Abstract We consider high-frequency observations from a fractional Brownian motion. Inspired by the work of Jean Jacod in a diffusion setting, we investigate the asymptotic behavior of various classical statistics related to the local times of the process. We show that as in the diffusion case, these statistics indeed converge to some local times up to a constant factor. As a corollary, we provide limit theorems for the quadratic variation of the absolute value of a fractional Brownian motion. Local times are fundamental objects in the theory of stochastic processes. The local time of the process X at time t and level x essentially measures the time spent by the process at point x between time 0 and time t. More precisely, let X=(Xt)t≥0 be defined on a filtered probability space (Ω,F,(Ft)t≥0,P). Recall that the occupation measure ν associated with X is defined via νt(A):=∫0t1A(Xs)ds, (1.1) for any Borel measurable set A⊂R. If the measure νt admits a Lebesgue density, we call it the local time of X and denote it by Lt(x):=dνtdx(x). (1.2) An immediate consequence of the existence of local times is the occupation times formula, which states the identity ∫0tf(Xs)ds=∫Rf(x)Lt(x)dx (1.3) with f being a Borel function on R. The theory of occupation measures and local times has been intensively studied in the literature for several decades. A general criterion for existence of local times is provided in Geman and Horowitz (1980). Properties of the local time of a Brownian motion, or more generally of a continuous diffusion process, have been particularly investigated. We refer to the monograph Revuz and Yor (1999) for a comprehensive study. In Berman (1969), the author focuses on sample paths properties in the case of stationary Gaussian processes. Then, a natural question is that of the estimation of local times based on discrete high-frequency observations of the process X over [0,t]: X0,X1/n,X2/n,…,X[nt]/n, where the asymptotic setting is that n goes to infinity. From Equation (1.2), a natural estimator of Lt(x) is given by the statistic unn∑i=1[nt]1{|Xi/n−x|≤1/un}, where un is a sequence of positive real numbers with un→∞. In general, we may consider a larger class of statistics, which have the form V(g)tn:=unn∑i=1[nt]g(un(Xi/n−x)) with g∈L1(R). (1.4) Indeed, the statistic V(g)tn is, up to a constant, a good estimator of Lt(x). This can be seen from the following intuitive argument, which follows from the occupation time formula: V(g)tn≈un∫0tg(un(Xs−x))ds=un∫Rg(un(y−x))Lt(y)dy=∫Rg(z)Lt(x+z/un)dz→PLt(x)∫Rg(z)dz, where the last convergence follows from the continuity of the local time in space. Despite this simple intuition, the first approximation in the above argument is not trivial to prove (in particular, it is wrong if un/n→∞). Such convergence results have been shown in the framework of Brownian motion in Borodin (1986, 1989) and they were extended in Jacod (1998) to the setting of continuous diffusion processes (in the latter article, the author has also shown the associated central limit theorems). In Jeganathan (2004, Theorem 4), the consistency of V(g)tn has been proved for linear fractional stable motions. In Jacod (1998), with the purpose of statistical applications, a more general class of functionals than V(g)tn is actually considered in the setting of continuous diffusion processes. More specifically, for a measurable function f:R2→R satisfying certain conditions to be specified below, the author studies statistics of the form V(f)tn:=unn∑i=0[nt]f(un(Xi/n−x),unΔinX), where ΔinX=X(i+1)/n−Xi/n. (1.5) In this work, we wish to somehow extend the approaches in Jacod (1998) and Jeganathan (2004). Indeed, we want to investigate the behavior of V(f)tn in the case where the underlying process X satisfies Xt=σBtH, (1.6) where σ>0 is a scale parameter and (BtH)t≥0 is a standard fractional Brownian motion with Hurst parameter H∈(0,1), that is a zero mean Gaussian process with covariance kernel given by E[BtHBsH]=12(t2H+s2H−|t−s|2H). The fractional Brownian motion has stationary increments and it is self-similar with parameter H: (BatH)t≥0=d(aHBtH)t≥0 for any a > 0. Furthermore, local times of BH exist for any H∈(0,1) and are known to be continuous in time and space, see Berman (1970). Studying high-frequency statistics in the context of fractional Brownian motion is very natural. Indeed, this process is probably the simplest non-trivial continuous process outside the semi-martingale world. Furthermore, beyond its obvious theoretical interest, the fractional Brownian motion is widely used in various applications, particularly in finance, see among others (Comte and Renault, 1996; Gatheral, Jaisson, and Rosenbaum, 2014; Czichowsky and Schachermayer, 2017). Finally, note that for the specification of the function f in Equation (1.5), there will be two cases of particular interest (both refer to un=nH): f1(y,z)=1{y(y+z)<0} and f2(y,z)=−2y(y+z)1{y(y+z)<0}. (1.7) We have that V(f1)tn counts the (scaled) number of crossings of level x and the statistics V(f2)tn will be useful for some applications. We give in the next section our assumptions and our main result on the behavior of V(f)tn. As a corollary, we provide a panorama of the different limit theorems obtained for the quadratic variation of the absolute value of a fractional Brownian motion, depending on the Hurst parameter. The proofs are gathered in Section 2. 1 The Setting and Main Results In order to formulate our main result, we will require some conditions on the function f: (A-γ) We assume that |f(y,z)|≤h1(y)h2(z), where h1, h2 are non-negative continuous functions such that h1 tends to zero at infinity, ∫R|y|γh1(y)dy<∞ (2.1) and h2 has polynomial growth. We notice that if condition (A-γ) is satisfied for some function h1, then it also holds for h1p for any p≥1. Furthermore, (A-γ) implies (A- γ′) for any 0≤γ′≤γ. The assumption (A-γ) might be relaxed (cf. Jacod, 1998, Hypothesis (B-γ)), but it suffices for applications we have in mind. Our main theorem is the following statement. Theorem 1.1 Assume Condition (A-2) is satisfied. If un→∞ and un/n→0, we obtain the uniform convergence in probability V(f)tnu.c.p.V(f)t:=Lt(x)∫R2f(y,z)φσ2(z)dydz, (2.2) where φσ2 denotes the density of N(0,σ2). We remark that the integral on the right-hand side of Equation (2.2) is indeed finite due to Assumption (A-2). We apply the result of Theorem 1.1 to the functions f1, f2 defined at Equation (1.7). First of all, we notice that they both satisfy Assumption (A-2). Indeed, the condition x(x+y)<0 implies that |x|<|y| and we obtain that |f1(x,y)|≤(max{1,|x|p})−1max{1,|y|p}, |f2(x,y)|≤4|x|(max{1,|x|p})−1|y|max{1,|y|p} for any p > 0. Choosing p > 4 ensures the validity of Equation (2.1) for γ = 2. We obtain the following corollary. Corollary 1.2 Define un=nH and let N∼N(0,σ2). Then, we obtain V(f1)tnu.c.p.E[|N|]Lt(x) and V(f2)tnu.c.p.13E[|N|3]Lt(x). Our next application of Theorem 1.1 is the weak limit theorem for the quadratic variation of the process |X|. Proposition 1.3 We define ρk=cov(B1H,Bk+1H−BkH) and v2=2σ4(1+2∑k=1∞ρk2). Let W=(Wt)t≥0 denote the standard Brownian motion. (i) If H < 1/2, we obtain the uniform convergence in probability nH−1∑i=0[nt]((nHΔin|X|)2−σ2)→u.c.p.13E[|N|3]Lt(0). (ii) If H = 1/2, we obtain the functional convergence for the Skorokhod topology 1n∑i=0[nt]((nHΔin|X|)2−σ2)⇒vWt+13E[|N|3]Lt(0), where W is independent of L. (iii) If H∈(1/2,3/4), we obtain the functional convergence for the Skorokhod topology 1n∑i=0[nt]((nHΔin|X|)2−σ2)⇒vWt. (iv) If H∈(3/4,1), we obtain the functional convergence for the Skorokhod topology n1−2H∑i=0[nt]((nHΔin|X|)2−σ2)⇒σ2Rt,where R is the Rosenblatt process. Hence, interestingly, we see that the asymptotic behavior of the quadratic variation of the absolute value of a fractional Brownian motion differs from that of a fractional Brownian motion when H≤1/2. In this case, the local time at zero appears in the limit. Note that in Robert (2017), the author considers max-stable processes whose spectral processes are exponential martingales associated to a Brownian motion. He introduces estimators of the integral of the extreme value index function and establishes asymptotic properties of these estimators thanks to results in Jacod (1998). Proposition 1.3 could then used to extend this framework to exponential martingales associated to a fractional Brownian motion, leading to a more general and useful class of max-stable processes. 2 Proofs of Theorem 1.1 and Proposition 1.3 Throughout this section all positive constants are denoted by C (or c), or by Cp if they depend on the external parameter p, although they may change from line to line. For the sake of exposition, we only consider the local time Lt(x) at x = 0 and set σ = 1. 2.1 Introduction for the Proof of Theorem 1.1 Before we proceed with formal proofs let us give important ideas for the strategy of proof. First, we note that it suffices to prove the pointwise result V(f)tn→PV(f)t for any fixed t, since the statistic V(f)tn is increasing in t and the limit V(f)t is continuous in t. Thus, without loss of generality we set t = 1. Next, we introduce the statistic V¯(f)n:=unn∑i=0nF(unXi/n) with F(x):=E[f(x,N(0,1))]. It follows from Jeganathan (2004, Theorem 4) that the convergence V¯(f)n→PL1(0)∫RF(x)dx holds as n→∞. Hence, we are left to proving V(f)1n−V¯(f)n→P0, (3.1) to show Theorem 1.1. 2.2 A Preliminary Result The following proposition will be used repeatedly in the proof of Theorem 1.1. It might be well-known in the literature, but nevertheless we provide a detailed proof. Below we denote by φΣ the density of Nd(0,Σ). Proposition 2.1 We consider random variables Z∼Nd(0,Σ) and Z′∼Nd(0,Σ′), where Σ,Σ′∈Rd×d are positive definite matrices. We assume that there exists a constant K > 0 with max1≤i,j≤d{|Σij|+|Σ′ij|}<K and min{det Σ,det Σ′}>1K. Let G:Rd→R be a function with polynomial growth. Then, there exist constants cK,CK>0 such that |E[G(Z)]−E[G(Z′)]|≤CK∫RdG(y)(1+||y||2)exp(−cK||y||2)dy×max1≤i,j≤d{|Σij−Σ′ij|}. Proof Let λ1≥⋯≥λd>0 denote the real eigenvalues of the matrix Σ. Since ∑i=1dλi=tr(Σ), we have that λi<dK for all i. The same inequality holds for the eigenvalues λ′1≥⋯≥λ′d>0 of Σ′. We also have that |det(Σ)−det(Σ′)|≤CKmax1≤i,j≤d{|Σij−Σ′ij|}, (3.2) because the determinant is a polynomial function. In the next step, we give an estimate on max1≤i,j≤d{|Σij−1−Σ′ij−1|}. Recall the Cramer’s rule: Σij−1=det(Σ(i,j))det(Σ), where the matrix Σ(i,j) is formed from Σ by replacing the ith column of Σ by the jth standard basis element ej∈Rd. Applying this formula and using the lower bound min{detΣ,detΣ′}>1/K, we conclude that max1≤i,j≤d{|Σij−1−Σ′ij−1|}≤CKmax1≤i,j≤d{|Σij−Σ′ij|} (3.3) as in Equation (3.2). We are now ready to obtain an upper bound for |φΣ(y)−φΣ′(y)|, y∈Rd. First, we note that |φΣ(y)−φΣ′(y)|≤CK(max1≤i,j≤d{|Σij−Σ′ij|}(exp(−y⋆Σ−1y/2)+exp(−y⋆Σ′−1/2))+|exp(−y⋆Σ−1y/2)−exp(−y⋆Σ′−1y/2)|), where ⋆ denotes the transpose operator. Now, using mean value theorem and Equation (3.3) we deduce that |exp(−y⋆Σ−1y/2)−exp(−y⋆Σ′−1y/2)|≤CK||y||2max1≤i,j≤d{|Σij−Σ′ij|}(exp(−y⋆Σ−1y/2)+exp(−y⋆Σ′−1y/2)). Next, we note that y⋆Σ−1y=||Σ−1/2y||2≥||y||2/λ1≥||y||2/dK (3.4) and the same inequality holds for the matrix Σ′−1. Since |E[G(Z)]−E[G(Z′)]|≤∫RdG(y)|φΣ(y)−φΣ′(y)|dy, we obtain the desired result.□ 2.3 Proof of Equation (3.1) We define the random variables rin:=f(unXi/n,nHΔinX)−F(unXi/n). We will now show that (un/n)2E[(∑i=0nrin)2]→0. We divide this proof into two parts. We will prove that un2n2∑i=0nE[(rin)2]→0, (3.5) un2n2∑j>inE[rinrjn]→0. (3.6) We start with the first statement. Using the covariance kernel of the fractional Brownian motion, we see that ρi:=corr(Xi/n,ΔinX)=12iH((i+1)2H−i2H−1) for i≥1. (3.7) We see immediately that |ρi|→0 as i→∞ and |ρi|<ρ for all i≥1 and some ρ<1. Let us now consider the correlation matrix Σi=(1ρiρi1). This matrix has two eigenvalues λi(1)=1+|ρi| and λi(2)=1−|ρi|. Hence, the matrix Σi−1/2 has eigenvalues λi(1)−1/2 and λi(2)−1/2, and we conclude that (x,z)⋆Σi−1(x,z)=||Σi−1/2(x,z)||2≥(x2+z2)/λi(1)≥(x2+z2)/(1+ρ). (3.8) For i = 0, we obviously have E[f2(0,nHΔ0nX)]≤C. On the other hand, for 1≤i≤n we obtain the estimate E[f2(unXi/n,nHΔinX)]=∫R2f2(un(i/n)Hx,z)φ0,Σi(x,z)dxdz=un−1(i/n)−H∫R2f2(y,z)φ0,Σi(un−1(i/n)−Hy,z)dydz≤Cun−1(i/n)−H∫R2f2(y,z)exp(−z2/2)dydz≤Cun−1(i/n)−H, where we have used Equation (3.8) and Condition (2.1) applied to h12 in the two last steps. Since the mapping x↦x−H is integrable around zero, we deduce that un2n2∑i=0nE[f2(unXi/n,nHΔinX)]≤Cunn→0. Similarly, we obtain the convergence un2n2∑i=0nE[F2(unXi/n)]→0, which concludes the proof of Equation (3.5). In order to prove Equation (3.6), we need several decompositions. Let us fix a δ∈(0,1) and write un2n2∑j>inE[rinrjn]=∑k=13Rn,δ(k), where Rn,δ(1):=un2n2∑i=0[nδ]∑j=i+1nE[rinrjn],Rn,δ(2):=un2n2∑i=[nδ]n∑j−i≤[nδ]nE[rinrjn],Rn,δ(3):=un2n2∑i=[nδ]n∑j−i>[nδ]nE[rinrjn]. We note that (Bi,n,Bj/n,nHΔinX,nHΔjnX)∼N4(0,Σi,jn) and we denote by Σ˜i,j the correlation matrix associated with Σi,jn, which does not depend on n. We obviously have that |Σi,jn(k,l)|≤C and |Σ˜i,j(k,l)|≤1 for any i, j and any 1≤k,l≤4. For any s∈(0,1) and 1≤i<j≤n, we have the identities cov(Xs,nHΔinX)=nH2((i+1n)2H−(in)2H+|in−s|2H−|i+1n−s|2H),cov(nHΔinX,nHΔjnX)=12((j−i−1)2H+(j−i+1)2H−2(j−i)2H). Using these identities, we observe the following properties: inf1≤i<j≤ndet(Σ˜i,j)>c>0, (3.9) det(Σi,jn)>cδ>0 for any i>[nδ] and j−i>[nδ], (3.10) |Σi,jn(k,l)|≤an,δ for any i>[nδ] and j−i>[nδ], (3.11) where an,δ→0 as n→∞ and (k,l)∈J with J={1,…,4}2∖(∪k=14{k,k}∪{1,2}∪{2,1}). Now, we start with the term Rn,δ(1). For 1≤i≤[nδ] and i<j≤n, we use Equations (3.9) and (3.4), and deduce that ( x=(x1,x2) and z=(z1,z2)) E[|f(unXi/n,nHΔinX)f(unXj/n,nHΔjnX)|]=∫R4|f(un(i/n)Hx1,z1)f(un(j/n)Hx2,z2)|φΣ˜i,j(x,z)dxdz≤C∫R4|f(un(i/n)Hx1,z1)f(un(j/n)Hx2,z2)|exp(−c(||x||2+||z||2))dxdz≤Cun−2(i/n)−H(j/n)−H∫R4|f(y1,z1)f(y2,z2)|exp(−c||z||2)dydz≤Cun−2(i/n)−H(j/n)−H. The last inequality follows from condition (A-2): ∫R4|f(y1,z1)f(y2,z2)|exp(−||z||2/2)dydz≤||h1||L1(R)2∫R2|h2(z1)h2(z2)|exp(−c||z||2)dz. The other terms in E[rinrjn] for 1≤i≤[nδ] and i<j≤n are treated in a similar way. We thus conclude that |Rn,δ(1)|≤Cδ1−H. (3.12) By the same arguments, we also deduce that |Rn,δ(2)|≤Cδ1−H. (3.13) Next, we compute the term Rn,δ(3). We define the matrix Σi,jn,1=(Σi,jn,1(k,l))1≤k,l≤4 via Σi,jn,1(k,l)=Σi,jn(k,l)1Jc. By properties (3.10) and (3.11) and Proposition 2.1, we deduce for any i>[nδ] and j−i>[nδ] E[f(unXi/n,nHΔinX)f(unXj/n,nHΔjnX)]=∫R4f(unx1,z1)f(unx2,z2)φΣi,jn,1(x,z)dxdz+wi,jn,1=:w¯i,jn,1+wi,jn,1, where |wi,jn,1|≤Cδan,δ∫R4f(unx1,z1)f(unx2,z2)(1+||(x,z)||2)exp(−cδ||(x,z)||2)dxdz.≤Cδan,δun−2(∫R(1+x12)h1(x1)dx1)2(∫Rh2(z1)z12exp(−cδz12)dz1)2. Now we define the matrix Σi,jn,2∈R3×3 (respectively, Σi,jn,3∈R3×3 and Σi,jn,4∈R2×2) through the matrix Σi,jn,1 by deleting the fourth row/column (respectively, the third row/column and the last two rows/columns). We similarly obtain the identities E[f(unXi/n,nHΔinX)F(unXj/n)]=∫R3f(unx1,z1)F(unx2)φΣi,jn,2(x,z1)dxdz1+wi,jn,2=:w¯i,jn,2+wi,jn,2,E[f(unXj/n,nHΔjnX)F(unXi/n)]=∫R3F(unx1)f(unx2,z2)φΣi,jn,3(x,z2)dxdz2+wi,jn,3=:w¯i,jn,3+wi,jn,3,E[F(unXi/n)F(unXj/n)]∫R2F(unx1)F(unx2)φΣi,jn,4(x)dx+wi,jn,4=:w¯i,jn,4+wi,jn,4 and |wi,jn,2|+|wi,jn,3|+|wi,jn,4|≤Cδan,δ. In fact, we obviously have w¯i,jn,1=w¯i,jn,2=w¯i,jn,3=w¯i,jn,4. Hence, we conclude that |Rn,δ(3)|≤Cδan,δ. (3.14) Thus, the statement of Theorem 1.1 follows from Equations (3.12)–(3.14) by letting n→∞ and then δ→0. 2.4 Proof of Proposition 1.3 Observe the identity (Δin|X|)2−(ΔinX)2=2|Xi/nX(i+1)/n|1{Xi/nX(i+1)/n<0}. Thus, we have 1n∑i=0[nt]((nHΔin|X|)2−σ2)=1n∑i=0[nt]((nHΔinX)2−σ2)+n−HV(f2)tn=:Mtn+n−HV(f2)tn It is well known that for H∈(0,3/4), we have the functional convergence nMtn⇒vWt, while for H∈(3/4,1) it holds that n2−2HMtn⇒σ2Rt, see Breuer and Major (1983), Dobrushin and Major (1979), and Taqqu (1979). Hence, we deduce the assertion of Proposition 1.3. Footnotes M.P. acknowledges the financial support from the project “Ambit fields: probabilistic properties and statistical inference” funded by Villum Fonden and from CREATES funded by the Danish National Research Foundation. M.R. acknowledges the financial support of the ERC 679836 Staqamof. References Berman S. M . 1969 . Local Times and Sample Function Properties of Stationary Gaussian Processes . Transactions of the American Mathematical Society 137 : 277 – 299 . Google Scholar Crossref Search ADS Berman S. M . 1970 . Gaussian Processes with Stationary Increments: Local Times and Sample Function Properties . The Annals of Mathematical Statistics 41 ( 4 ): 1260 – 1272 . Google Scholar Crossref Search ADS Borodin A. N . 1986 . On the Character of Convergence to Brownian Local Time . Probability Theory and Related Fields 72 : 251 – 278 . Google Scholar Crossref Search ADS Borodin A. N . 1989 . Brownian Local Time . Russian Mathematical Surveys 44 ( 2 ): 1 – 51 . Google Scholar Crossref Search ADS Breuer P. , and Major P. . 1983 . Central Limit Theorems for Non-linear Functionals of Gaussian Fields . Journal of Multivariate Analysis 13 ( 3 ): 425 – 441 . Google Scholar Crossref Search ADS Comte F. , and Renault E. . 1996 . Long Memory Continuous Time Models . Journal of Econometrics 73 ( 1 ): 101 – 149 . Google Scholar Crossref Search ADS Czichowsky C. , and Schachermayer W. . 2017 . Portfolio Optimisation beyond Semimartingales: Shadow Prices and Fractional Brownian Motion . The Annals of Applied Probability 27 : 1414 – 1451 . Google Scholar Crossref Search ADS Dobrushin R. L. and Major P. . 1979 . Non-central Limit Theorems for Non-linear Functional of Gaussian Fields . Probability Theory and Related Fields 50 ( 1 ): 27 – 52 . Gatheral J. , Jaisson T. , and Rosenbaum M. . 2014 . Volatility Is Rough . Quantitative Finance. arXiv : 1410.3394. Geman D. , and Horowitz J. . 1980 . Occupation Densities . The Annals of Probability 8 ( 1 ): 1 – 67 . Google Scholar Crossref Search ADS Jacod J . 1998 . Rates of Convergence to the Local Time of a Diffusion . Annales de l’Institut Henri Poincaré 34 : 505 – 544 . Google Scholar Crossref Search ADS Jeganathan P . 2004 . Convergence of Functionals of Sums of R.V.s to Local Times of Fractional Stable Motions . The Annals of Probability 32 ( 3 ): 1771 – 1795 . Google Scholar Crossref Search ADS Revuz D. , and Yor M. . 1999 . Continuous Martingales and Brownian Motion , 3rd edn . Springer , Heidelberg . Robert C. Y . 2017 . Infill Asymptotics for Extreme Value Estimators of the Integral of the Index Function in C[0,1] . Preprint. Taqqu M . 1979 . Convergence of Integrated Processes of Arbitrary Hermite Rank . Probability Theory and Related Fields 50 ( 1 ): 53 – 83 . © The Author, 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
Journal of Financial Econometrics – Oxford University Press
Published: Sep 1, 2018
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