Large Deviations for Statistics of Jacobi Processes N. Demni (Paris - - PowerPoint PPT Presentation

Large Deviations for Statistics of Jacobi Processes

• N. Demni (Paris VI), M. Zani (Paris XII)

14 septembre 2007 Journ´ ees de Probabilit´ es 2007 La Londe

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Sketch of talk

◮ Jacobi process = Unique strong solution on [−1, 1] of SDE

• dYt =
• 1 − Y 2

t dWt + (bYt + c)dt

Y0 = y0

◮ Aim: derive a LDP for estimate of b in ultraspherical case:

c = 0

◮ Question: handable form for the semi-group density p ?

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification of Bakry & Mazet

◮ µ << λ measure on I interval

∃λ > 0 ;

• I eλ|y|µ(dy) < ∞ ⇒ orthonormal base (Rn)n of

polynomials in L2(I)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification of Bakry & Mazet

◮ µ << λ measure on I interval

∃λ > 0 ;

• I eλ|y|µ(dy) < ∞ ⇒ orthonormal base (Rn)n of

polynomials in L2(I)

◮ Symetric Markov diffusion semi-groups on L2(I) having µ as

stationary measure

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification of Bakry & Mazet

◮ µ << λ measure on I interval

∃λ > 0 ;

• I eλ|y|µ(dy) < ∞ ⇒ orthonormal base (Rn)n of

polynomials in L2(I)

◮ Symetric Markov diffusion semi-groups on L2(I) having µ as

stationary measure

◮ Require Rn as e.v. of spectral decomposition

∀n , PtRn = e−λntRn

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification of Bakry & Mazet

◮ µ << λ measure on I interval

∃λ > 0 ;

• I eλ|y|µ(dy) < ∞ ⇒ orthonormal base (Rn)n of

polynomials in L2(I)

◮ Symetric Markov diffusion semi-groups on L2(I) having µ as

stationary measure

◮ Require Rn as e.v. of spectral decomposition

∀n , PtRn = e−λntRn

◮

⇒ L = (Ax2 + Bx + C) d2 dx2 + (ax + b) d dx

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification

◮

L = d2 dx2 − x d dx Ornstein-Uhlenbeck semi-group ; Rn= Hermite

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification

◮

L = d2 dx2 − x d dx Ornstein-Uhlenbeck semi-group ; Rn= Hermite

◮

L = x d2 dx2 + (γ + 1 − x) d dx squared Ornstein-Uhlenbeck semi-group ; Rn= Laguerre

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Classification

◮

L = d2 dx2 − x d dx Ornstein-Uhlenbeck semi-group ; Rn= Hermite

◮

L = x d2 dx2 + (γ + 1 − x) d dx squared Ornstein-Uhlenbeck semi-group ; Rn= Laguerre

◮

L = (1 − x2) d2 dx2 + (β − γ − (β + γ + 2)x) d dx Jacobi semi-group ; Rn= Jacobi

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Caracterisation :

◮ Infinitesimal generator

L = (1 − x2) ∂2 ∂2x + (px + q) ∂ ∂x , x ∈ [−1, 1] p = 2b , q = 2c.

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Caracterisation :

◮ Infinitesimal generator

L = (1 − x2) ∂2 ∂2x + (px + q) ∂ ∂x , x ∈ [−1, 1] p = 2b , q = 2c.

◮

LPα,β

n

= −n(n + α + β + 1)Pα,β

n

and p = −(β + α + 2) and q = β − α Jacobi polynomial of parameters α, β > −1 Pα,β

n

(x) = (α + 1)n n!

2F1

• −n, n + α + β + 1, α + 1; 1 − x

2

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Mehler type formula ?

(Wong 1964) pt(x, y) =  

n≥0

(Rn)−1e−λntPα,β

n

(x)Pα,β

n

(y)   W (y), x, y ∈ [−1, 1] λn = n(n + α + β + 1) B Beta function, W (y) = (1 − y)α(1 + y)β 2α+β+1B(α + 1, β + 1) , Rn = ||Pα,β

n

||2

L2([−1,1],W (y)dy)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

O-U and squared O-U cases: λn = n Jacobi λn quadratic → computation of pt ?

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Subordinated process

◮ Bµ t := Bt + µt, µ > 0,

T µ,δ

t

= inf{s > 0; Bµ

s = δt},

δ > 0.

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Subordinated process

◮ Bµ t := Bt + µt, µ > 0,

T µ,δ

t

= inf{s > 0; Bµ

s = δt},

δ > 0.

◮ Martingale methods, t > 0, u ≥ 0,

E(e−uT µ,δ

t

) = e−tδ(√

2u+µ2−µ)

density of Tt: t > 0 νt(s) = δt √ 2π eδtµs−3/2 exp

• −1

2(t2δ2 s + µ2s)

• 1{s>0}
• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Mehler type formula

Consider subordinated process YT µ,δ

t

• f semi-group density qt

Fix δ, µ and write λn = (n + γ)2 − γ2, qt(x, y) = ∞ ps(x, y)νt(s)ds = W (y)

• n≥0

(Rn)−1E(e−λnT µ,δ

t

)Pα,β

n

(x)Pα,β

n

(y) = W (y)

• n≥0

(Rn)−1e−ntPα,β

n

(x)Pα,β

n

(y)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Inverse Laplace

◮ Besides

qt(x, y) = t eγt 2√π ∞ ps(x, y) s−3/2e−γ2s e− t2

4s ds

= t eγt 2 √ 2π ∞ p2/r(x, y) r−1/2e−2γ2/r e− t2

8 rdr

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Inverse Laplace

◮ Besides

qt(x, y) = t eγt 2√π ∞ ps(x, y) s−3/2e−γ2s e− t2

4s ds

= t eγt 2 √ 2π ∞ p2/r(x, y) r−1/2e−2γ2/r e− t2

8 rdr

◮ From Biane, Pitman & Yor we know some Laplace transform

∞ e− t2

8 sfCh(s) ds

=

• 1

cosh(t/2) h , h > 0 (1) ∞ e− t2

8 sfTh(s) ds

= tanh(t/2) (t/2) h , h > 0 (2) (Ch) and (Th) two families of L´ evy processes

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Expression of pt

◮

pt(x, y) = √πW (y) 2α+β eγ2t √t

• n≥0

(a)2n (α + 1)n(β + 1)n Pα,β

n

(z) (1 + xy) 8 n fT1 ⋆ fC2n+α+β+1

• (2

t ).

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Expression of pt

◮

pt(x, y) = √πW (y) 2α+β eγ2t √t

• n≥0

(a)2n (α + 1)n(β + 1)n Pα,β

n

(z) (1 + xy) 8 n fT1 ⋆ fC2n+α+β+1

• (2

t ).

◮ and ultraspherical case α = β

pt(x, y) = √πKα eγ2t √t W (y)

• n,k≥0

Γ(ν(n, k, α) + 1)(xy)k k!n!Γ(α + n + 1) (1 − x2)(1 − y2) 4 n fT1⋆fCν(n,k,α)( 1 2t )

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi

◮

• dYt =
• 1 − Y 2

t dWt + bYtdt

Y0 = 0

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi

◮

• dYt =
• 1 − Y 2

t dWt + bYtdt

Y0 = 0

◮ b < −1 ⇒ Yt ∈] − 1, 1[ p. s.

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi

◮

• dYt =
• 1 − Y 2

t dWt + bYtdt

Y0 = 0

◮ b < −1 ⇒ Yt ∈] − 1, 1[ p. s. ◮ From Girsanov formula, the generalized densities are given by

dQb

a

dQb0

a Ft

= exp

• (b − b0)

t Ys 1 − Y 2

s

dYs − 1 2(b2 − b02) t Y 2

s

1 − Y 2

s

ds

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b :

ˆ bt = t Ys 1 − Y 2

s

dYs t Y 2

s

1 − Y 2

s

ds

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b :

ˆ bt = t Ys 1 − Y 2

s

dYs t Y 2

s

1 − Y 2

s

ds

◮

St,x = t Ys 1 − Y 2

s

dYs − x t Y 2

s

1 − Y 2

s

ds so that for x > b P(ˆ bt ≥ x) = P(St,x ≥ 0)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b :

ˆ bt = t Ys 1 − Y 2

s

dYs t Y 2

s

1 − Y 2

s

ds

◮

St,x = t Ys 1 − Y 2

s

dYs − x t Y 2

s

1 − Y 2

s

ds so that for x > b P(ˆ bt ≥ x) = P(St,x ≥ 0)

◮

Λt,x(φ) = 1 t log Eb(eφSt,x)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ From Itˆ

• formula,

F(Yt) = −1 2 log(1−Y 2

t ) =

t Ys 1 − Y 2

s

dYs + 1 2 t 1 + Y 2

s

1 − Y 2

s

ds.

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ From Itˆ

• formula,

F(Yt) = −1 2 log(1−Y 2

t ) =

t Ys 1 − Y 2

s

dYs + 1 2 t 1 + Y 2

s

1 − Y 2

s

ds.

◮ + Girsanov

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

◮ From Itˆ

• formula,

F(Yt) = −1 2 log(1−Y 2

t ) =

t Ys 1 − Y 2

s

dYs + 1 2 t 1 + Y 2

s

1 − Y 2

s

ds.

◮ + Girsanov ◮

Λt,x(φ) = 1 t log Eb(φ,x)(exp({φ+b−b(φ, x))[F(Yt)−F(y0)−t/2]}) b(φ, x) = −1 −

• (b + 1)2 + 2φ(x + 1)
• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Λt,x(φ) = −φ + b − b(φ, x) 2 + 1 t log Eb(φ,x)((1 − Y 2

t )−(φ+b−b(φ,x))/2)

= Λx(φ) + 1 t log √ 2πKα(φ,x)Rt(φ, x) √t Λt,x(φ) − →t→∞ Λx(φ)

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Large deviations

Theorem: When b ≤ −1, the family {ˆ bt}t satisfies a LDP with speed t and good rate function Jb(x) =    −1 4 (x − b)2 x + 1 if x ≤ x0 x + 2 +

• (b − x)2 + 4(x + 1)

if x > x0 > b where x0 is the unique solution x < −1 of the equation (b − x)2 = 4x(x + 1) .

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Observation

Other cases:

◮ O-U case: (Florens & Pham, Bercu & Rouault):

J1(x) =    −1 4 (x − b)2 x if x ≤ b/3 2x − b if x > x0 > b/3

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Observation

Other cases:

◮ O-U case: (Florens & Pham, Bercu & Rouault):

J1(x) =    −1 4 (x − b)2 x if x ≤ b/3 2x − b if x > x0 > b/3

◮ squared Bessel case: (M.Z.):

I(x) =      (x − ν)2 4x if x ≥ x1 1 − x +

• (ν − x)2 − 4x

if x < x1 .

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Observation

Other cases:

◮ O-U case: (Florens & Pham, Bercu & Rouault):

J1(x) =    −1 4 (x − b)2 x if x ≤ b/3 2x − b if x > x0 > b/3

◮ squared Bessel case: (M.Z.):

I(x) =      (x − ν)2 4x if x ≥ x1 1 − x +

• (ν − x)2 − 4x

if x < x1 .

◮ squared O-U case: ( M.Z.):

Jδ(x) = δJ1

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Bibliography

• D. Bakry, O. Mazet. Characterization of Markov Semi-groups on R

Associated to Some Families of Orthogonal Polynomials. Sem.

• Proba. XXXVI. Lecture Notes in Maths. Springer. Vol. 1832, 2002.

60-80.

• P. Biane, J. Pitman, M. Yor. Probability Laws Related To The

Jacobi Theta and Riemann Zeta Functions, and Brownian

• Excursions. Bull. Amer. Soc. 38, no. 4, 2001, 435-465.
• O. Mazet. Classification des semi-groupes de diffusion sur R associ´

es ` a une famille de polynˆ

• mes orthogonaux. S´

eminaire de probabilit´ es XXI, Lecture notes in mathematics, vol 1655, pp40–54, Springer, 1997.

• E. Wong. The construction of a class of stationnary Markov.
• Proceedings. The 16th Symposium. Applied Math. AMS. Providence.
• RI. 1964. 264-276.
• M. Zani. Large deviations for squared radial Ornestein-Uhlenbeck
• processes. Stoch. Proc. App. 102, no. 1, 2002, 25-42.
• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Caracterisation

◮ Wong (1964) : principal solution of the Fokker-Planck eq. :

∂2

y[(B(y)pt(x, y)] − ∂y[(A(y)pt(x, y)] = ∂t(pt(x, y)).

deg(B) = 2, deg(A) = 1.

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes

Caracterisation

◮ Wong (1964) : principal solution of the Fokker-Planck eq. :

∂2

y[(B(y)pt(x, y)] − ∂y[(A(y)pt(x, y)] = ∂t(pt(x, y)).

deg(B) = 2, deg(A) = 1.

◮ Class of stationary Markov

lim

t→∞ pt(x, y) =

y2

y1

pt(x, y)W (y)dy = W (x) , W density function solution of the corresponding Pearson equation

• N. Demni (Paris VI), M. Zani (Paris XII)

Large Deviations for Statistics of Jacobi Processes