − Ys For n ≥ 1, take tn

i = iτ n

Estimator: ˆ Hn consistent and asymptotically normal with Rn =

2n

k=1

• δYt2n

k−1t2n k

2 n

k=1

• δYtn

k−1tn k

2 ,

and ˆ Hn = 1 2 − ln(Rn) 2 ln(2) Extensions: Joint estimation of (σ, H) Use of filters (weights on increments δYtn

k−1tn k )

Contributors: León-Berzin, Kubilius-Mishura, Brouste-Iacus

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Drift estimation for H > 1/2 known

Fractional Ornstein-Uhlenbeck: For θ ∈ R, set Yt = θ

t

0 Ys ds + Bt

A fractional kernel: Define kH(t, s) = cHs1/2−H(t − s)1/2−H Fundamental semi-martingale and tilted drift: Set Zt =

t

0 kH(t, s) dYs,

and Qt = cH

• t2H−1 Zt +

t

0 r 2H−1 dZr

• Estimator:

ˆ θt = cH

t

0 Qs dZs

t

0 Q2 s s1−2H ds

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Drift estimation for H > 1/2 known (2)

FOU case: ˆ θt 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 kH Estimator without weights: for FOU, ˆ θn = cH

2

n

n

k=1 Y 2 k

σ2

− 1

2H

is consistent and asymptotically Gaussian (Hu-Nualart).

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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

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Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

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Setting

Equation: Yt = y0 +

t

0 b(Ys; θ0) ds +

d

j=1 σjBj t

Observation: {Y kτ

nα ; k ≤ n} with α < 1 and unknown θ0

Main assumption 1: b(x; θ) − b(y; θ), x − y ≤ −α |x − y|2 (2) Ergodic behavior: Under Hypothesis (2), Unique invariant measure νθ for L(Yt) Ergodic convergence towards stationary solution ¯ Y = ¯ Y (θ) Main assumption 2: Identifiability, For all θ ∈ Θ, νθ = νθ0 ⇐ ⇒ θ = θ0

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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 Define ˆ θn as: ˆ θn = argmin

  dTV  1

n

n

• k=1

δY kτ

nα ,

1 Mn

Mn

• k=1

δX γ,θ

kγ

  ; θ ∈ Θ   

Under previous assumptions we have a.s− limn→∞,γ→0 ˆ θn = θ0. Theorem 2. Remark: In fact we use a discretized or smoothed version of dTV

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Numerical experiments

Quadratic test function: n = Mn = 1000 Step: T = γ =

τ nα = 1

FOU with θ0 = 1

• n

k=1(YkT)2 − (X γ,θ kT )2

• Smoothed TV distance:

n = Mn Step: T = γ =

τ nα

FOU with θ0 = 1 L1 distance for densities

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Numerical experiments (2)

Equation: dYt = −(1 + cos(θYt))dt + dBt Parameters: θ0 = 1 and H = 2

3 Figure: TV type distance Figure: With

• n

k=1(YkT )2 − (X γ,θ kT )2

• Open questions: Convexity, gradient descent

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Outline

1

Introduction Setting of the problem Estimation for systems driven by fBm: brief review

2

Generalized least squares estimator

3

LAN property

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Setting

Equation: Yt = y0 +

t

0 b(Ys; θ) ds + d j=1 σjBj t

Observation: {Yt; 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 νθ for L(Yt) Ergodic convergence towards stationary solution ¯ Y = ¯ Y (θ)

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LAN property: definition

LAN property for {Pτ

θ, θ ∈ Θ, τ > 0} satisfied if there exists:

ϕτ ∈ R such that limτ→∞ ϕτ = 0 Γ(θ) ∈ Rq,q positive definite matrix such that for any u ∈ Rq, as τ → ∞: log

dPτ

θ+ϕτu

dPτ

θ

• L(Pθ)

− − − → uTN(0, Γ(θ)) − 1 2uTΓ(θ)u Definition 3. Interpretation: Statistical model behaves locally like a Gaussian i.i.d model

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Cramer-Rao type bound

Suppose: LAN property is satisfied We have a family of estimators (ˆ θτ)τ≥0 Then: lim inf

τ→∞ Eθ

 

• ˆ

θτ − θ ϕτ

• 2

 ≥ Tr(Γ(θ))

Theorem 4.

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LAN for fBm systems (with E. Nualart)

Consider: Our system with coercive hypothesis (2) Then as τ → ∞ we have: log dPτ

θ+ u

√τ

dPτ

θ L(Pθ)

− − − → uTN(0, Γ(θ)) − 1 2uTΓ(θ)u, where the quantity Γ(θ) is defined by ( ¯ Y ergodic limit of Y ):

• R2

+

dr1dr2 r −(1/2+H)

1

r −(1/2+H)

2

Eθ

• σ−1(ˆ

b( ¯ Y0; θ) − ˆ b( ¯ Yr1; θ))(ˆ b( ¯ Y0; θ) − ˆ b( ¯ Yr2; θ))T(σ−1)T Theorem 5.

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Perspective: rate of convergence

A consequence of LAN: Best convergence rate for ˆ θτ of order τ −1/2 (does not depend on H) Case of fractional Ornstein-Uhlenbeck: Rate τ −1/2 achieved Other cases: No rate of convergence!

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