The rate of convergence of estimate for Hurst index of fractional Brownian motion involved into stochastic differential equation
aa r X i v : . [ m a t h . P R ] N ov The rate of convergence of estimate for Hurst index offractional Brownian motion involved into stochasticdifferential equation ✩ K. Kubilius a, ∗ , Y. Mishura b a Vilniaus university Institute of Mathematics and Informatics, Akademijos 4, LT-08663Vilnius, Lithuania b National Taras Shevchenko Kyiv University, Volodymyrska 64, 01601 Kiev, Ukraine
Abstract
We consider stochastic differential equation involving pathwise integral with re-spect to fractional Brownian motion. The estimates for the Hurst parameterare constructed according to first- and second-order quadratic variations of ob-served values of the solution. The rate of convergence of these estimates to thetrue value of a parameter is established.
Keywords:
Fractional Brownian motion; stochastic differential equation; first-and second-order quadratic variations; estimates of Hurst parameter; rate ofconvergence.
1. Introduction
Consider stochastic differential equation X t = ξ + Z t f ( X s ) ds + Z t g ( X s ) dB Hs , t ∈ [0 , T ] , T > , (1)where f and g are measurable functions, B H is a fractional Brownian motion(fBm) with Hurst index 1 / < H < ξ is a random variable. It is well-knownthat almost all sample paths of B H have bounded p -variations for p > /H. Therefore it is natural to define the integral with respect to fractional Brownianmotion as pathwise Riemann-Stieltjes integral (see, e.g., [16] for the originaldefinition and [5] for the advanced results).A solution of stochastic differential equation (1) on a given filtered probabil-ity space (Ω , F , P , F = {F t } , t ∈ [0 , T ]), with respect to the fixed fBm ( B H , F ) , ✩ ∗ Corresponding author
Preprint submitted to Elsevier June 20, 2018 / < H < F -measurable initial condition ξ is an adapted to thefiltration F continuous process X = { X t : 0 t T } such that X = ξ a.s., P (cid:18) Z t | f ( X s ) | ds + (cid:12)(cid:12) Z t g ( X s ) dB Hs (cid:12)(cid:12) < ∞ (cid:19) = 1 for every 0 t T, and its almost all sample paths satisfy (1).For 0 < α C α ( R ) denotes the set of all C -functions g : R → R suchthat sup x | g ′ ( x ) | + sup x = y | g ′ ( x ) − g ′ ( y ) || x − y | α < ∞ . Let f be a Lipschitz function and let g ∈ C α ( R ), H − < α
1. Thenthere exists a unique solution of equation (1) with almost all sample paths inthe class of all continuous functions defined on [0 , T ] with bounded p -variationfor any p > H (see [6], [12], [13] and [10]). Different (but similar in many fea-tures) approach to the integration with respect to fractional Brownian motionbased on the integration in Besov spaces and corresponding stochastic differ-ential equations were studied in [15], see also [4] and [14] where the differentapproaches to stochastic integration and to stochastic differential equations in-volving fractional Brownian motion are summarized.The main goal of the present paper is to establish the rate of convergenceof two estimates of Hurst parameter to the true value of a parameter. Theestimates are based on the two types of the quadratic variations of the observedsolution to stochastic differential equation involving the integral with respectto fractional Brownian motion and considered on the fixed interval [0 , T ]. Thepaper is organized as follows: Section 2 contains some preliminary information.More precisely, subsection 2.1 describes the properties of p -variations and of theintegrals with respect to the functions of bounded p -variations while subsec-tion 2.2 contains the results on the asymptotic behavior of the normalized first-and second-order quadratic variations of fractional Brownian motion. Section 3describes the rate of convergence of the first- and second-order quadratic varia-tions of the solution to stochastic differential equation involving fBm. Section 4contains the main result concerning the rate of convergence of the constructedestimates of Hurst index to its true value when the diameter of partitions of theinterval [0 , T ] tends to zero. Section 5 contains simulation results.
2. Preliminaries
First, we mention some information concerning p -variation and the functionsof bounded p -variation. It is containing, e.g., in [5] and [16]. Let interval[ a, b ] ⊂ R . Consider the following class of functions: W p (cid:0) [ a, b ] (cid:1) := (cid:8) f : [ a, b ] → R : v p (cid:0) f ; [ a, b ] (cid:1) < ∞ (cid:9) , v p (cid:0) f ; [ a, b ] (cid:1) = sup π n X k =1 (cid:12)(cid:12) f ( x k ) − f ( x k − ) (cid:12)(cid:12) p . Here π = { x i : i = 0 , . . . , n } stands for any finite partition of [ a, b ] such that a = x < x i < · · · < x n = b . Denote Π([ a, b ]) the class of such partitions. Wesay that function f has bounded p -variation on [ a ; b ] if v p ( f ; [ a, b ]) < ∞ .Let V p ( f ) := V p ( f ; [ a, b ]) = v /pp ( f ; [ a, b ]). Then for any fixed f we have that V p ( f ) is a non-increasing function of p . It means that for any 0 < q < p therelation V p ( f ) V q ( f ) holds.Let a < c < b and let f ∈ W p ([ a, b ]) for some p ∈ (0 , ∞ ) . Then v p (cid:0) f ; [ a, c ] (cid:1) + v p (cid:0) f ; [ c, b ] (cid:1) v p (cid:0) f ; [ a, b ] (cid:1) ,V p (cid:0) f ; [ a, b ] (cid:1) V p (cid:0) f ; [ a, c ] (cid:1) + V p (cid:0) f ; [ c, b ] (cid:1) . Let f ∈ W q ([ a, b ]) and h ∈ W p ([ a, b ]), where p − + q − >
1. Then thewell-known Love-Young inequality states that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ba f dh − f ( y ) (cid:2) h ( b ) − h ( a ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C p,q V q (cid:0) f ; [ a, b ] (cid:1) V p (cid:0) h ; [ a, b ] (cid:1) , (2)whence V p Z · a f dh ; [ a, b ] ! C p,q V q, ∞ (cid:0) f ; [ a, b ] (cid:1) V p (cid:0) h ; [ a, b ] (cid:1) . (3)Here V q, ∞ ( f ; [ a, b ]) = V q ( f ; [ a, b ]) + sup a x b | f ( x ) | , C p,q = ζ ( p − + q − ) and ζ ( s ) = P n > n − s is the Riemann zeta function. Further, for any y ∈ [ a, b ] V p Z · a [ f ( x ) − f ( y )] dh ( x ); [ a, b ] ! C p,q (cid:2) V q (cid:0) f ; [ a, b ] (cid:1) + sup a x b | f ( x ) − f ( y ) | (cid:3) V p (cid:0) h ; [ a, b ] (cid:1) C p,q V q (cid:0) f ; [ a, b ] (cid:1) V p (cid:0) h ; [ a, b ] (cid:1) . (4)Denote | A | ∞ = sup x ∈ R | A ( x ) | , | A | α = sup x,y ∈ R | A ( x ) − A ( y ) || x − y | α . Let F be a Lipschitz function and let G ∈ C α ( R ) with 0 < α p < α . Then for any h ∈ W p ([ a, b ]) V p, ∞ (cid:0) F ( h ); [ a, b ] (cid:1) LV p (cid:0) h ; [ a, b ] (cid:1) + sup a x b | F ( h ( x )) − F ( h ( a )) | + | F ( h ( a )) | LV p (cid:0) h ; [ a, b ] (cid:1) + | F ( h ( a )) | , (5) V p/α, ∞ (cid:0) G ( h ); [ a, b ] (cid:1) V p, ∞ (cid:0) G ( h ); [ a, b ] (cid:1) | G ′ | ∞ V p (cid:0) h ; [ a, b ] (cid:1) + | G ( h ( a )) | , (6)3nd V p/α, ∞ (cid:0) G ′ ( h ); [ a, b ] (cid:1) | G ′ | α V αp (cid:0) h ; [ a, b ] (cid:1) + sup a x b | G ′ ( h ( x )) − G ′ ( h ( a )) | + | G ′ ( h ( a )) | | G ′ | α V αp (cid:0) h ; [ a, b ] (cid:1) + | G ′ ( h ( a )) | . (7)Let f ∈ W p ([ a, b ]) and p > p > . Then V p (cid:0) f ; [ a, b ] (cid:1) Osc( f ; [ a, b ]) ( p − p ) /p V p/p p (cid:0) f ; [ a, b ] (cid:1) , (8)where Osc( f ; [ a, b ]) = sup {| f ( x ) − f ( y ) | : x, y ∈ [ a, b ] } .Take functions f , f ∈ W p ([ a, b ]), 0 < p < ∞ . Then f f ∈ W p ([ a, b ]) and V p (cid:0) f f ; [ a, b ] (cid:1) ≤ V p, ∞ (cid:0) f f ; [ a, b ] (cid:1) C p V p, ∞ (cid:0) f ; [ a, b ] (cid:1) V p, ∞ (cid:0) f ; [ a, b ] (cid:1) . (9)Let f ∈ W q ([ a, b ]) and f ∈ W p ([ a, b ]) . Then it follows from Young’s version ofH¨older’s inequality that for any partition π ∈ Π([ a, b ]) and for any p − + q − > X i V q (cid:0) f ; [ x i − , x i ] (cid:1) V p (cid:0) f ; [ x i − , x i ] (cid:1) V q (cid:0) f ; [ a, b ] (cid:1) V p (cid:0) f ; [ a, b ] (cid:1) . (10)Second, we state some facts from the theory of Riemann-Stieltjes integration.Let f ∈ W q ([ a, b ]) and h ∈ W p ([ a, b ]) with 0 < p < ∞ , q > , /p + 1 /q > . Letsymbol (R) stands for the Riemann integration, and (RS) stands for Riemann-Stieltjes integration. Then integral ( RS ) R ba f d h exists under the additionalassumption that f and h have no common discontinuities. Proposition 1.
Let f : [ a, b ] → R be such function that for some p < f ∈ CW p ([ a, b ]) . Also, let F : R → R be a differentiable function with locallyLipschitz derivative F ′ . Then composition F ′ ( f ) is Riemann-Stieltjes integrablewith respect to f and F ( f ( b )) − F ( f ( a )) = ( RS ) Z ba F ′ ( f ( x )) df ( x ) . Furthermore, the following substitution rule holds.
Proposition 2.
Let f , f and f be functions from CW p ([ a, b ]) , p < .Then ( RS ) Z ba f ( x ) d (cid:18) ( RS ) Z xa f ( y ) df ( y ) (cid:19) = ( RS ) Z ba f ( x ) f ( x ) df ( x ) . Finally, assume that F ( x ) = ( R ) Z xa f ( y ) dy and F ( x ) = ( RS ) Z xa f ( y ) df ( y ) , f is continuous function, f , f ∈ CW p ([ a, b ]) for some 1 p <
2, and Q is a differentiable function with locally Lipschitz derivative q . It follows fromPropositions 1 and 2 that Q ( F ( x ) + F ( x )) − Q (0) = Z xa q ( F ( y ) + F ( y )) d ( F ( y ) + F ( y ))= Z xa q ( F ( y ) + F ( y )) f ( y ) dy + Z xa q ( F ( y ) + F ( y )) f ( y ) df ( y ) . (11) Consider the fractional Brownian motion (fBm) B H = { B Ht , t ∈ [0 , T ] } withHurst index H ∈ ( , H . Moreover, for any 0 < γ < H we have that L H,γT := sup s = ts,t T | B Ht − B Hs || t − s | γ is finite a.s. and even more, E (cid:0) L H,γT (cid:1) k < ∞ for any k > . The followingestimate for the p -variation of fBm is evident: V p (cid:0) B H ; [ s, t ] (cid:1) L H, /pT ( t − s ) /p , (12)where s < t T, p > /H .Let π n = { t n < t n < · · · < t nn = T } , T >
0, be a sequence of uniformpartitions of interval [0 , T ] with t nk = kTn for all n ∈ N and all k ∈ { , . . . , n } ,and let X be some real-valued stochastic process defined on the interval [0 , T ]. Definition 3.
The normalized first- and second-order quadratic variations of X taking along the partitions ( π n ) n ∈ N and corresponding to the value / < H < are defined as V (1) n ( X,
2) = n H − n X k =1 (cid:0) ∆ (1) k,n X (cid:1) , ∆ (1) k,n X = X ( t nk ) − X ( t nk − ) , and V (2) n ( X,
2) = n H − n − X k =1 (cid:0) ∆ (2) k,n X (cid:1) , ∆ (2) k,n X = X (cid:0) t nk +1 (cid:1) − X (cid:0) t nk (cid:1) + X (cid:0) t nk − (cid:1) . For simplicity, we shall omit index n for points t nk of partitions π n .It is known (see, e.g., Gladyshev [7]) that V (1) n ( B H , → T a.s. as n → ∞ .Also, it was proved in Benasi et al. [3] and Istas et al. [9] that V (2) n ( B H , → (4 − H ) T a.s. as n → ∞ .Denote V (1) n ( B H , t = n H − r ( t ) X k =1 (∆ (1) k,n B H ) ,V (2) n ( B H , t = n H − r ( t ) − X k =1 (∆ (2) k,n B H ) , t ∈ [0 , T ] , r ( t ) = max { k : t k ≤ t } = [ tnT ]. It is evident that E V (1) n ( B H , t = ρ ( t ) and E V (2) n ( B H , t = (4 − H ) ρ ( t ) , (13)where ρ ( t ) = max { t k : t k t } .The following classical result will be used in the proof of Theorem 5. Lemma 4. (L´evy-Octaviani inequality) Let X , . . . , X n be independent randomvariables. Then for any fixed t, s > P (cid:18) max i n (cid:12)(cid:12)(cid:12)(cid:12) i X j =1 X j (cid:12)(cid:12)(cid:12)(cid:12) > t + s (cid:19) P (cid:0)(cid:12)(cid:12) P nj =1 X j (cid:12)(cid:12) > t (cid:1) − max i n P (cid:0)(cid:12)(cid:12) P nj = i X j (cid:12)(cid:12) > s (cid:1) . (cid:3) Theorem 5.
The following asymptotic property holds for the first- and second-order quadratic variations of fractional Brownian motion: sup t T (cid:12)(cid:12) V ( i ) n ( B H , t − E V ( i ) n ( B H , t (cid:12)(cid:12) = O (cid:0) n − / ln / n (cid:1) a.s. , i = 1 , . (14) Remark 6.
It follows from (13) and (14) that sup t T (cid:12)(cid:12) V ( i ) n ( B H , t (cid:12)(cid:12) = O (1) a.s., i = 1 , . . Proof.
We can consider V ( i ) n (cid:0) B H , (cid:1) as the square of the Euclidean norm ofthe n -dimensional Gaussian vector X n with the components n H − / ∆ ( i ) k,n B H , k n − ( i − . Obviously, one can get a new n -dimensional Gaussian vector e X n with inde-pendent components applying the linear transformation to X n . It means thatthere exist nonnegative real numbers ( λ ( i )1 ,n , . . . , λ ( i ) n − ( i − ,n ) and such n − ( i − Y n with independent Gaussian N (0 , V ( i ) n (cid:0) B H , (cid:1) = n − ( i − X j =1 λ j,n (cid:0) Y ( j ) n (cid:1) . The numbers ( λ ( i )1 ,n , . . . , λ ( i ) n − ( i − ,n ) are the eigenvalues of the symmetric n − ( i − × n − ( i − (cid:16) n H − E (cid:2) ∆ ( i ) j,n B H ∆ ( i ) k,n B H (cid:3)(cid:17) j,k n − ( i − . Now we can apply the Hanson and Wright’s inequality (see Hanson et al. [8]or Begyn [1]), and it yields that for ε > P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) n − ( i − X j = k λ ( i ) j,n (cid:2)(cid:0) Y ( j ) n (cid:1) − (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:19) (cid:18) − min (cid:20) C ελ ∗ ( i ) k,n , C ε P n − ( i − j = k ( λ ( i ) j,n ) (cid:21)(cid:19) (cid:18) − min (cid:20) C ελ ∗ ( i ) n , C ε P n − ( i − j =1 ( λ ( i ) j,n ) (cid:21)(cid:19) , (15)6here C , C are nonnegative constants, λ ∗ ( i ) k,n = max k j n − ( i − λ ( i ) j,n , λ ∗ ( i ) n =max j n − ( i − λ ( i ) j,n .The evident equality holds: n − ( i − X j =1 λ ( i ) j,n = E V ( i ) n (cid:0) B H , (cid:1) . Furthermore, it follows from (13) that the sequence E V ( i ) n (cid:0) B H , (cid:1) , n ≥ P n − ( i − j =1 λ ( i ) j,n are bounded as well. It is easy to checkthat n − ( i − X j =1 ( λ ( i ) j,n ) λ ∗ ( i ) n n − ( i − X j =1 λ ( i ) j,n . Therefore for any 0 < ε P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) n − ( i − X j = i λ j,n (cid:2)(cid:0) Y ( j ) n (cid:1) − (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:19) (cid:18) − Kε λ ∗ ( i ) n (cid:19) , (16)where K is a positive constant.Now we use L´evy-Octaviani inequality (see Lemma 4) and evident inequality x − x x for 0 < x / P (cid:18) max k n − ( i − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ ( i ) j,n (cid:2)(cid:0) Y ( j ) n (cid:1) − (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) (cid:16) − Kε λ ∗ ( i ) n (cid:17) − (cid:16) − Kε λ ∗ ( i ) n (cid:17) (cid:18) − Kε λ ∗ ( i ) n (cid:19) , assuming that exp (cid:18) − Kε λ ∗ ( i ) n (cid:19) / < ε ≤ . So, for the values of parameters mentioned above, P n H − max k n − ( i − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 (∆ ( i ) j,n B H ) − k X j =1 E (∆ ( i ) j,n B H ) (cid:12)(cid:12)(cid:12)(cid:12) > ε ! (cid:18) − Kε λ ∗ ( i ) n (cid:19) . Furthermore, λ ∗ ( i ) n Kn H − max k n − ( i − n − ( i − X j =1 | d ( i ) jkn | , d ( i ) jkn = E ∆ ( i ) j,n B H ∆ ( i ) k,n B H . From Gladyshev [7] and Begyn [1] we get λ ∗ ( i ) n Cn − . (17)Now we set ε n = 2 CK n − ln n and conclude that P n H − max k n − ( i − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 (∆ ( i ) j,n B H ) − k X j =1 E (∆ ( i ) k,n B H ) (cid:12)(cid:12)(cid:12)(cid:12) > ε n ! (cid:18) − n (cid:19) = 4 n . It means that ∞ X n =0 P n H − max k n − ( i − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 (∆ ( i ) j,n B H ) − k X j =1 E (∆ ( i ) j,n B H ) (cid:12)(cid:12)(cid:12)(cid:12) > ε n ! < ∞ . Finally, we get the statement of the present theorem from the Borel-Cantellilemma and the evident equalitysup t T (cid:12)(cid:12) V ( i ) n ( B H , t − E V ( i ) n ( B H , t (cid:12)(cid:12) = n H − max k n − ( i − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 (∆ ( i ) j,n B H ) − k X j =1 E (∆ ( i ) j,n B H ) (cid:12)(cid:12)(cid:12)(cid:12) .
3. The rate of convergence of the first- and second-order quadraticvariations of the solution of stochastic differential equation
First, we formulate the following result from [11] about convergence of first-and second-order quadratic variation.
Theorem 7.
Consider stochastic differential equation (1) , where function f isLipschitz and g ∈ C α for some < α < . Let X be its solution. Then V ( i ) n ( X, → c ( i ) Z T g (cid:0) X ( t ) (cid:1) dt a.s. as n → ∞ , (18) where c ( i ) = ( for i = 1 , (4 − H ) for i = 2 . Second, we prove the following auxiliary result.8 emma 8.
Let X be a solution of stochastic differential equation (1) . Definea step-wise process X π that is a discretization of process X : X πt = ( X ( t k ) for t ∈ [ t k , t k +1 ) , k = 0 , , . . . , n − ,X ( t n − ) for t ∈ [ t n − , t n ] . Then for any p > H we have that sup t T | X πt − X t | = O (cid:0) n − /p (cid:1) . Proof.
Consider t ∈ [ t k , t k +1 ). We get immediately from the Love-Younginequality (2) that | X πt − X t | = (cid:12)(cid:12)(cid:12)(cid:12) Z tρ n ( t ) f ( X s ) ds + Z tρ n ( t ) g ( X s ) dB Hs (cid:12)(cid:12)(cid:12)(cid:12) T n − sup t k t t k +1 | f ( X t ) | + C p,p V p, ∞ ( g ( X ); [ t k , t k +1 ]) V p ( B H ; [ t k , t k +1 ]) . Further,sup t k t t k +1 | f ( X t ) | sup t T | f ( X t ) − f ( ξ ) | + | f ( ξ ) | LV p ( X ; [0 , t ]) + | f ( ξ ) | , (19)where L is a Lipschitz constant for f , and V p, ∞ ( g ( X ); [ t k , t k +1 ]) | g ′ | ∞ V p ( X ; [ t k , t k +1 ]) + sup t T | g ( X t ) − g ( ξ ) | + | g ( ξ ) | | g ′ | ∞ V p ( X ; [0 , T ]) + | g ( ξ ) | . (20)We get the statement of the lemma from (19), (20) and inequality (12) .Now we prove the main result of this section which specifies the rate ofconvergence in Theorem 7. Theorem 9.
Let the conditions of Theorem 7 hold and, in addition, α > H − .Then V ( i ) n ( X, − c ( i ) Z T g (cid:0) X ( t ) (cid:1) dt = O (cid:0) n − / ln / n (cid:1) . (21) Proof.
Decompose the left-hand side of (21) into three parts: I ( i ) n := V ( i ) n ( X, − c ( i ) Z T g (cid:0) X ( t ) (cid:1) dt = I (1 ,i ) n + I (2 ,i ) n + I (3 ,i ) n , I (1 ,i ) n = n H − n − ( i − X k =1 (cid:0) ∆ ( i ) k,n X (cid:1) − n − ( i − X k =1 g ( X k − i − ) (cid:0) ∆ ( i ) k,n B H (cid:1) ,I (2 ,i ) n = n H − n − ( i − X k =1 g ( X k − i − ) (cid:0) ∆ ( i ) k,n B H (cid:1) − n − ( i − X k =1 g ( X k − i − ) E (cid:0) ∆ ( i ) k,n B H (cid:1) ,I (3 ,i ) n = n H − n − ( i − X k =1 g ( X k − i − ) E (cid:0) ∆ ( i ) k,n B H (cid:1) − c ( i ) Z T g ( X s ) ds, and X k = X ( t k ). We start with the most simple term I (3 ,i ) n and get immediately,similarly to bounds contained in (6), that for any p > H | I (3 ,i ) n | c ( i ) n − ( i − X k =1 − ( i − Z t k +( i − t k − i − (cid:12)(cid:12) g ( X ( t k − i − )) − g ( X s ) (cid:12)(cid:12) ds c ( i ) T | g ′ | ∞ sup t T | X πt − X t | · (cid:2) | g ′ | ∞ V p ( X ; [0 , T ]) + | g ( ξ ) | (cid:3) . In order to estimate I (2 ,i ) n , denote S ( i ) t = n H − r ( t ) − ( i − X k =1 (cid:0) ∆ ( i ) k,n B H (cid:1) , t ∈ [0 , T ] , i = 1 , . Then n H − n − ( i − X k =1 g ( X k − i − ) (cid:0) ∆ ( i ) k,n B H (cid:1) = Z T g ( X t ) dS ( i ) t and n H − n − ( i − X k =1 g ( X k − i − ) (cid:2)(cid:0) ∆ ( i ) k,n B H (cid:1) − E (cid:0) ∆ ( i ) k,n B H (cid:1) (cid:3) = Z T g ( X t ) d (cid:2) S ( i ) t − E S ( i ) t (cid:3) . /p + 1 / > H < p <
2. Therefore, we obtain from theLove-Young inequality and from (8)-(9) that | I (2 ,i ) n | = (cid:12)(cid:12)(cid:12)(cid:12) Z T g ( X t ) d (cid:2) S ( i ) t − E S ( i ) t (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) C p, V p, ∞ (cid:0) g ( X ); [0 , T ] (cid:1) V (cid:0) S ( i ) − E S ( i ) ; [0 , T ] (cid:1) C p, (cid:8) Osc (cid:0) S ( i ) − E S ( i ) ; [0 , T ] (cid:1)(cid:9) / V p, ∞ (cid:0) g ( X ); [0 , T ] (cid:1) × V / (cid:0) S ( i ) − E S ( i ) ; [0 , T ] (cid:1) C p, (cid:16) sup t T (cid:12)(cid:12) S ( i ) t − E S ( i ) t (cid:12)(cid:12)(cid:17) / V p, ∞ (cid:0) g ( X ); [0 , T ] (cid:1) × (cid:20) n H − n − ( i − X k =1 (cid:0) ∆ ( i ) k,n B H (cid:1) + c ( i ) T (cid:21) / . It follows from Theorem 5, Remark 6, and (6) that the rate of convergenceof I (2 ,i ) n is O ( n − / ln / n ).It still remains to estimate I (1 ,i ) n . Consider only i = 2, the proof for i = 1 issimilar.Denote J k = Z t k +1 t k [ f ( X s ) − f ( X k )] ds − Z t k t k − [ f ( X s ) − f ( X k )] ds,J k = Z t k t k − (cid:18) g ( X k ) − g ( X s ) − Z t k s g ′ ( X k ) f ( X k ) du − Z t k s g ′ ( X k ) g ( X k ) dB Hu (cid:19) dB Hs ,J k = Z t k +1 t k (cid:18) g ( X s ) − g ( X k ) − Z st k g ′ ( X k ) f ( X k ) du − Z st k g ′ ( X k ) g ( X k ) dB Hu (cid:19) dB Hs ,J k = g ′ ( X k ) f ( X k ) (cid:16) Z t k +1 t k ( s − t k ) dB Hs + Z t k t k − ( t k − s ) dB Hs (cid:17) ,J k = 12 g ′ ( X k ) g ( X k ) (cid:16)(cid:0) ∆ (1) k,n B H (cid:1) + (cid:0) ∆ (1) k +1 ,n B H (cid:1) (cid:17) , J k = g ( X k )∆ (2) k,n B H . Equalities Z t k t k − (cid:18) Z t k s g ′ ( X k ) g ( X k ) dB Hu (cid:19) dB Hs = 12 g ′ ( X k ) g ( X k ) (cid:0) ∆ (1) k B H (cid:1) , Z t k +1 t k (cid:18) Z st k g ′ ( X k ) g ( X k ) dB Hu (cid:19) dB Hs = 12 g ′ ( X k ) g ( X k ) (cid:0) ∆ (1) k +1 ,n B H (cid:1) . (see Proposition 1) imply ∆ (2) k,n X = X l =1 J lk . f and Lemma 8, we can conclude that | f ( X t ) − f ( X k ) | L sup t T | X πt − X t | = O (cid:0) n − /p (cid:1) . Therefore n − X k =1 ( J k ) T n − n − X k =1 Z t k +1 t k [ f ( X s ) − f ( X k )] ds + 2 T n − n − X k =1 Z t k t k − [ f ( X s ) − f ( X k )] ds T n − L (cid:16) sup t T | X πt − X t | (cid:17) = O (cid:0) n − − /p (cid:1) . Consider J k . It follows from equality (11) that for any fixed t ∈ [ t k − , t k ] g ( X k ) − g ( X t ) = Z t k t g ′ ( X s ) f ( X s ) ds + Z t k t g ′ ( X s ) g ( X s ) dB Hs . (22)Substituting equality (22) into J k we get | J k | (cid:12)(cid:12)(cid:12)(cid:12) Z t k t k − Z t k s (cid:2) g ′ ( X u ) f ( X u ) − g ′ ( X k ) f ( X k ) (cid:3) du dB Hs (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z t k t k − Z t k s (cid:2) g ′ ( X u ) g ( X u ) − g ′ ( X k ) g ( X k ) (cid:3) dB Hu dB Hs (cid:12)(cid:12)(cid:12)(cid:12) . (23)Transforming identically the first term in the right-hand side of (23) andapplying to it Love-Young inequality (2), we conclude that for any p > H (cid:12)(cid:12)(cid:12)(cid:12) Z t k t k − Z t k s (cid:2) g ′ ( X u ) f ( X u ) − g ′ ( X k ) f ( X k ) (cid:3) du dB Hs (cid:12)(cid:12)(cid:12)(cid:12) C p, V (cid:18) Z t k · (cid:2) g ′ ( X u ) f ( X u ) − g ′ ( X k ) f ( X k ) (cid:3) du ; [ t k − , t k ] (cid:19) V p (cid:0) B H ; [ t k − , t k ] (cid:1) C p, V p (cid:0) B H ; [ t k − , t k ] (cid:1) Z t k t k − (cid:12)(cid:12) g ′ ( X u ) f ( X u ) − g ′ ( X k ) f ( X k ) (cid:12)(cid:12) du. (24)Henceforth we consider the following interval of the values of p : H < p < α .Then it follows from inequality (4) that the second term in the right-hand side12f (23) admits the bound: (cid:12)(cid:12)(cid:12)(cid:12) Z t k t k − Z t k s (cid:2) g ′ ( X u ) g ( X u ) − g ′ ( X k ) g ( X k ) (cid:3) dB Hu dB Hs (cid:12)(cid:12)(cid:12)(cid:12) C p,p/α V p/α (cid:18) Z t k · (cid:2) g ′ ( X u ) g ( X u ) − g ′ ( X k ) g ( X k ) (cid:3) dB Hu ; [ t k − , t k ] (cid:19) × V p (cid:0) B H ; [ t k − , t k ] (cid:1) C p,p/α V p/α (cid:0) g ′ ( X u ) g ( X u ); [ t k − , t k ] (cid:1) V p (cid:0) B H ; [ t k − , t k ] (cid:1) . (25)We conclude from (23)–(25) that | J k | T n − C p, V p/α (cid:0) g ′ ( X ) f ( X ); [ t k , t k +1 ] (cid:1) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1) + 2 C p,p/α V p/α (cid:0) g ′ ( X ) g ( X ); [ t k , t k +1 ] (cid:1) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1) . Applying inequalities (9) and (10) we immediately obtain that n X k =1 (cid:0) J k (cid:1) T C p, n − max k n − (cid:2) V p/α (cid:0) g ′ ( X ) f ( X ); [ t k , t k +1 ] (cid:1) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1)(cid:3) × V p/α (cid:0) g ′ ( X ) f ( X ); [0 , T ] (cid:1) V p (cid:0) B H ; [0 , T ] (cid:1) + 4 C p,p/α max k n − (cid:2) V p/α (cid:0) g ′ ( X ) g ( X ); [ t k , t k +1 ] (cid:1) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1)(cid:3) × V p/α (cid:0) g ′ ( X ) g ( X ); [0 , T ] (cid:1) V p (cid:0) B H ; [0 , T ] (cid:1) T C p, n − max k n − (cid:2) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1)(cid:3) V p/α, ∞ (cid:0) g ′ ( X ); [0 , T ] (cid:1) × V p, ∞ (cid:0) f ( X ); [0 , T ] (cid:1) V p (cid:0) B H ; [0 , T ] (cid:1) + 4 C p,p/α max k n − (cid:2) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1)(cid:3) × V p/α, ∞ (cid:0) g ′ ( X ); [0 , T ] (cid:1) V p/α, ∞ (cid:0) g ( X ); [0 , T ] (cid:1) V p (cid:0) B H ; [0 , T ] (cid:1) . It follows from the inequalities (5)–(7) that the values of the variations V p, ∞ (cid:0) f ( X ); [0 , T ] (cid:1) , V p/α, ∞ (cid:0) g ( X ); [0 , T ] (cid:1) , and V p/α, ∞ (cid:0) g ′ ( X ); [0 , T ] (cid:1) are finite. Therefore we get from (12) that n X k =1 (cid:0) J k (cid:1) = O ( n − − /p ) + O ( n − /p ) = O ( n − /p ) . The similar reasonings lead to the similar bound for J k , and we concludethat n − X k =0 [ J k + J k ] = O (cid:0) n − /p (cid:1) . Consider J k . It consists of two terms that can be estimated in a similar way.Applying inequalities (2) and (12), we obtain the following bound for the first13erm: n − X k =1 (cid:2) g ′ ( X k ) f ( X k ) (cid:3) (cid:18) Z t k +1 t k ( s − t k ) dB Hs (cid:19) C p, n − X k =1 (cid:2) g ′ ( X k ) f ( X k ) (cid:3) ( t k +1 − t k ) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1) n − T C p, max k n − (cid:2) g ′ ( X k ) f ( X k ) V p (cid:0) B H ; [ t k , t k +1 ] (cid:1)(cid:3) = O (cid:0) n − − /p (cid:1) . As a consequence, n − X k =1 (cid:0) J k (cid:1) = O (cid:0) n − − /p (cid:1) . Furthermore, note that under our assumptions sup s ∈ [0 ,T ] | g ( X s ) | < ∞ andsup s ∈ [0 ,T ] | g ′ ( X s ) | < ∞ a.s. Therefore we have for the first term in J k that n − X k =1 h g ′ ( X k ) g ( X k ) (cid:0) ∆ (1) k +1 ,n B H (cid:1) (cid:3) max k n − (cid:2) g ′ ( X k ) g ( X k ) (cid:3) n − X k =0 (cid:0) ∆ (1) k +1 ,n B H (cid:1) = O (cid:0) n − /p (cid:1) . The second term is bounded in a similar way, and we conclude that n − X k =1 (cid:0) J k (cid:1) = O (cid:0) n − /p (cid:1) . Thus (cid:18) X l =1 J lk (cid:19) = O (cid:0) n − − /p ∨ n − /p ∨ n − /p (cid:1) = O (cid:0) n − /p (cid:1) . At last, n H − n − X k =1 (cid:2) ∆ (2) k,n X − g ( X k )∆ (2) k,n B H (cid:3) = O (cid:0) n − /p +2 H − (cid:1) = O (cid:0) n − /p +2 H (cid:1) (26)for any H < p < α . Set 1 /p = H − ε for ε < ( H/ − / ∧ ( H − α ).Then V (2) n ( X, − c (2) Z T g ( X s ) ds = O (cid:0) n − /p +2 H (cid:1) + O (cid:0) n − / ln / n (cid:1) + O (cid:0) n − /p (cid:1) = O (cid:0) n − H +4 ε (cid:1) + O (cid:0) n − / ln / n (cid:1) + O (cid:0) n − H + ε (cid:1) = O (cid:0) n − / ln / n (cid:1) . (27)14 . The rate of convergence of estimators of Hurst index Consider the following statistics: R ( i ) n = P n − ( i − k =1 (∆ ( i ) k, n X ) P n − ( i − k =1 (∆ ( i ) k,n X ) and construct the following estimate of Hurst index H : b H ( i ) n = (cid:18) −
12 ln 2 ln R ( i ) n (cid:19) e C n , where e C n = (cid:26) − (cid:0) − n − / (ln n ) / β (cid:1) R ( i ) n n − / (ln n ) / β (cid:27) , β > . Further, introduce the following notation: g ( i ) ( T ) = c ( i ) R T g ( X s ) ds . Theorem 10.
Let conditions of Theorem 7 hold with α > H − . Also, let X be a solution of (1) and assume that random variable g ( i ) ( T ) is separated fromzero: there exists a constant c > such that g ( i ) ( T ) > c a.s. Then b H ( i ) n isa strongly consistent estimator of the Hurst index H and the following rate ofconvergence holds : | b H ( i ) n − H | = O (cid:0) n − / (ln n ) / β (cid:1) a.s. , for any β > . Proof.
Consider a sequence 1 > δ n ↓ n → ∞ . It will be specified later on.Introduce the events C n = (cid:26)
12 (1 − δ n ) R ( i ) n δ n (cid:27) . Also, introduce the notations A ( i ) n = V ( i )2 n ( X,
2) and B ( i ) n = V ( i ) n ( X, H − R ( i ) n = A ( i ) n B ( i ) n . Then C n = (cid:26) H − (1 − δ n ) A ( i ) n B ( i ) n H − (1 + δ n ) (cid:27) , and estimate b H ( i ) n has a form b H ( i ) n = (cid:18) −
12 ln 2 ln R ( i ) n (cid:19) C n .
15t is easy to see that C n := Ω \ C n has a form C n = (cid:26) A ( i ) n B ( i ) n < H − (1 − δ n ) (cid:27) [ (cid:26) A ( i ) n B ( i ) n > H − (1 + δ n ) (cid:27) ⊂ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) A ( i ) n B ( i ) n − (cid:12)(cid:12)(cid:12)(cid:12) > δ n (cid:27) . (28)Then b H ( i ) n = H C n −
12 ln 2 ln (2 n ) H − V ( i )2 n ( X, n H − V ( i ) n ( X, C n = H C n −
12 ln 2 ln A ( i ) n B ( i ) n C n . The latter representation implies that (cid:12)(cid:12) b H ( i ) n − H (cid:12)(cid:12) H (cid:8)(cid:12)(cid:12) A ( i ) nB ( i ) n − (cid:12)(cid:12) >δ n (cid:9) + 12 ln 2 (cid:12)(cid:12)(cid:12)(cid:12) ln A ( i ) n B ( i ) n (cid:12)(cid:12)(cid:12)(cid:12) (cid:8) − δ n A ( i ) nB ( i ) n δ n (cid:9) − (cid:18)
12 ln 2 ln A ( i ) n B ( i ) n (cid:19) (cid:8) H − (1 − δ n ) A ( i ) nB ( i ) n < − δ n (cid:9) + (cid:18)
12 ln 2 ln A ( i ) n B ( i ) n (cid:19) (cid:8) δ n A ( i ) nB ( i ) n < H − (1+ δ n ) (cid:9) := X l =1 L ln . (29)In what follows we need an elementary inequalities: − ln(1 − x ) ≤ x ) x provided that 0 x / L n . We divide it in two parts. As to the first part, it is obviousthat (cid:18) ln A ( i ) n B ( i ) n (cid:19) (cid:8) − δ n A ( i ) nB ( i ) n < (cid:9) = (cid:18) ln (cid:20) − (cid:18) − A ( i ) n B ( i ) n (cid:19)(cid:21)(cid:19) (cid:8) − δ n A ( i ) nB ( i ) n < (cid:9) , and 1 − δ n A ( i ) n B ( i ) n < < − A ( i ) n B ( i ) n δ n . Applying inequality − ln(1 − x ) x , 0 x /
2, we deduce that for δ n / (cid:18) − ln A ( i ) n B ( i ) n (cid:19) (cid:8) − δ n A ( i ) nB ( i ) n < (cid:9) (cid:18) − A ( i ) n B ( i ) n (cid:19) (cid:8) − δ n A ( i ) nB ( i ) n < (cid:9) δ n (cid:8) − δ n A ( i ) nB ( i ) n < (cid:9) . As to the second part, (cid:18) ln A ( i ) n B ( i ) n (cid:19) (cid:8) A ( i ) nB ( i ) n δ n (cid:9) = (cid:18) ln (cid:20) (cid:18) A ( i ) n B ( i ) n − (cid:19)(cid:21)(cid:19) (cid:8) A ( i ) nB ( i ) n δ n (cid:9) (cid:18) A ( i ) n B ( i ) n − (cid:19) (cid:8) A ( i ) nB ( i ) n δ n (cid:9) δ n (cid:8) A ( i ) nB ( i ) n δ n (cid:9) . L n . From here we easy deduce that − (cid:18)
12 ln 2 ln A ( i ) n B ( i ) n (cid:19) (cid:8) H − (1 − δ n ) A ( i ) nB ( i ) n < − δ n (cid:9) −
12 ln 2 (cid:2) ln (cid:0) H − (1 − δ n ) (cid:1)(cid:3) (cid:8) H − (1 − δ n ) A ( i ) nB ( i ) n < − δ n (cid:9) (cid:18) (1 − H ) − ln(1 − δ n )2 ln 2 (cid:19) (cid:8) H − (1 − δ n ) A ( i ) nB ( i ) n < − δ n (cid:9) (cid:18) − H + δ n ln 2 (cid:19) (cid:8) H − (1 − δ n ) A ( i ) nB ( i ) n < − δ n (cid:9) (1 + 2 δ n ) (cid:8) A ( i ) nB ( i ) n < − δ n (cid:9) . The term L n is estimated similarly as the second part of L n . Thus we get (cid:18) ln A ( i ) n B ( i ) n (cid:19) (cid:8) δ n A ( i ) nB ( i ) n H − (1+ δ n ) (cid:9) (cid:18) A ( i ) n B ( i ) n − (cid:19) (cid:8) δ n A ( i ) nB ( i ) n H − (1+ δ n ) (cid:9) (1 + 2 δ n ) (cid:8) A ( i ) nB ( i ) n > δ n (cid:9) . Summarizing, we conclude that | b H ( i ) n − H | (1 + 2 δ n ) (cid:8)(cid:12)(cid:12) A ( i ) nB ( i ) n − (cid:12)(cid:12) >δ n (cid:9) + 2 δ n (cid:8) − δ n A ( i ) nB ( i ) n δ n (cid:9) (1 + 2 δ n ) (cid:8)(cid:12)(cid:12) A ( i ) nB ( i ) n − (cid:12)(cid:12) >δ n (cid:9) + 2 δ n . Now, let β >
0. Note that (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) A ( i ) n B ( i ) n − (cid:12)(cid:12)(cid:12)(cid:12) > δ n (cid:27) ⊂ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) A ( i ) n B ( i ) n − (cid:12)(cid:12)(cid:12)(cid:12) > δ n , B ( i ) n > (ln n ) − β (cid:27) [ (cid:8) B ( i ) n < (ln n ) − β (cid:9) = (cid:8) | A ( i ) n − B ( i ) n | > δ n B ( i ) n , B ( i ) n > (ln n ) − β (cid:9) ∪ (cid:8) B ( i ) n < (ln n ) − β (cid:9) ⊂ (cid:8)(cid:12)(cid:12) A ( i ) n − B ( i ) n (cid:12)(cid:12) > δ n (ln n ) − β (cid:9) ∪ (cid:8) B ( i ) n < (ln n ) − β (cid:9) . Therefore | b H ( i ) n − H | (1 + 2 δ n ) {| A ( i ) n − B ( i ) n | >δ n (ln n ) − β }∪{ B ( i ) n < (ln n ) − β } + 2 δ n . It follows from (27) that | A ( i ) n − B ( i ) n | = O (cid:0) n − / ln / n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) B ( i ) n − c ( i ) Z T g ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) n − / ln / n (cid:1) . Obviously, for any n > exp { (cid:0) c (cid:1) β } we have that g ( i ) ( T ) > c ≥ n ) − β a.s.and (cid:8) B ( i ) n < (ln n ) − β (cid:9) = (cid:8) B ( i ) n < (ln n ) − β , g ( i ) ( T ) > n ) − β (cid:9) . Now, let δ n < (ln n ) − β . Then it is not hard to deduce that (cid:8) B ( i ) n < (ln n ) − β , g ( i ) ( T ) > n ) − β (cid:9) = (cid:8) B ( i ) n < (ln n ) − β , g ( i ) ( T ) > n ) − β , B ( i ) n < g ( i ) ( T ) − δ n (cid:9) ⊂ (cid:8) | B ( i ) n − g ( i ) ( T ) | > δ n (cid:9) . Therefore, (cid:8) B ( i ) n < (ln n ) − β (cid:9) ⊂ (cid:8) | B ( i ) n − g ( i ) ( T ) | > δ n (cid:9) if n > exp { (cid:0) c (cid:1) β } .Finally, specify δ n . More precisely, set δ n = n − / (ln n ) / β , β >
0. Notethat δ n < (ln n ) − β for sufficiently large n and, moreover, O (cid:0) n − / ln / n (cid:1) δ n (ln n ) − β = O (cid:0) n − / ln / n (cid:1) n − / (ln n ) / β −→ n → ∞ . The latter relation together with Theorem 9 imply that for any ω ∈ Ω ′ with P (Ω ′ ) = 1 there exists n = n ( ω ) such that for any n > n (cid:8) | A ( i ) n − B ( i ) n | >δ n (ln n ) − β (cid:9) ∪ (cid:8) B ( i ) n < (ln n ) − β (cid:9) = 0 a.s. , and we obtain the proof.
5. Simulation results
Consider fractional Ornstein-Uhlenbeck process that is the solution of thelinear stochastic differential equation dX t = − X t dt + dB Ht , X = 0 . with the step 0 .
05 and for increasing (in the logarithmic scale) number n ofpoints from n = 10 to n = 10 . Table 1 presents the values of the difference | b H (1) n − H | for the values of H from 0 .
55 to 0 .
95. We can conclude that thedifference | b H (1) n − H | decreases rapidly in n and for fixed value of n increases in H . Table 2 demonstrates that the rate of convergence agrees with Theorem 10,at least, for β = 0 .
05. Moreover, we can see from Table 3 that in the case ofthe linear equation the rate of convergence for H ∈ (0 . , .
7) can be estimatedby n − / (ln n ) / . 18 able 1: | b H (1) n − H | n points H
100 250 1000 2500 10 . · . · | b H (1) n − H | · n . (ln n ) − . n points H
100 250 1000 2500 10 . · . · | b H (1) n − H | · n . (ln n ) − . n points H
100 250 1000 2500 10 . · . · References [1] A. B´egyn, Quadratic variations along irregular subdivisions for Gaussianprocesses,
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