Noliear Aalysis: Modellig ad Cotrol, 2010, Vol 15, No 4, 445 450 O the covergece rates of Gladyshev s Hurst idex estimator K Kubilius 1, D Melichov 2 1 Istitute of Mathematics ad Iformatics, Vilius Uiversity Aademijos str 4, LT-08663 Vilius, Lithuaia estutisubilius@miivult 2 Vilius Gedimias techical uiversity Saulėteio ave 11, LT-10223 Vilius, Lithuaia Received: 2010-11-11 Revised: 2010-11-18 Published olie: 2010-11-29 Abstract This paper presets the covergece rates for a modified Gladyshev s estimator of the Hurst idex of the fractioal Browia motio Keywords: fractioal Browia motio, Gladyshev s Hurst idex estimator, covergece rate 1 Itroductio The fractioal Browia motio fbm) ad processes based o it have foud may applicatios i fields as diverse as ecoomics ad fiace, physics, chemistry, medicie ad evirometal studies The fbm is a well-ow example of a process with log-rage depedece property Recetly much attetio has bee give to the study of the Hurst parameter, or of the other parameters associated to log rag depedece By fbm B H = {Bt H : t 0}, 0 < H < 1, we uderstad a cetered Gaussia process with B0 H = 0 ad covariace cov B H t, B H s ) 1 = t 2H + s 2H t s 2H), t, s 0 2 The parameter H is called the Hurst idex of the process I 1961, E Gladyshev [1] derived a limit theorem for a statistic based o the first order quadratic variatios for a class of Gaussia processes The fbm B H belogs to this class of processes Gladyshev proposed a estimator of H which was strogly cosistet, but ot asymptotically ormal I 1997, aother estimator was itroduced by J Istas ad G Lag [2] which agai employed the first order quadratic variatios ad it was asymptotically ormal for H 1/2; 3/4) I 2005, A Bégy [3, 4] cosidered the Supported by the Research Coucil of Lithuaia cotract No MIP-66/2010 445
K Kubilius, D Melichov secod order quadratic variatios for processes with Gaussia icremets I 2008 2010, K Kubilius ad D Melichov [7 10] studied the behavior of the first ad secod order quadratic variatios of the pathwise solutio of certai stochastic differetial equatios drive by fbm It was show that the quadratic variatio based estimators remai strogly cosistet i that case as well I this ote we defie the modified Gladyshev s estimator of fbm parameter H ad derive the rate of covergece of it to its real value To our owledge, this problem is ew ad iterestig from the practical poit of view Let π = {0 = t 0 < t 1 < < t = 1} be a sequece of subdivisios of the iterval [0, 1] such that t = for all N ad all {0,, }, where ) is a icreasig sequece of atural umbers Such subdivisio π is called regular For a real-valued process X = {X t, t [0, 1]} taig values at the poits t, = 0,,, the first order quadratic variatio is defied as V 1) ) X, 2) = X 2, X = X t ) ) X t 1 =1 Let B H be the fractioal Browia motio with the Hurst idex H Set t = 2, = 1,, 2 It is ow see Gladyshev [1]) that 2 2H 1) V B H, 2 ) as 1 as This result yields that H = 1 1) l V B H, 2) 2 2 l 2 is a strogly cosistet estimator of H Let us defie a modified Gladyshev s estimator of fbm parameter H by 1 Ĥ = 2 l [ 1) V B H, 2) ] ) 1 C, 2 l for a regular subdivisio π, where C = { V B H, 2 ) N 2 } The estimate Ĥ is strogly cosistet Moreover, we ca derive the rate of covergece of it to H This follows from the followig theorem Theorem 1 Let B H, 1/2 < H < 1, be the fractioal Browia motio Ĥ is a strogly cosistet estimator of the Hurst idex H ad the followig rates of covergece hold: ) Ĥ H = O N 1 l as if N 2 < 1) ad ) E Ĥ H = O N 1 l 2) =1 446
O the covergece rates of the discrete variatio based Hurst idex estimators of fbm 2 Proof of Theorem 1 First we have Ĥ = H1 C l B 2 l 1 C, where B = N 2H 1 B H, 2) Thus Ĥ H H1 C + l B 2 l 1 {B N 2 V 1) } H1 {B<N 1 } l B 1 2 l N {N 2 B + l B <1} 1 {B 1} 2 l Let δ ) be a sequece of positive umbers such that δ < 1 ad δ 0 The iequality l1 x) 20x, 0 x 19/20, gives l B )1 {1 δ B <1} = l [ 1 1 B ) ]) 1 {1 δ B <1} if δ 19/20 So, we have Thus 201 B )1 {1 δ B <1}, l B 1 2 l N {N 2 B 1 <1} {N 2 B + 10 1 B <1 δ } l 1 {1 δ B <1} 1 {N 2 B + 10δ <1 δ } 1 {1 δ B l <1} 3) A applicatio of iequality l1 + x) x, x 0, yields l B )1 {B 1} = l [ 1 + B 1) ]) 1 {B 1} B 1)1 {B 1} l B 1 {B 1} B 1 1 {1 B 1+δ 2 l 2 l } + B 1 1 {B>1+δ 2 l } δ 1 {1 B 1+δ 2 l } + B 1 1 {B>1+δ 2 l } 4) Iequalities 3), 4) implies that Ĥ H 2 + B ) 1 1 { B 1 >δ 2 l } + 10δ 5) l To complete the proof, it suffices to estimate the first term i iequality 5) by usig Haso ad Wright iequality [5] Note that N 2H 1 V B H, 2 ) is the square of the Euclidea orm of oe -dimesioal Gaussia vector X with compoets N 2H 1 B H,, 1 447
K Kubilius, D Melichov By liear trasformatio of X oe ca get a ew Gaussia vector Y with idepedet compoets So there exists oegative real umbers λ 1,,, λ N, ) ad oe - dimesioal Gaussia vector Y, such that its compoets are idepedet Gaussia variables N 0, 1) ad N 2H 1 V B H, 2 ) ) = λ j,n Y j) 2 j=1 Numbers λ 1,,, λ N, ) are the eigevalues of the symmetric -matrix N 2H 1 E [ B H, j B H, ]) 1 j, With the argumets of [6] ad [3] oe ca get iequality P N 2H 1 V 1) B H, 2 ) EV B H, 2 ) ) ε 2 exp Kε 2 ), 6) which follows directly from Haso ad Wright iequality, where 0 < ε 1, K is a positive costat Set δ 2 = 2 l K From iequality 6) it follows that P ) 2 B 1 > δ N 2 7) Obviously, P 2 + B 1 2 l ) ) 1 { B 1 >δ } > 0 P ) 2 B 1 > δ N 2 Uder coditio of the theorem ad from the Borel Catelli lemma it follows that { 1 P lim sup 2 + B ) }) 1 1 { B 1 >δ 2 l } > 0 = 0, ie, 2 + B ) 1 1 { B 1 >δ 2 l } = 0 for sufficietly large From what has bee said above ad iequality 5) it follows that ) Ĥ H = O N 1 l as 448
O the covergece rates of the discrete variatio based Hurst idex estimators of fbm which completes the proof of 1) Note that from the iequalities 5) ad 7) we get E Ĥ H 2 N 2 + E B 1 1 { B 1 >δ 2 l } + 10δ l We ow estimate the secod term o the right side of the previous iequality Note that Thus E B 1 1 { B 1 >δ } E 1/2 B 1 2 P ) 2 B 1 > δ E 1/2 B 2 + 1 ) ) 2 N 2H 1/2 E 1/2 2 3 N 1/2 + 1 ) B H, 4 + 1 =1 E Ĥ H 2 1/2 3 + 1 N 2 + + 10δ l l The proof of 2) is completed Refereces 1 EG Gladyshev, A ew Limit theorem for stochastic processes with Gaussia icremets, Theory Probab Appl, 61), pp 52 61, 1961 2 J Istas, G Lag, Quadratic variatios ad estimatio of the local Hölder idex of a Gaussia process, A Ist Heri Poicaré, Probab Stat, 33, pp 407 436, 1997 3 A Bégy, Quadratic variatios alog irregular subdivisios of Gaussia processes, Electro J Probab, 10, pp 691 717, 2005 4 A Bégy, Geeralized quadratic variatios of Gaussia processes: limit theorems ad applicatios to fractioal processes, PhD thesis, 2006 5 DI Haso, FT Wright, A boud o tail probabilities for quadratic forms i idepedet radom variables, A Math Stat, 42, pp 1079 1083, 1971 6 R Klei, E Gie, O quadratic variatios of processes with Gaussia icremets, A Probab, 34), pp 716 721, 1975 7 K Kubilius, D Melichov, O estimatio of the Hurst idex of solutios of stochastic itegral equatios, Liet Mat Ri, LMD Darbai, 48/49, pp 401 406, 2008 8 K Kubilius, D Melichov, Estimatig the Hurst idex of the solutio of a stochastic itegral equatio, Liet Mat Ri, LMD Darbai, 50, pp 24 29, 2009 449
K Kubilius, D Melichov 9 K Kubilius, D Melichov, Quadratic variatios ad estimatio of the Hurst idex of the solutio of SDE drive by a fractioal Browia motio, Lith Math J, 504), pp 401 417, 2010 10 K Kubilius, D Melichov, O compariso of the estimators of the Hurst idex of the solutio of SDEs drive by a fbm, Iformatica i press) 450