Power Variation and Gaussian Processes with Stationary Increments - PowerPoint PPT Presentation

Power Variation and Gaussian Processes with Stationary Increments José M. Corcuera Faculty of Mathematics University of Barcelona Stochastics in Turbulence and Finance, Sandbjerg, 31 January 2008 fblanc1 José M. Corcuera (University of Barcelona) 1 / 53

Outline Power variation for some class of processes 1 The power variation Sequences of functionals of Gaussian processes Gaussian processes with stationary increments Integral processes Convergence in probability Ideas for the proof. Central limit theorem I Ideas for the proof Central limit theorem II Ideas for the proof fblanc1 José M. Corcuera (University of Barcelona) 2 / 53

Outline Power variation for some class of processes 1 The power variation Sequences of functionals of Gaussian processes Gaussian processes with stationary increments Integral processes Convergence in probability Ideas for the proof. Central limit theorem I Ideas for the proof Central limit theorem II Ideas for the proof More general functionals 2 Convergence in probability Ideas for the proof Central limit theorems fblanc1 José M. Corcuera (University of Barcelona) 2 / 53

Outline Power variation for some class of processes 1 The power variation Sequences of functionals of Gaussian processes Gaussian processes with stationary increments Integral processes Convergence in probability Ideas for the proof. Central limit theorem I Ideas for the proof Central limit theorem II Ideas for the proof More general functionals 2 Convergence in probability Ideas for the proof Central limit theorems fblanc1 References 3 José M. Corcuera (University of Barcelona) 2 / 53

Power variation for some class of processes The power variation For any p > 0, a natural number n and for any stochastic process Z = { Z t , t ∈ [ 0 , T ] } the (normalized) power variation of order p is defined as [ nt ] � � � 1 p � � V n p ( Z ) t := � Z i n − Z i − 1 , � n τ p n n i = 1 where τ n is a normalization factor. fblanc1 José M. Corcuera (University of Barcelona) 3 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Consider a complete probability space (Ω , F , P ) and a Gaussian subspace H 1 of L 2 (Ω , F , P ) whose elements are zero-mean Gaussian random variables . Let be I H a separable Hilbert space with scalar product denoted by �· , ·� I H and norm || · || I H , we will assume there is an isometry W : I H → H 1 h �→ W ( h ) in the sense that E ( W ( h 1 ) W ( h 2 )) = � h 1 , h 2 � I H . fblanc1 José M. Corcuera (University of Barcelona) 4 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes For any m ≥ 2, we denote by H m the m -th Wiener chaos, that is, the closed subspace of L 2 (Ω , F , P ) generated by the random variables H m ( X ) , where X ∈ H 1 , E ( X 2 ) = 1, and H m is the m -th Hermite x 2 dx m ( e − x 2 d m polynomial, i.e. H 0 ( x ) = 1 and H m ( x ) = ( − 1 ) m e 2 ) . 2 fblanc1 José M. Corcuera (University of Barcelona) 5 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Suppose that I H is infinite-dimensional and let { e i , i ≥ 1 } be an orthonormal basis of I H . Denote by Λ the set of all sequences a = ( a 1 , a 2 , ... ) , a i ∈ N , such that all the terms except a finite number of i = 1 a i ! and | a | = � ∞ them, vanish. For a ∈ Λ we set a ! = Π ∞ i = 1 a i . For any multiindex a ∈ Λ we define 1 Π ∞ Φ a = √ i = 1 H a i ( W ( e i )) . a ! fblanc1 José M. Corcuera (University of Barcelona) 6 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes The family of random variables { Φ a , a ∈ Λ } is an orthonormal system. In fact � � Π ∞ i = 1 H a i ( W ( e i ))Π ∞ E i = 1 H b i ( W ( e i )) = δ ab a ! . And { Φ a , a ∈ Λ , | a | = m } is a complete orthonormal system in H m . fblanc1 José M. Corcuera (University of Barcelona) 7 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Let a ∈ Λ , with | a | = m , the mapping H ⊙ m I m : I → H m ∞ i = 1 e ⊗ a i Π ∞ � ⊗ �→ i = 1 H a i ( W ( e i )) , i where � ⊗ denotes the symmetrization of the tensor product ⊗ , between H ⊙ m , equipped with the norm the symmetric tensor product I √ m ! �·� H ⊗ m and the m -th chaos, is a linear isometry. We also define I 0 as the identity in R . fblanc1 José M. Corcuera (University of Barcelona) 8 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes H ⊗ m , we define the For any h = h 1 ⊗ · · · ⊗ h m and g = g 1 ⊗ · · · ⊗ g m ∈ I p -th contraction of h and g , denoted by h ⊗ p g , as the element of H ⊗ 2 ( m − p ) given by I h ⊗ p g = < h m , g 1 > I H · · · < h m − p + 1 , g p > H h 1 ⊗· · ·⊗ h m − p ⊗ g p + 1 ⊗· · ·⊗ g m . I H ⊗ m . Note that if h This can be extended by linearity to any element of I H ⊙ m , h ⊗ p g does not necessarily belong to I H ⊙ ( 2 m − p ) . and g belong to I fblanc1 José M. Corcuera (University of Barcelona) 9 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes We have the following properties I m ( e i ⊗ m ) = H m ( W ( e i )) . � p � � q � I p ( h ) I q ( g ) = � p ∧ q I p + q − 2 r ( h � r = 0 r ! ⊗ r g ) r r fblanc1 José M. Corcuera (University of Barcelona) 10 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Note that if we take h = e i ⊗ p , g = e i ⊗ q we obtain � p � � q � p ∧ q � H p ( W ( e i )) H q ( W ( e i )) = r ! H p + q − 2 r ( W ( e i )) r r r = 0 fblanc1 José M. Corcuera (University of Barcelona) 11 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Theorem Let G the σ -field generated by the random variables { W ( h ) , h ∈ I H } . Any square integrable random variable F ∈ L 2 (Ω , G , P ) can be expanded as ∞ � F = I m ( f m ) . m = 0 fblanc1 José M. Corcuera (University of Barcelona) 12 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Consider a sequence of d -dimensional random vectors F n = ( F 1 n , F 2 n , ..., F d n ) , such that F k n ∈ L 2 (Ω , G , P ) and ∞ � F k I m ( f k n = m , n ) m = 0 fblanc1 José M. Corcuera (University of Barcelona) 13 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Theorem Assume the following conditions hold: (i) For k , l = 1 , . . . , d we have ∞ � m ! || f k m , n || 2 lim H ⊗ m = Σ kk . I n →∞ m = 1 ∞ � n →∞ � f k m , n , f l lim m , n � = Σ kl , k � = l , m = 1 (ii) For any m ≥ 1 , k = 1 , . . . , d and r = 1 , . . . , m − 1 n →∞ || f k m , n ⊗ r f k m , n || 2 lim H ⊗ 2 ( m − r ) = 0 . I Then we have D F n − f 0 , n − → N d ( 0 , Σ) . (1) fblanc1 as n tends to infinity. José M. Corcuera (University of Barcelona) 14 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes Consider the simple case where we have a family of stationary, centered, Gaussian random variables { X i } i ≥ 1 , with E ( X 2 1 ) = 1, and we want to know the behavior of the sequence � n � � 1 Y n = √ n H ( X i ) − E ( H ( X 1 )) i = 1 when n goes to infinity. We assume that E ( H ( X 1 ) 2 ) < ∞ . We can take H 1 = span { X i , i ≥ 1 } , and I H ≡ H 1 !, then fblanc1 José M. Corcuera (University of Barcelona) 15 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes ∞ � H ( x ) = c m H m ( x ) m = 0 and ∞ n � � 1 = √ n c m H m ( X i ) Y n m = 1 i = 1 ∞ n � � 1 c m I m ( X ⊗ m = √ n ) i m = 1 i = 1 � � ∞ n � � 1 c m X ⊗ m √ n = I m i m = 1 i = 1 then n � 1 c m X ⊗ m f m , n = √ n i i = 1 fblanc1 and José M. Corcuera (University of Barcelona) 16 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes ∞ ∞ n � � � m ! c 2 m ! || f m , n || 2 m ρ ( i − j ) m = H ⊗ m I n m = 1 m = 1 i , j = 1  � � ∞ n − 1 � � 1 − j m ! c 2  1 + ρ ( j ) m  , = m n m = 1 j = 1 n � f m , n ⊗ r f m , n = c 2 ρ ( i − j ) r X ⊗ ( m − r ) ⊗ X ⊗ ( m − r ) m , i j n i , j = 1 and fblanc1 José M. Corcuera (University of Barcelona) 17 / 53

Power variation for some class of processes Sequences of functionals of Gaussian processes � f m , n ⊗ r f m , n � 2 H ⊗ 2 ( m − r ) I n � c 4 m ρ ( i − j ) r ρ ( k − l ) r ρ ( i − k ) m − r ρ ( j − l ) m − r = n 2 i , j , k , l = 1 n − 1 � c 4 ρ ( i ) r ρ ( j − k ) r ρ ( j ) m − r ρ ( i − k ) m − r ( 1 − i ∨ j ∨ k m = ) n n i , j , k = 0 fblanc1 José M. Corcuera (University of Barcelona) 18 / 53