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 � � 1 − Y 2 dY t = t dW t + ( bY t + c ) dt Y 0 = y 0 ◮ 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 I e λ | y | µ ( dy ) < ∞ ⇒ orthonormal base ( R n ) n of � ∃ λ > 0 ; polynomials in L 2 ( 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 I e λ | y | µ ( dy ) < ∞ ⇒ orthonormal base ( R n ) n of � ∃ λ > 0 ; polynomials in L 2 ( I ) ◮ Symetric Markov diffusion semi-groups on L 2 ( 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 I e λ | y | µ ( dy ) < ∞ ⇒ orthonormal base ( R n ) n of � ∃ λ > 0 ; polynomials in L 2 ( I ) ◮ Symetric Markov diffusion semi-groups on L 2 ( I ) having µ as stationary measure ◮ Require R n as e.v. of spectral decomposition ∀ n , P t R n = e − λ n t R n 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 I e λ | y | µ ( dy ) < ∞ ⇒ orthonormal base ( R n ) n of � ∃ λ > 0 ; polynomials in L 2 ( I ) ◮ Symetric Markov diffusion semi-groups on L 2 ( I ) having µ as stationary measure ◮ Require R n as e.v. of spectral decomposition ∀ n , P t R n = e − λ n t R n ◮ ⇒ L = ( Ax 2 + Bx + C ) d 2 dx 2 + ( ax + b ) d dx N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Classification ◮ L = d 2 dx 2 − x d dx Ornstein-Uhlenbeck semi-group ; R n = Hermite N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Classification ◮ L = d 2 dx 2 − x d dx Ornstein-Uhlenbeck semi-group ; R n = Hermite ◮ L = x d 2 dx 2 + ( γ + 1 − x ) d dx squared Ornstein-Uhlenbeck semi-group ; R n = Laguerre N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Classification ◮ L = d 2 dx 2 − x d dx Ornstein-Uhlenbeck semi-group ; R n = Hermite ◮ L = x d 2 dx 2 + ( γ + 1 − x ) d dx squared Ornstein-Uhlenbeck semi-group ; R n = Laguerre ◮ L = (1 − x 2 ) d 2 dx 2 + ( β − γ − ( β + γ + 2) x ) d dx Jacobi semi-group ; R n = Jacobi N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Caracterisation : ◮ Infinitesimal generator L = (1 − x 2 ) ∂ 2 ∂ 2 x + ( px + q ) ∂ ∂ x , x ∈ [ − 1 , 1] p = 2 b , q = 2 c . N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Caracterisation : ◮ Infinitesimal generator L = (1 − x 2 ) ∂ 2 ∂ 2 x + ( px + q ) ∂ ∂ x , x ∈ [ − 1 , 1] p = 2 b , q = 2 c . ◮ LP α,β = − n ( n + α + β + 1) P α,β n n and p = − ( β + α + 2) and q = β − α Jacobi polynomial of parameters α, β > − 1 ( x ) = ( α + 1) n � − n , n + α + β + 1 , α + 1; 1 − x � P α,β 2 F 1 n n ! 2 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Mehler type formula ? (Wong 1964)   � ( R n ) − 1 e − λ n t P α,β ( x ) P α,β  W ( y ) , p t ( x , y ) = ( y ) x , y ∈ [ − 1 , 1] n n n ≥ 0 λ n = n ( n + α + β + 1) B Beta function, (1 − y ) α (1 + y ) β R n = || P α,β || 2 W ( y ) = 2 α + β +1 B ( α + 1 , β + 1) , L 2 ([ − 1 , 1] , W ( y ) dy ) n N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

O-U and squared O-U cases: λ n = n Jacobi λ n quadratic → computation of p t ? N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Subordinated process ◮ B µ t := B t + µ t , µ > 0, T µ,δ B µ = inf { s > 0; s = δ t } , δ > 0 . t N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Subordinated process ◮ B µ t := B t + µ t , µ > 0, T µ,δ B µ = inf { s > 0; s = δ t } , δ > 0 . t ◮ Martingale methods, t > 0, u ≥ 0, ) = e − t δ ( √ E ( e − uT µ,δ 2 u + µ 2 − µ ) t density of T t : t > 0 2( t 2 δ 2 δ t � − 1 � e δ t µ s − 3 / 2 exp + µ 2 s ) ν t ( s ) = √ 1 { s > 0 } s 2 π N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Mehler type formula Consider subordinated process Y T µ,δ of semi-group density q t t Fix δ, µ and write λ n = ( n + γ ) 2 − γ 2 , � ∞ q t ( x , y ) = p s ( x , y ) ν t ( s ) ds 0 ( R n ) − 1 E ( e − λ n T µ,δ � ) P α,β ( x ) P α,β = W ( y ) ( y ) t n n n ≥ 0 � ( R n ) − 1 e − nt P α,β ( x ) P α,β = W ( y ) ( y ) n n n ≥ 0 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Inverse Laplace ◮ Besides � ∞ q t ( x , y ) = t e γ t p s ( x , y ) s − 3 / 2 e − γ 2 s e − t 2 4 s ds 2 √ π 0 � ∞ = t e γ t p 2 / r ( x , y ) r − 1 / 2 e − 2 γ 2 / r e − t 2 8 r dr √ 2 2 π 0 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Inverse Laplace ◮ Besides � ∞ q t ( x , y ) = t e γ t p s ( x , y ) s − 3 / 2 e − γ 2 s e − t 2 4 s ds 2 √ π 0 � ∞ = t e γ t p 2 / r ( x , y ) r − 1 / 2 e − 2 γ 2 / r e − t 2 8 r dr √ 2 2 π 0 ◮ From Biane, Pitman & Yor we know some Laplace transform � ∞ � h � e − t 2 1 8 s f C h ( s ) ds = , h > 0 (1) cosh( t / 2) 0 � ∞ � h � tanh( t / 2) e − t 2 8 s f T h ( s ) ds = , h > 0 (2) ( t / 2) 0 ( C h ) and ( T h ) two families of L´ evy processes N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Expression of p t ◮ √ π W ( y ) e γ 2 t √ t p t ( x , y ) = 2 α + β � n � � (1 + xy ) ( a ) 2 n (2 � P α,β � ( z ) f T 1 ⋆ f C 2 n + α + β +1 t ) . n ( α + 1) n ( β + 1) n 8 n ≥ 0 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Expression of p t ◮ √ π W ( y ) e γ 2 t √ t p t ( x , y ) = 2 α + β � n � � (1 + xy ) ( a ) 2 n (2 � P α,β � ( z ) f T 1 ⋆ f C 2 n + α + β +1 t ) . n ( α + 1) n ( β + 1) n 8 n ≥ 0 ◮ and ultraspherical case α = β e γ 2 t p t ( x , y ) = √ π K α √ t W ( y ) � n Γ( ν ( n , k , α ) + 1)( xy ) k � (1 − x 2 )(1 − y 2 ) f T 1 ⋆ f C ν ( n , k ,α ) ( 1 � 2 t ) k ! n !Γ( α + n + 1) 4 n , k ≥ 0 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi ◮ � � 1 − Y 2 dY t = t dW t + bY t dt Y 0 = 0 N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi ◮ � � 1 − Y 2 dY t = t dW t + bY t dt Y 0 = 0 ◮ b < − 1 ⇒ Y t ∈ ] − 1 , 1[ p. s. N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

Statistics of Jacobi ◮ � � 1 − Y 2 dY t = t dW t + bY t dt Y 0 = 0 ◮ b < − 1 ⇒ Y t ∈ ] − 1 , 1[ p. s. ◮ From Girsanov formula, the generalized densities are given by dQ b a F t dQ b 0 a � t � t Y 2 � dY s − 1 � Y s 2( b 2 − b 02 ) s = exp ( b − b 0 ) ds 1 − Y 2 1 − Y 2 0 s 0 s N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b : � t Y s dY s 1 − Y 2 ˆ 0 s b t = � t Y 2 s ds 1 − Y 2 0 s N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b : � t Y s dY s 1 − Y 2 ˆ 0 s b t = � t Y 2 s ds 1 − Y 2 0 s ◮ � t � t Y 2 Y s s S t , x = dY s − x ds 1 − Y 2 1 − Y 2 0 0 s s so that for x > b P (ˆ b t ≥ x ) = P ( S t , x ≥ 0) N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

◮ Maximum Likelihood Estimate of b : � t Y s dY s 1 − Y 2 ˆ 0 s b t = � t Y 2 s ds 1 − Y 2 0 s ◮ � t � t Y 2 Y s s S t , x = dY s − x ds 1 − Y 2 1 − Y 2 0 0 s s so that for x > b P (ˆ b t ≥ x ) = P ( S t , x ≥ 0) ◮ Λ t , x ( φ ) = 1 t log E b ( e φ S t , x ) N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes

◮ From Itˆ o formula, � t � t 1 + Y 2 F ( Y t ) = − 1 Y s dY s + 1 2 log(1 − Y 2 s t ) = ds . 1 − Y 2 1 − Y 2 2 0 0 s s N. Demni (Paris VI), M. Zani (Paris XII) Large Deviations for Statistics of Jacobi Processes