The rate of convergence of the Hurst index estimate for a stochastic differential equation

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1 Noliear Aalysis: Modellig ad Cotrol, Vol. 22, No. 2, ISSN The rate of covergece of the Hurst idex estimate for a stochastic differetial equatio Kęstutis Kubilius a, Viktor Skoriakov a,b, Kostiaty Ralcheko c a Istitute of Mathematics ad Iformatics, Vilius Uiversity, Akademijos str. 4, LT-8663 Vilius, Lithuaia kestutis.kubilius@mii.vu.lt b Faculty of Mathematics ad Iformatics, Vilius Uiversity, Naugarduko str. 24, LT-3225 Vilius, Lithuaia c Taras Shevcheko Natioal Uiversity of Kyiv, Volodymyrska str. 64, 161, Kyiv, Ukraie Received: October 14, 216 / Revised: December 2, 216 / Published olie: Jauary 19, 217 Abstract. We cosider a estimator of the Hurst parameter of stochastic differetial equatio with respect to a fractioal Browia motio ad establish the rate of covergece of this estimator to the true value of H whe the diameter of partitio of observatio iterval teds to zero. Keywords: fractioal Browia motio, stochastic differetial equatio, secod-order quadratic variatios, estimates of Hurst parameter, rate of covergece. 1 Itroductio Cosider a stochastic differetial equatio t X t = ξ + t fs, X s ds + gs, X s db H s, t [; T ], 1 where T > is fixed, B H t t [;T ] is a fbm with the Hurst idex 1/2 < H < 1 defied o a complete probability space Ω, F, P, ξ is a iitial r.v., f, g : [; T ] R R are measurable fuctios. This research was fuded by a grat No. MIP-48/214 from the Research Coucil of Lithuaia. c Vilius Uiversity, 217

2 274 K. Kubilius et al. Such equatios are very frequetly met i differet applicatios. The list, though beig far from complete, icludes the followig fractioal versios of well-kow models see [6 8,1,12,14,15] ad refereces therei with correspodig fields of applicatios give i the brackets: Verhulst equatio X t = ξ + t λx s Xs 2 ds + σ t X s dbs H demography, biology; Orstei Uhlebeck equatio X t = X λ t X s ds + σbt H physics, fiace, etworkig; Ladau Gizburg equatio X t = ξ + t λx s X 3 s ds + σ t X s db H s physics; Black Scholes equatio X t = ξ + λ t X s ds + σ t Fractioal Browia Traffic equatio X t = at + σbt H X s dbs H fiace; etworkig. It is therefore clear that a area of applicatios is very wide ad there are may results devoted to estimatio problems i models of this type. O the other had, to our best kowledge there are o a lot of moographs treatig subject i a systematic way. A recet oe to metio is that of C. Berzi, A. Latour ad J.R. Leó see [1]. Moreover, most results devoted to estimatio problems deal with costructio of estimators ad ivestigatio of usual asymptotic properties such as cosistecy ad ormality. Our goal is differet. We assume that oe kows a discrete set {X kt/2, k =, 1,..., 2} of observatios of X t t [;T ] ad cosider a estimator of H based o the secodorder icremets 2 X kt/2 = X kt/2 2X k 1T/2 + X k 2T/2, k = 2, 3,..., 2, which is kow, i most cases, to possess the properties metioed above, ad establish the rate of covergece of the estimator the true value of H. The same problem was treated i [9]. Preset paper improves results of [9] i two directios. First of all, equatio 1 is more geeral tha that of [9]. Secodly, the order of the rate of covergece give here is sharper. The paper is orgaized i the followig way. I Sectio 2, we preset the mai result of the paper ad compare it to that of [9]. Sectio 3 is devoted to several auxiliary facts eeded for the proofs. Sectio 4 cotais the proof of the mai result together with several auxiliary statemets groudig the mai result. 2 Mai result 2.1 Statemet Before proceedig to the statemet of the mai results, we provide several commets regardig the solutio of 1. The coditios esurig existece ad uiqueess of X t t [;T ], which satisfies 1 were established i [13] see also [11]. We assume them to hold. However, ote that cosiderig particular models, oe ca relax or eve drop some of them. For the sake of coveiece, we restate the result of [13] i a oe dimesioal form which applies to our settig ad herewith puts the assumptios made. Note that costats K f,n, K g,n o the right-had side of bouds below may deped o ω. If this is the case, the correspodig

3 O the rate of covergece of the Hurst idex estimate 275 relatios are assumed to hold with probability 1. Here ad further o C λ [; T ]; R, λ ; 1], stads for a space of Hölder cotiuous fuctios equipped with a orm f λ := f + sup s<t T ft fs t s λ, f = sup ft. t [;T ] Theorem 1. See [13, Thm. 2.1]. Let the followig cotiuity costraits o f ad g hold: c1 For all x, y R, sup t [;T ] gt, x gt, y K g, x y uiform Lipschitz cotiuity i x; c2 gs, x is differetiable i x; c3 For all N>, there exist δ 1/H 1; 1] ad K g,n such that sup t [;T ] g xt, x g xt, y K g,n x y δ for all x, y [ N; N] local uiform Hölder cotiuity i x; c4 For all t, s [; T ], there exists β 1 H; 1] such that sup x R gs, x gt, x + g xs, x g xt, x K g, t s β uiform Hölder cotiuity i t; c5 For all N >, there exists K f,n such that sup t [;T ] ft, x ft, y K f,n x y for all x, y [ N; N] local uiform Lipschitz cotiuity i x; c6 For p 1/κ, where κ 1 H; mi{β, δ/1 + δ}, there exists f L p [; T ]; R ad K f, such that ft, x K f, x + f t for all x R ad t [; T ]. The there exists uique solutio of 1 havig property X ω C 1 κ [; T ]; R a.s. Remark. I the statemet above, we have omitted coditio H3 appearig i the origial statemet of Theorem 2.1 of [13]. This is due to the fact that the latter coditio is used i the secod part of the Theorem 2.1 devoted to boudedess of momets of orm of X t t [;T ] ad is irrelevat i our cotext. I what follows, we add two additioal costraits to the set of those imposed by the Theorem 1. First of all, we assume that f satisfies aalog of c4 with the same β 1 H; 1]. To be more precise, we assume c7 For all t, s [; T ], sup x R fs, x ft, x K f, t s β with β give i c4 uiform Hölder cotiuity i t. Secodly, we assume that T g2 t, X t dt > a.s. Our mai result is give below 1. Theorem 2. Suppose that all coditios of Theorem 1 hold. Let θ = mi{1 κ, β}, Ĥ = l 2 l 2 X kt/2 2 2 X kt/, 2 1 Symbols O ω, o ω beig used i a stochastic cotext should be uderstood i the usual sese. The oly differece, as compared to determiistic versios, is that validity of the correspodig relatioships holds with probability oe. Subscript ω idicates possible depedece o ω. Noliear Aal. Model. Cotrol, 222:

4 276 K. Kubilius et al. 2 X kt/ = X kt/ 2X k 1T/ + X k 2T/, k = 2,...,, 2 X kt/2 = X kt/2 2X k 1T/2 + X k 2T/2, k = 2, 3,..., 2. The Ĥ = H + O ω l / θ/2. I the rest of the paper, we retai otios of coefficiets β, δ, κ, θ reserved for the quatities itroduced i the theorems above. 2.2 Compariso with a result of paper [9] We have already metioed i the itroductio that the same problem was treated i [9]. The authors cosidered equatio t X t = ξ + t fx s ds + gx s db H s, t [; T ], ad the same statistic Ĥ as give i the Theorem 2. Uder assumptios that f : R R is Lipschitz, g : R R is differetiable with bouded derivative g C α R; R for some α H 1 1; 1] ad that T g2 X t dt c > a.s., they have proved relatioship Ĥ = H + O ω l 1/4+γ 1/4 where γ ca take ay positive value but is assumed to be fixed. Specializig our result to their case, we see that: Omittig a argumet of time i fuctios f, g yields almost the same set of restrictios required for a existece ad uiqueess of solutio 2 ; T g2 X t dt c > a.s. is replaced by T g2 X t dt > a.s.; Suppose that β, κ, δ satisfies the assumptios of Theorem 1. Set β = δ = 1. The θ 1/2, H. Sice H > 1/2, takig θ = 1/2 + ε with < ε < H 1/2, we get l / 1/4+ε/2. 3 Auxiliary facts The proof of Theorem 2 is preceded by proofs of several techical statemets. To make all expositio easier to follow, we itroduce some otios ad remid several kow facts used i the sequel. I what follows, λ 1 stads for restrictio of the Lebesgue measure o a iterval [; 1], i.e. for all A BR, λ 1 A = λa [; 1], where BR deotes the Borel σ-field o the lie R equipped with a stadard metric fuctio d 1 x, y = x y, x, y R. 2 We say almost the same because i our case Theorem 1 does ot impose boudedess of g ad therefore is a bit less restrictive.,

5 O the rate of covergece of the Hurst idex estimate 277 Let W p [a; b] deotes the class of fuctios o [a; b] with bouded p-variatio for details o p-variatio, cosult [5] ad V p h; [a; b] stads for correspodig variatio. For each r W q ad h W p with p, q,, 1/p + 1/q > 1, a itegral b r dh exists as the Riema Stieltjes itegral provided r ad h have o a commo discotiuities. I such a case, the Love Youg iequality b a r dh ry [ hb ha ] C p,q V q r; [a; b] Vp h; [a; b] holds for all y [a; b], where C p,q = ζp 1 + q 1 ad ζs = 1 s. fbm B H t t is a cetered Gaussia process with a covariace fuctio give by The fbm has the followig properties: EB H t B H s = 1 2 t 2H + s 2H t s 2H. For each H ; 1, almost all sample paths of B H t t [;T ] are locally Hölder of order strictly less tha H. I other words, for ay fixed < γ < H ad ay fixed T >, there exists a oegative a.s. fiite r.v. G γ,t such that Bt H Bs H sup s t T t s γ G γ,t a.s. 3 Squared secod-order icremets of B H t t [;T ] satisfy some uiform LLN the precise statemet of which is give by the theorem below. Theorem 3. See [9, Thm. 3]. For ay t [; T ], defie 3 r t = [t/t ], ρ t = T r t /. The a.s. sup t [;T ] 2 V t ρ t = Oω 1/2 l 1/2, 4 where V 2 t = 2H 1 r t T 2H H 2 BkT/ H 2, 2 B H kt/ = BH kt/ 2BH k 1T/ + BH k 2T/. 2 4 Proofs As already metioed previously, the proof of the mai theorem is preceded by several techical statemets, which are give below. 3 [x] deotes a iteger part of x R. Noliear Aal. Model. Cotrol, 222:

6 278 K. Kubilius et al. Lemma 1. Let < a < b <, p, q ;, h, r : [a; b] R, ε >, x a; b be such that: The 1/p + 1/q > 1; h C 1/p [a; b]; R, r C 1/q [a; b]; R; x ± ε [a; b]. x+ε x r dh x x ε r dh = rx 2 h x,ε + θ x,ε ψε, 5 where 2 h x,ε = hx + ε 2hx + hx ε, θ x,ε [ 1; 1] ad ψε = Oε 1/p+1/q, ε +. Proof. By the Love Youg iequality ad Hölder cotiuity of h, r, x+ε x r dh rx hx + ε hx C p,q V q r; [x; x + ε] Vp h; [x; x + ε] C p,q K r K h ε 1/p+1/q, where K h, K r are such that sup a s<t b rt rs K r t s 1/q, sup a s<t b ht hs K h t s 1/p. Therefore x+ε x r dh = rx hx + ε hx + θ + x,εc p,q K r K h ε 1/p+1/q with some θ + x,ε [ 1; 1]. Usig the same argumet, x x ε r dh = rx hx hx ε + θ x,εc p,q K r K h ε 1/p+1/q, θ x,ε [ 1; 1]. Settig θ x,ε = θ + x,ε θ x,ε/2, ψε = 2C p,q K r K h ε 1/p+1/q, oe obtais 5. Lemma 2. For ay fixed γ ; H, 2 X kt/ = X kt/ 2X k 1T/ + X k 2T/ k 1 = g T, X k 1T/ 2 B H kt/ + O ω θ+γ 1. 6 Proof. Fix γ ; H ad ω {ω: X ω C 1 κ [; T ]; R} {ω: B H ω C γ [; T ]; R}. Let N = Nω = sup t [;T ] X t. The, sice θ = mi{1 κ, β},

7 O the rate of covergece of the Hurst idex estimate 279 f, X, g, X C θ [; T ]; R. Ideed, by c5, c7, fs, X s ft, X t fs, X s ft, X s + ft, X s ft, X t K f, s t β + K f,n X s X t K f, s t β + K f,n K X s t 1 κ s t θ K f, T β θ + K f,n K X T 1 κ θ for s, t [; T ] ad K X = sup s<t T X t X s /t s 1 κ. Hece, the claim holds for f. The case of g is hadled i the same way. Next, ote that 2 X kt/ = kt/ ft, X t dt k 1T/ ft, X t dt + k 1T/ T k/ k 1T/ k 2T/ T k 1/ gt, X t dbt H gt, X t dbt H k 2T/, k = 2,...,. The take x = k 1T/, ε = T/ ad apply Lemma 1 to differeces i the brackets to coclude that k 1 2 X kt/ = g T, X k 1T/ 2 BkT/ H θ+1 θ+γ O ω + O ω. Lemma 3. Let α ; 1], ad let h : Ω [; T ] R be a radom fuctio, which is Hölder cotiuous of order α, i.e. for almost each ω Ω, hs ω h t ω Kh ω s t α with some a.s. fiite ad positive r.v. K h. The 2H 1 h kt/ 2 B kt/ 2 = 4 2 2H T h t dt + O ω α/2 l α/2. 7 T Proof. For clarity, sake we split the proof ito three steps. Step 1. Let Ω = [; 1] =, B 1 = BR [; 1] = {A [; 1]: A BR}, P = λ 1 ad L 1 deotes a set of r.vs. o Ω, B 1, P supported o, i.e. L 1 = { Z: Ω I1 Z is B 1, B 1 measurable }. Noliear Aal. Model. Cotrol, 222:

8 28 K. Kubilius et al. For each τ ; 1], defie a metric d τ o as follows: d τ x, y = x y τ. The ay d τ iduces the same topology o ad correspodig Borel σ-fields coicide with B 1. Therefore it does ot matter whether we treat as a metric space, d α or as a metric space, d 1. I each case, the set L 1 remais the same. Let M 1 deotes the set of probability measures o BR correspodig to r.vs. of L 1, i.e. M 1 = { µ: BR [; 1] Z L1 : P Z = PZ = µ }. Defie o M 1 two Wasserstei metrics: d Wτ µ, ν = if Ed τ Y, Z = if E Y Y µ, Z ν Y µ, Z ν Z τ, τ {α, 1}. Take arbitrary Y, Z L 1 such that Y µ, Z ν. By Jese s iequality, E Y Z α E Y Z α. Thus, d Wα µ, ν = ifỹ µ, Z ν E Ỹ Z α E Y Z α. Cosequetly, d Wα d W1 α. Step 2. Let V 2 t, t [; T ], be the same as i Theorem 3. Deote 2H 1 2 BkT/ H p k = 2 k, P T 4 2 2H V 2 k = p j = V 2 kt, k = 2,...,, T j=2 V 2 T ad µ A = p k δ k/ A for A BR, 8 where δ a deotes the Dirac measure, i.e. for each measurable set A, δ a A = 1 A a. The a.s. µ is a discrete measure from M 1. Let F, F be distributio fuctios correspodig to the measures µ, λ 1 accordigly. For defiiteess, here ad further o we use rightcotiuous versios. By 8,, x < 2 ; F x = P k, x [ k ; k+1, k = 2,..., 1; 1, x 1. Sice F x = x 11 ; x, it follows that F x F x x, x [; 2/; =, x [; 1; x P k, x [ k ; k+1, k = 2,..., 1. Equality 4 implies relatioship V 2 t = t + O ω 1/2 l 1/2, sice deotig by {x} [; 1 a fractioal part of x R + oe has ρ t = t T { t T } 1 T = t + O.

9 O the rate of covergece of the Hurst idex estimate 281 Cosequetly, for all x [k/; k + 1/, k = 2,..., 1, F x F x k = x Pk = x T + O ω l 1/2 T + O ω l 1/2 k = x + O ω l 1/2 1 + O ω l 1/2 = x k + O ω l 1 + O ω l 1/2, = O ω l 1/2 1/2 ad d K µ, λ 1 = sup x F x F x = O ω 1/2 l 1/2, where d K deotes the Kolmogorov metric o the set of probability measures o BR. Step 3. Let ϕt = h tt /T α K h, t [; 1]. Retaiig otatios itroduced i the previous steps, 2H 1 h kt/ 2 2 B kt/ T = 4 2 2H V 2 k T T α K h ϕ p k = 4 2 2H T + O ω 1/2 l 1/2 T α K h ϕ dµ = 4 2 2H T 1+α K h ϕ dµ + O ω 1/2 l 1/2. 9 By Katorovich duality theorem see [4, p. 421], µ, ν M 1, d Wα µ, ν = sup ψ C α 1 ψ dµ ψ dν, where C1 α = {ψ: R ψs ψt d α s, t = s t α }. Sice ϕ C1 α, ϕ dµ d W α µ, λ 1. 1 ϕ dλ 1 Now, recall that there is aother explicit formula for d W1 o M 1 see [3, p. 271]: d W1 µ, ν = 1 Fµ x F ν x dx. Thus, results of the previous steps yield α d Wα µ, λ 1 d α W 1 µ, λ 1 d K µ, λ 1 dx = O ω α/2 l α/2, 11 Noliear Aal. Model. Cotrol, 222:

10 282 K. Kubilius et al. ad from 9 11 it follows 2H 1 h kt/ 2 B H 2 kt/ T = 4 2 2H T 1+α K h ϕdλ 1 + O ω α/2 l α/2 = 4 2 2H T 1 h tt dt + O ω α/2 l α/2 = T 4 2 2H h t dt + O α/2 l α/2. Proof of Theorem 2. Fix γ ; H, which satisfies γ + θ/4 > H. It was already show 4 that g, X C θ [; T ]; R. Therefore g 2, X C θ [; T ]; R. Sice T g2 t, X t dt > a.s. ad B H C γ [; T ]; R a.s., Lemmas 2, 3 yield T 2H X kt/ = T Cosequetly, 2H 1 = T 4 2 2H = T 4 2 2H 2 T k g 2 T, X k T 2 BkT/ H Oω g 2 t, X t dt + O ω l θ/2 θ/2 l g 2 t, X t dt 1 + O ω. 2H 1 2 = T 4 2 2H 2 X kt/2 2 θ/2 l g 2 t, X t dt 1 + O ω, θ 2H γ 4 See proof of Lemma 2.

11 O the rate of covergece of the Hurst idex estimate 283 ad by Maclauri s expasio, l 2 2 X kt/2 2 2 X kt/ 2 = l 2 2H H T g2 t, X t dt[1 + O ω l θ/2 ] 4 2 2H T g2 t, X t dt[1 + O ω l θ/2 ] = 2H 1 l l 1 + O ω l θ/2 1 + O ω l θ/2 θ/2 l = 2H 1 l l 1 + O ω = 2H 1 l O ω l θ/2. Refereces 1. C. Berzi, A. Latour, J.R. Leó, Iferece o the Hurst Parameter ad the Variace of Diffusios Drive by Fractioal Browia Motio, Lect. Notes Stat., Vol. 216, Spriger, F. Biagii, Y. Hu, B. Øksedal, T. Zhag, Stochastic Calculus for Fractioal Browia Motio ad Applicatios, Probability ad Its Applicatios, Spriger, M.M. Deza, E. Deza, Ecyclopedia of Distaces, 3rd ed., Spriger, R.M. Dudley, Real Aalysis ad Probability, Cambridge Uiversity Press, R.M. Dudley, R. Norvaiša, Cocrete Fuctioal Calculus, Spriger Moographs i Mathematics, Spriger, New York, C. Grimm, G. Schlüchterma, IP-Traffic Theory ad Performace, Spriger Series o Sigals ad Commuicatio Techology, Spriger, O.C. Ibe, Elemets of Radom Walk ad Diffusio Processes, 1st ed., Wiley Series i Operatios Research ad Maagemet Sciece, Wiley, Hoboke, NJ, K. Kubilius, D. Melichov, O compariso of the estimators of the Hurst idex of the solutios of stochastic differetial equatios drive by the fractioal Browia motio, Iformatica, 221:97 114, K. Kubilius, Y. Mishura, The rate of covergece of estimate for Hurst idex of fractioal Browia motio ivolved ito stochastic differetial equatio, Stochastic Processes Appl., 12211: , Y. Mishura, Stochastic Calculus for Fractioal Browia Motio ad Related Processes, Lect. Notes Math., Vol. 1929, Spriger, Y. Mishura, G. Shevcheko, Mixed stochastic differetial equatios with log-rage depedece: Existece, uiqueess ad covergece of solutios, Comput. Math. Appl., 641: , 212. Noliear Aal. Model. Cotrol, 222:

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