On estimation for the fractional Ornstein-Uhlembeck process observed at discrete time Stefano M. Iacus Department of Economics, Management and Quantitative Methods University of Milan & Core Team joint work with A. Brouste (Univ. Le Mans, FR) ISI Tokyo Satellite Meeting, Tokyo, 2-9-2013
Fractional Brownian Motion � � Let W H = W H t , t ≥ 0 be a normalized fractional Brownian motion (fBM), i.e. the zero mean Gaussian processes with covariance function � | s | 2 H + | t | 2 H − | t − s | 2 H � = 1 E W H s W H t 2 with Hurst exponent H ∈ (0 , 1) . the process is self-similar ( W H at ∼ a H W H t ) ■ presents long range dependence (persistency - antipersistency) ■ has dependent increments (apart for H = 1 2 ). ■ For H = 1 2 , W H = W t is a standard Brownian motion, i.e. independent t increments. 2 / 33
Fractional Brownian Motion 0 < H < 1 H = 1 1 2 < H < 1 2 2 antipersistent independence persistent negative Brownian positive correlated motion correlated 3 / 33
fractional Ornstein-Uhlenbeck (fOU) Let X = ( Y t , t ≥ 0) be a fractional Ornstein-Uhlenbeck process (fOU), i.e. the solution of � t Y s ds + σW H Y t = y 0 − λ t > 0 , Y 0 = y 0 , (1) t , 0 where unknown parameter ϑ = ( λ, σ, H ) belongs to an open subset Θ of (0 , Λ) × [ σ, σ ] × (0 , 1) , 0 < Λ < + ∞ , 0 < σ < σ < + ∞ and W H = ( W H t , t ≥ 0) is a standard fractional Brownian motion [10, 12] of Hurst parameter H ∈ (0 , 1) , The fOU process is not Markovian nor a semimartingale for H � = 1 2 but nevertheless Gaussian and ergodic. ([2]) We denote discrete observations of Y t by X j = Y t j = Y ( t j ) , where 0 = t 0 < t 1 < · · · < t N = T is a grid of deterministic times. 4 / 33
Auxiliary known facts about fBm Let H ∈ (0 , 1) , t > s , { W t , t ∈ [0 , T ] } a standard Brownian motion and � t 1 ( u − s ) H − 3 2 u H − 1 2 − H K H ( t, s ) = c H s 2 du s � � 1 2 H (2 H − 1) with c H = . The fBM can be written as follows: Beta ( 2 − 2 H,H − 1 2 ) � t W H K H ( t, s ) dW s , t ∈ [0 , T ] = t 0 The following is called random walk approximation to fBM, t ∈ [0 , T ] , i K H � � [ Nt ] √ � � N [ Nt ] B H,N = ξ i ’s i.i.d., E ( ξ i ) = 0 , Var ( ξ i ) = 1 . N N , s ds ξ i t i =1 i − 1 N w Then, as N → ∞ , B H,N → W H in Skorohod topology [14]. t t 5 / 33
fOU estimation: QMLE Bertin et al. (2011) [1] considered the following statistical problem dY t = λdt + dW H t with λ ∈ R unknown and H ∈ ( 1 2 , 1) known. Using random walk approximation & Euler scheme for the fOU, with X j = Y t j , t j = j ∆ , j = 0 , 1 , . . . , N , N ∆ = T � � B H,N t j +1 − B H,N X j +1 = X j + a ∆ + t j the following QMLE estimator of λ N α − 1 � (1+ α j )( X j +1 − X j − h j ( X 1 ,...,X j )) F 2 j j =0 ˆ λ N = N N α − 1 � (1+ α j ) 2 F 2 j j =0 where α j , F j and h j ( · · · ) are explicit functions on the data. 6 / 33
fOU estimation: true MLE Bertin et al. (2011) proved that, under the asymptotic: N → ∞ , T = N ∆ → ∞ N α with α < 1 the QMLE estimator ˆ 1 and ∆ = λ N is unbiased and consistent for λ given the known H ∈ ( 1 2 , 1) . � � 2 N − 1 � and given that [15] N 2 H − 1 S N ∼ 1 , for large N Let S N = N − X i X i +1 N i =0 if H is estimated from the data with H N = 1 + log S N ˆ log N by simulation results only it has been shown that the estimator ˆ λ N is consistent and its variance is an increasing function of H . 7 / 33
fOU estimation: true MLE Hu et al. (2011) [6] considered the following statistical problem dY t = λdt + σdW H t with λ ∈ R , σ ∈ R + unknown and H ∈ (0 , 1) known. Let t = (∆ , 2∆ , . . . , N ∆) ′ , X = ( X 1 , X 2 , . . . , X N ) ′ , and Γ H = [ Cov ( W H i ∆ , W H j ∆ )] i,j =1 , 2 ,...,N Then, by Malliavin calculus, it is possible to prove that the true MLE estimators µ N = t ′ Γ − 1 ( X ′ Γ − 1 H X )( t ′ Γ − 1 H t ) − ( t ′ Γ − 1 H X ) 2 σ N = 1 H X ˆ and ˆ t ′ Γ − 1 t ′ Γ − 1 N H t H t N σ 2 ] N ) = N − 1 σ 2 are strongly consistent as N → ∞ [ though E (ˆ 8 / 33
fOU estimation: true MLE Central Limit Theorems. Further, it is possible to prove that � � 0 , σ 2 � µ N − µ ) d t ′ Γ − 1 → N H t (ˆ and � � N − σ 2 � d � 0 , σ 2 � 1 N σ 2 ˆ → N σ 2 2 as N → ∞ . Simulations show that the empirical variance of ˆ µ N increases with H . 9 / 33
fOU estimation: contrast functions Ludena (2004) [11] Let H ∈ ( 1 2 , 3 4 ) be known and consider the “vanishing drift” fOU process � t σ ( θ, Y s )d B H Y t = y 0 + s 0 � N Let U N ( θ ) = 1 h ( θ, X k ∆ , ∆ X k n H ) with h = h ( θ, x, y ) at most of polynomial N k =1 growth in x and y . The minimum contrast estimator ˆ θ N = arg min θ ∈ Θ U N ( θ ) √ θ N − θ ) d N (ˆ is asymptotically Gaussian, i.e. → N . Result extends to the following fOU model dY t = − λY t dt + σ ( θ ) dW H t with λ > 0 known. 10 / 33
fOU estimation: contrast functions Neuenkirch and Tindel (2011) [13] Let H ∈ ( 1 2 , 1) be known and consider the fSDE � t m � σ j W ( j ) H Y t = y 0 + b ( Y s ; θ )d s + t 0 j =1 ) ′ is an m -dimensional fBM, σ j , t = ( W (1) H , W (2) H , . . . , W ( m ) H where W H t t t j = 1 , 2 , . . . , m and b ( y, θ ) are known (up to θ ). Let ∆ = κN − α , α ∈ (0 , 1) and κ > 0 . Let N − 1 m � � 1 | ∆ X k − b ( X k ; θ )∆ | 2 − | σ j | 2 ∆ 2 H Q N ( θ ) = N ∆ 2 j =1 k =0 then, the least squares estimator ˆ θ N = arg min θ ∈ Θ | Q N ( θ ) | is strongly consistent. t , ˆ For the special case dY t = θY t dt + dW H θ N is explicit. 11 / 33
fOU estimation: plug-in Xiao et al. (2011) [16] Let H ∈ ( 1 2 , 1) be known and consider the fOU process dY t = − λY t dt + σdW H t The estimator Γ(3 − 2 H ) σ 2 2 )( N ∆) 2 − 2 H × ˆ N = 2 H Γ 3 ( 3 2 − H )Γ( H + 1 � � 2 j j � � � N 1 1 1 1 2 − H ( j ∆ − ∆ − i ∆) 2 − H ∆ X i − 2 − H ( j ∆ − i ∆) 2 − H ∆ X i ( i ∆) ( i ∆) j =1 i =1 i =1 is strongly consistent for σ 2 . 4 ) the estimator ˆ Moreover, for H ∈ ( 1 2 , 3 λ N (with σ known) � � − 1 N � 2 H 1 ˆ X 2 λ N = i σ 2 H Γ(2 H ) N i =0 is also strongly consistent for λ . 12 / 33
Our proposal The present work exposes an estimation procedure for estimating all three components of ϑ = ( λ, σ, H ) given the regular discretization of the sample path Y T = ( Y t , 0 ≤ t ≤ T ) dY t = λY t dt + σdW H t ∈ [0 , T ] t , from discrete observations ( X n := Y n ∆ N , n = 0 , 1 , . . . , N ) , where T = T N = N ∆ N − → + ∞ and ∆ N − → 0 as N − → + ∞ . Goal: estimate all three elements of ϑ . As H and σ can be efficiently estimated without the knowledge of λ we propose a two stage procedure. 13 / 33
Quadratic generalized variations Let a = ( a 0 , . . . , a K ) be a discrete filter of length K + 1 , K ∈ N , and of order L ≥ 1 , K ≥ L , i.e. K K � � a k k L � = 0 . a k k ℓ = 0 0 ≤ ℓ ≤ L − 1 for and (2) k =0 k =0 Let it be normalized with K � ( − 1) 1 − k a k = 1 . (3) k =0 In the following, we will also consider dilatated filter a 2 associated to a defined by � a k ′ if k = 2 k ′ a 2 0 ≤ k ≤ 2 K . k = for 0 otherwise. 2 K � � K k k r = 2 r k r a k , filter a 2 as the same order than a . a 2 Since k =0 k =0 14 / 33
Quadratic generalized variations estimators of H and σ Denote by � K � 2 N − K � � V N, a = a k X i + k i =0 k =0 the generalized quadratic variations associated to the filter a (see for instance [7]). Then, the estimators of H and σ are as follows V N, a 2 H N = 1 � 2 log 2 V N, a and 1 2 V N, a − 2 · � σ N = � H N ∆ 2 � k,ℓ a k a ℓ | k − ℓ | 2 � H N N 15 / 33
Properties of estimators of H and σ Theorem 1. Let a be a filter of order L ≥ 2 . Then, both estimators � H N and � σ N are strongly consistent, i.e. σ N ) a.s. ( � − → ( H, σ ) as N − → + ∞ . H N , � Moreover, we have asymptotical normality property, i.e. as N → + ∞ , for all H ∈ (0 , 1) , √ L N ( � H N − H ) − → N (0 , Γ 1 ( ϑ, a )) and √ N L σ N − σ ) − → N (0 , Γ 2 ( ϑ, a )) log N ( � where Γ 1 ( ϑ, a ) and Γ 2 ( ϑ, a ) symmetric definite positive matrices depending on σ , H , λ and the filter a (see next slide). Proof: based on an application of [7, Theorem 3(i)]. 16 / 33
Asymptotic variances of ˆ H N and ˆ σ N Let � � mK nK ℓ | mk − nℓ + i | 2 H a m k a n ρ a m , a n m =0 ℓ =0 ( mn ) H � ( i ) = H a k a ℓ | k − ℓ | 2 H k,ℓ � ( i ) 2 � � 1 H ( i ) 2 + ρ a 2 , a 2 ( i ) 2 − 2 ρ a , a 2 ρ a , a Γ 1 ( ϑ, a ) = H H 2 log(2) 2 j ∈ Z and Γ 2 ( ϑ, a ) = σ 2 4 Γ 1 ( ϑ, a ) see also [3]. 17 / 33
Recommend
More recommend
