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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


  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. ◮ 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

  25. ◮ 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

  26. ◮ 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

  27. ◮ 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

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