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  • 1. On estimating the integrated co-volatility using noisy high frequency data with jumps Bing-Yi JINGa,b Cui-Xia LIa Zhi LIUc,∗ a School of Mathematics and Statistics, Lanzhou University. b Department of Mathematics, Hong Kong University of Science and Technology, c Department of Statistics and Wangyanan Institute for Studies in Economics, Xiamen University.Abstract: In this paper, we consider the estimation of covariation of two assetprices which contain jumps and microstructure noise, based on high frequency data.We propose a realized covariance estimator, which combines pre-averaging methodto remove the microstructure noise and the threshold method to reduce the jumpseffect. The asymptotic properties, such as consistency and asymptotic normality,are investigated. The estimator allows very general structure of jumps, for example,infinity activity or even infinity variation. Simulation is also included to illustrate theperformance of the proposed procedure.Keywords: Ito semi-martingale; High frequency data; Microstructure noise; Co-volatility; Jumps; Central limit theorem. ∗ The corresponding author 1
  • 2. 1 IntroductionSuppose that we have p-multiple underlying log price processes of assets, X1t , ..., Xpt .Denote Zt = (Z1t , ..., Zpt )τ for any generic processes Zit ’s. Then a widely used modelfor Xt is the following semi-martingale: Xt = Xc + Xd , t t (1.1)where Xc and Xd are, respectively, the continuous and discontinuous components, t twhose forms are given in (2.3)-(2.4) later. From Protter (1990) and Jacod andShiryaev (2003), it is known that the covariance matrix [X, X]T has the followingdecomposition: [X, X]T = [Xc , Xc ]T + [Xd , Xd ]T . Given discrete high frequency observations, Xti at 0 = t0 < t1 < ... < tn = T ,both for pricing and hedging purposes and for financial econometrics applications,it is important to separate the contributions of the diffusion part and jump part ofX (see Andersen et. al (2001); Barndorff-Nielsen and Shephard (2002b); Mancini(2009)). Define the realized covariance matrix (RCV ) as ∑ n RCV =: [X, X]T = ∆n X(∆n X)τ i i i=1where ∆n X i = Xti − Xti−1 is a column vector and Aτ defines the transpose of matrixA. It is well known (see, e.g., Protter (1990)) that [X, X]T →P [X, X]T . Therefore,it suffices to focus on the estimation of the continuous part [Xc , Xc ]T , which is thecenter of our attention in the present paper. However, it is widely accepted that the observed prices are contaminated by mi-crostructure noise due to bid-ask spreads and/or rounding errors etc. Hence, insteadof observing Xti = (X1ti , ..., Xpti ), we observe Yti = (Y1ti , ..., Ypti ), where Yti = Xti + ϵti , i = 0, 1, ..., n, (1.2) 2
  • 3. where Xt are the latent semi-martingale price processes, the perturbation term ϵti =(ϵ1ti , ..., ϵpti ) is the microstructure noise at time ti . Our objectives concern the in-ference of [Xc , Xc ]T (and [Xd , Xd ]T ) for the latent price process X, based on thecontaminated observations Yti ’s. The covariation matrix is of strong interest in financial applications, such as port-folio risk and hedging of funds management; see, e.g., A¨ ıt-Sahalia, Fan, and Xiu(2010), A¨ ıt-Sahalia, Mykland and Zhang (2011), Bai, Liu and Wong (2009), Zhengand Li (2010) among others. For p = 1, there has been a huge literature on the integrated volatility estimation.In the absence of microstructure noise (ϵ = 0), we refer the reader to Andersen,et al. (2001), Barndorff-Nielsen and Shephard (2002a, 2002b), Jacod and Protter(1998), Mykland and Zhang (2006), Mancini (2009) among others. In the presence ofmicrostructure noise, the references include A¨ ıt-Sahalia, Mykland and Zhang (2005,2011) and Bandi and Russell (2006), Zhang, Mykland, and A¨ ıt-Sahalia (2005) andZhang (2006), Fan and Wang (2007), Podolskij and Vetter (2009a, 2009b), Barndorff-Nielsen et al. (2008), and Jacod, Podolskij and Vetter (2010), among others. For p ≥ 2, there has been an increasing literature recently. For instance, Gobbiand Mancini (2009, 2010) derived an asymptotically unbiased estimator of the con-tinuous part of the covariation process as well as of the co-jumps, and consideredits asymptotic behavior as well. However, these works assume that there is no mi-crostructure noise in the underlying price process Xt . In this paper, we attempt to develop a procedure that gives a consistent estimatorof the integrated co-volatility in the simultaneous presence of microstructure noiseand jumps. The method combines the two approaches: the pre-averaging method andthreshold technique. The former is employed to reduce the effect of microstructurenoise while the latter is used to remove the jumps. The central limit theorem is alsodeveloped. The study of the p × p covariation matrix has some distinct features, 3
  • 4. compared with that of variation in the one-dimensional case. As will be seen later,its studies depend on the assumptions of how the jumps of components related toeach other, and yield different estimators under different assumptions. The remainder of this paper is organized as follows. In Section 2, we give someassumptions on the model. We introduce the asymptotic theorems and central limittheorem in Section 3. Simulation study is put to Section 4. Section 5 features theconclusions and all the technical proofs are postponed to the appendix.2 Preliminaries2.1 Model assumptionsSince there is no essential difference between the two cases: p = 2 and p > 2, so forsimplicity, we shall assume that p = 2 from now on. In this case, [Xc , Xc ]t involves c c c c c cthree [X1 , X1 ]t , [X2 , X2 ]t , and [X1 , X2 ]t . The first two variation processes are well c cstudied in the literature. Below, we will focus on the covariation process [X1 , X2 ]t . c d We now describe model (1.2) in details. Assume that Xrt = Xrt + Xrt , r = 1, 2,defined on a stochastic basis (Ω, F, Ft , P ), are Itˆ semi-martingales of the form o ∫ t ∫ t c Xrt = Xr0 + brt dt + σrt dWrt (2.3) ∫ t∫ 0 0 ∫ t∫ d Xrt = x(µr − νr )(ds, dx) + xµr (ds, dx), (2.4) 0 |x|≤1 0 |x|>1where b and σ are locally bounded optional processes, µr is a jump measure compen-sated by νr , and νr has the form dtFrt (dx), where Frt (dx) is a transition measure form ∫Ω × R+ endowed with the predictable σ-field into R/0. If |x|≤1 |x|Frt (dx) < ∞, we ∫say that Xr has finite variation. Let βr =: inf{s : |x|≤1 |x|s Frt (dx) < ∞}, which is dcalled jump activity index in the literature. Further Wr = (Wrt ) are standard Wiener 4
  • 5. processes such that d[W1 , W2 ]t = ρdt. The integrated covariation is defined as ∫ T c c [X1 , X2 ]T = ρσ1s σ2s ds. (2.5) 0 We will impose some condition on the microstructure noise. For more details, seeJacod, Podolskij and Vetter (2010).Assumption 1 (Microstructure noise) The microstructure noise  = (ϵ1 , ϵ2 ) are  ϵ 2 ω1 0iid processes with E[ϵt ] = 0 and E[ϵt ϵ′t ] = ω, where ω =   . We further 2 0 ω2assume X ⊥ ϵ (here, ⊥ denotes stochastic independence).2.2 Notations • Let g be a continuous weight function on [0,1] with a piecewise Lipschitz deriva- ∫1 tive g ′ with g(0) = g(1) = 0 and 0 g 2 (s)ds > 0. Define ∫1 – g (2) = ¯ 0 g 2 (s)ds, ∫1 ∫1 – h(s) = s g(u)g(u − s)du, Φ = 0 h2 (s)ds. • For a generic process Z = {Zt , t ≥ 0}, the one-step increment is ∆n Z = Zti − i Zti−1 , and kn -step pre-averaged increment is defined as (kn is an integer) ∑ kn −1 ∆n n Z i,k = g(j/kn )∆n Z. i+j j=1 • Yt = Xc + ϵt , Yt = Xd = Xt − Yt . ∆n = T /n and ∆s X = Xs − Xs− . c t d t c • ∥x∥ represents the Euclidean distance. 5
  • 6. 3 Main results c cWe now consider the estimation of the integrated covariation [X1 , X2 ]T in (2.5). The d destimators depend on how the jumps from the two processes, X1t and X2t , are relatedto each other. We consider two separated cases below.Case I: Two jump processes are independentFirst we suppose that jump parts of two processes are independent. This is appro-priate when the jumps are mainly due to effects of individual stocks but not thesystematic influence. In this case, we define an estimator as ∑ n−kn +1 V1n (Y) = (∆n n Y1 )(∆n n Y2 ). i,k i,k (3.6) i=0 √Theorem 1 Assume that E∥ϵ∥4 < ∞. Fix θ > 0 and choose kn such that kn ∆n = 1/4θ + O(∆n ). Then, under Assumption 1, we have ∫ T 1 ∆ V1n (Y) → 1/2 P ρσ1s σ2s ds. θ¯(2) n g 0Remark 1 Although V1n (Y) contains jumps from both processes, their effects disap-pear in the limit. It is not surprising because two independent purely discontinuousprocesses never jump together almost surely. It is therefore easy to understand why theestimator takes the same form as that by Podolskij and Vetter (2009a), who considera similar problem but without jumps.Case II: Two jump processes are dependentWe now consider a more general case, which allows the two jump process to becorrelated. The two individual stocks may jump together since the common news 6
  • 7. such as the government announcements may affect all the individual stocks. In thiscase, the estimator V1n (Y) in (3.6) will not work as all the common jumps will beincluded in the limit. We need to get rid of the effect of the jumps from V1n (Y). For ease of exposition, let us take kn = θ n1/2 from now on unless otherwise stated.After the pre-averaging procedure, the smoothed increments from the diffusion part 1/4and the smooth noise part, are both of size ∆n , while the smoothed increments from 1/4the jump part may still be larger than ∆n . Following the idea of Mancini (2009) orJacod (2008), we can propose the following threshold estimator of [X c , X c ]T : ∑ n−kn +1 V2n (Y) = (∆n n Y1 )(∆n n Y2 )1{|∆n (1) 1{|∆n (2) (3.7) i,kn Y1 |≤un } i,kn Y2 |≤un } i,k i,k i=0 (r)where, un satisfy un /∆ϖ1 → 0, u(r) /∆ϖ2 → ∞, for some 0 ≤ ϖ1 < ϖ2 < 1/4, r = 1, 2. (r) n n n (3.8) (r)The threshold level un is chosen such that those (smoothed) increments larger than (r)un will be gradually excluded as n → ∞, and essentially only those increments dueto continuous part are included for the calculation of the integrated co-volatility sincewe already smooth the data by pre-averaging. Hence we have the next theorem.Theorem 2 Assume that E∥ϵ∥4 < ∞. Under Assumption 1, we have ∫ T 1 ∆ V2n (Y) → 1/2 P ρσ1s σ2s ds. (3.9) θ¯(2) n g 0 We now establish the central limit theorem. To do that, a structural assumptionon the volatility process σ is needed for technical reasons.Assumption 2 The volatility process σ = {σt , t ≥ 0} satisfies the equation ∫ t ∫ t ∫ t ′ ′ ′ ′ σrt = σr0 + ars ds + σrs dWrs + νrs dBrs , (3.10) 0 0 0where a′ , σ ′ and ν ′ are adapted c`dl`g processes, with a′ being predictable and locally a abounded, and B ′ is a standard Brownian motion, independent of W , for r = 1, 2. 7
  • 8. The concept of stable convergence is needed in the next theorem. A sequenceof random variables Xn converges stably in law to X defined on the appropriateextension of the original probability space, written as Xn →S X, if lim P (Xn ≤ x, A) = P (X ≤ x, A), for any A ∈ F and any x ∈ R. n→∞If Xn →S X, then for any F−measurable random variable σ, we have the jointweak convergence (Xn , σ) =⇒ (X, σ). So stable convergence is slightly stronger thanconvergence in law.Theorem 3 Let X1t and X2t be given in (2.3) and (2.4). Under Assumptions 1-2,E∥ϵ∥8 < ∞, and X1 and X2 are of finite variation, we have d d ( ∫ T ) √ ∫ T 1 √ −1/4 S 2 θΦ ∆n ∆n V2n (Y) − 1/2 ρσ1s σ2s ds → 1 + ρ2 σ1s σ2s dWs′ , θ¯(2) g 0 g (2) 0 ¯where W ′ is a standard Brownian motion which may be defined on an extension ofthe original space and independent of F. To do inference for the integrated covariance, we need to estimate the asymptotic ∫Tconditional variance 0 (1 + ρ2 )(σ1s σ2s )2 ds. Inspired by Mancini (2009), we have ∑ n−kn +1 Γ2 n = ∆n [(△n n Y1 )(△n n Y2 )]2 1{|∆n i,k i,k (1) 1{|∆n (2) i,kn Y1 |≤un } i,kn Y2 |≤un } i=0 ∫ T ( ∫ T ∫ T ) →P 3θ2 g (2)2 ¯ (1 + ρ2 )σ1s σ2s ds + g (2)g ′ (2) ω2 2 2 ¯ ¯ 2 2 2 σ1s ds + ω1 2 σ2s ds 0 0 0 ¯ 2 g ′ (2) 2 2 + 2 ω1 ω2 T. θ ∑n 2Remark 2 Zhang, et al. (2005) showed ωr = 1 2n n i=1 (△i Yr ) 2 →p ωr when X has 2continuous path. Jing, Liu and Kong (2010) showed that this is still true when Xcontains jumps. Thus, we obtain a consistent estimator for integrated volatility as ∫ T 1 ¯ g ′ (2) 2 Σr = △ U (Yr , g)t − 2 1/2 n ω T → P 2 σrs ds, θ¯(2) n g θ g (2) r ¯ 0 8
  • 9. ∑n−kn +1where, U (Yr , g)n = (∆n n Yr )2 1{|∆n . From these and using similar (r) i,kn Yr |≤un } t i=0 i,k ∫Tprocedure as theorem 2, we obtain a consistent estimator of 0 (ρσ1s σ2s )2 ds: 1 ¯ g ′ (2)2 2 2 ¯ g ′ (2) 2 Cn =: Γ2 − 4 ω1 ω2 T − 2 2 (ω 2 Σ1 + ω1 Σ2 ) 3θ2 g (2)2 n 3θ g (2)2 ¯ ¯ 3θ g (2) 2 ¯ ∫ T →P (1 + ρ2 )σ1s σ2s ds. 2 2 0Consequently, we have a studentized version of central limit theorem.Corollary 1 Under the same assumptions as Theorem 3, we have −1/4 ( ∫ T ) g (2)∆n ¯ 1 √ ∆1/2 V2n (Y) − ρσ1s σ2s ds →S N (0, 1), 2 θ¯(2) n g 0 2 θΦCnwhere N (0, 1) is a standard normal variable, independent of F.4 Simulation StudyIn the simulation, we take n = 23, 400. This corresponds to the number of transac-tions observed every second within one day (T = 1) consisting of 6.5 trading hours.The latent values are drawn from two Ornstein-Uhlenbeck processes with drift addedby two different symmetric stable Levy processes, respectively, namely, ∫t ∫t • X1t = 0 cos(s)ds + 0 e−2(t−s) dW1s + X1t , d ∫t ∫t • X2t = 0 cos(2s)ds + 0 e−3(t−s) dW2s + X2t , dwhere W1s and W2s are two standard Brownian motions with correlation ρ, and dXit ’s are symmetric βi -stable Levy process. We consider the weight function g(x) =min{x, 1 − x}. The parameters will be given when we report the simulation results.The microstructure noise ϵi ’s are independent and identical distributed N (0, ω) with (r) 1/4ω = 0.1, and ω = 0.2 for comparison. We take the threshold un = 5∆n , r = 1, 2. 9
  • 10. The variance of noise is chosen to match the size of integrated co-volatility. The sameprocedure is repeated 5000 times and results including relative biases, standard errorsand mean square errors are displayed in several tables, and some of the correspondinghistograms and QQ plots are displayed as well, which verify our central limit theorem. We make the following observations from the simulation results. (1). In all cases, as n increases, all the biases, standard errors and mean square errors tend to decrease. This is in line with our theoretical results. (2). The larger β is, the larger the biases and stand errors are. This is because, as β gets larger, the jumps are more frequent and jump sizes are smaller, hence it is more difficult to distinguish the increments of diffusion and β-stable process. (3). The estimator is insensitive (robust) to the variance of noise.Figure 1: Histogram and QQ plot of 5000 values of the estimator, with β1 = 1, β2 = 1,VAR(ϵ) = 0.2, ρ = 0 and n = 23400. QQ Plot of Sample Data versus Standard Normal 300 0.8 0.6 250 0.4 Quantiles of Input Sample 200 0.2 0 150 −0.2 100 −0.4 −0.6 50 −0.8 0 −1 −1 −0.5 0 0.5 1 −4 −2 0 2 4 Standard Normal Quantiles 10
  • 11. Table 1: Simulation result: Var(ϵ) =0.1β1 = 1β2 = 1 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias, s.e., mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.1029, 0.1350, 0.0286) (0.0307, 0.1350, 0.0190) (0.0865, 0.1560, 0.0316) 4680 (-0.0670, 0.0970, 0.0138) (-0.0062, 0.0949, 0.0089) (0.0510, 0.1008, 0.0127) 7800 (-0.0573, 0.0814, 0.0098) (-0.0034, 0.0875, 0.0076) (-0.0491, 0.0926, 0.0109) 23400 (-0.0376, 0.0674, 0.0059) (-0.0021, 0.0595, 0.0036) (-0.0466, 0.0621, 0.0063)β1 = 1.5β2 = 1.5 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias,s.e.,mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.1236, 0.1695, 0.0437) (0.0372, 0.1809, 0.0338) (0.1251, 0.1936, 0.0528) 4680 (-0.1200, 0.1049, 0.0253) (-0.0199, 0.1170, 0.0139) (0.0919, 0.1202, 0.0227) 7800 (-0.0986, 0.1038, 0.0208) (-0.0056, 0.1128, 0.0126) (-0.0917, 0.1133, 0.0211) 23400 (-0.0825, 0.0792, 0.0130) (-0.0039, 0.0749, 0.0056) (-0.0820, 0.0841, 0.0137)β1 = 1β2 = 1.5 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias, s.e., mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.1090, 0.1528, 0.0350) (0.0292, 0.1564, 0.0251) (0.1091, 0.1693, 0.0403) 4680 (-0.0958, 0.1019, 0.0195) (-0.0095, 0.1047, 0.0109) (0.0747, 0.1098, 0.0175) 7800 (-0.0785, 0.0941, 0.0149) (-0.0089, 0.1013, 0.0102) (-0.0646, 0.0989, 0.0139) 23400 (-0.0562, 0.0692, 0.0079) (-0.0080, 0.0639, 0.0041) (-0.0612, 0.0696, 0.0090) 11
  • 12. Table 2: Simulation result: Var(ϵ) =0.2β1 = 1β2 = 1 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias, s.e., mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.0163,0.3263,0.1067) (0.0053,0.2517,0.0634) (0.0346, 0.2475, 0.0625) 4680 (0.0077,0.1933,0.0374) (-0.0034,0.1618,0.0262) (0.0126, 0.1616, 0.0263) 7800 (-0.0169,0.1640,0.0272) ( -0.0006,0.1319,0.0174) (0.0235, 0.1401, 0.0202) 23400 (-0.0097,0.1161,0.0136) (-0.0013,0.0982,0.0097) (0.0082,0.0990,0.0099)β1 = 1.5β2 = 1.5 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias,s.e.,mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.0192,0.3024,0.0918) (-0.0132,0.3078,0.0949) (0.0211,0.3297,0.1091) 4680 (-0.0053,0.2065,0.0427) (-0.0075,0.2025,0.0411) (0.0154,0.2109,0.0447) 7800 (-0.0198,0.1834,0.0340) (-0.0040,0.1763,0.0311) (0.0176,0.1813,0.0332) 23400 (-0.0135,0.1285,0.0167) (0.0035,0.1274,0.0163) (0.0092,0.1320,0.0175)β1 = 1β2 = 1.5 ρ = 0.5 ρ=0 ρ = −0.5 n (relative bias, s.e., mse) (relative bias, s.e., mse) (relative bias, s.e., mse) 1170 (-0.0354,0.2844,0.0822) (0.0085,0.2722,0.0742) (0.0132,0.2814,0.0793) 4680 (-0.0192,0.1881,0.0357) (-0.0099,0.1825,0.0334) (0.0236,0.1863,0.0353) 7800 (-0.0164,0.1513,0.0232) (-0.0008,0.1493,0.0223) (0.0106,0.1619,0.0263) 23400 (-0.0180,0.1133,0.0132) (0.0033,0.1115,0.0124) (0.0092,0.1112,0.0115) 12
  • 13. Figure 2: Histogram and QQ plot of 5000 values of the estimator, with β1 = 1, β2 =1.5, VAR(ϵ) = 0.2 and ρ = −0.5. QQ Plot of Sample Data versus Standard Normal 250 0.8 0.6 200 Quantiles of Input Sample 0.4 150 0.2 0 100 −0.2 50 −0.4 0 −0.6 −1 −0.5 0 0.5 1 −4 −2 0 2 4 Standard Normal Quantiles5 ConclusionIn this paper, we propose a threshold type estimator of integrated co-volatility inthe simultaneous presence of microstructure noise and jumps. The estimator can beapplied to a general semi-martingale which contains jump part, regardless of the typesof the jump and the result can be employed to deal with the statistical inference of co-volatility. The future work includes estimation of regression and correlation betweentwo processes under the same settings as in this paper, and also the generalization ofproposed methodology to the non-synchronous data.6 Appendix: ProofsBy a standard localization procedure, we can replace the local boundedness in assump-tions by a boundedness assumption, and also assume that the process Yi , i = 1, 2, 13
  • 14. dand thus the jump process Xit , are bounded. That is, for all results which need theassumption about volatility and L´vy measure, we may assume, almost surely, e max{|bit |, |σit |, |Xit |} ≤ C, for some constant C > 0 Proof of Theorem 1. The proof is similar to the case without jump compo-nent. Rewrite (3.6) as ∑ n−kn +1 ∑ n−kn +1∆1/2 V1n (Y) n = ∆1/2 n (∆n n Y1c )(∆n n Y2c ) i,k i,k + ∆1/2 n (∆n n Y1c )(∆n n X2 ) i,k i,k d i=0 i=0 ∑ n−kn +1 ∑ n−kn +1 +∆1/2 n (∆n n X1 )(∆n n Y2c ) + ∆1/2 i,k d i,k n (∆n n X1 )(∆n n X2 ) i,k d i,k d i=0 i=0 =: T1 + T2 + T3 + T4 .Firstly, from Podolskij and Vetter (2009a), we have ∫ t T1 → θ¯(2) P g ρσ1s σ2s ds. 0Secondly, by the independence of Yic and Xj for i ̸= j, we can show easily Tj →L 0, d 2hence Tj →P 0 for j = 2, 3. Finally, as X1 and X2 are independent, by Proposition d d5.3 of Cont and Tankov (2004), there is no intersection between the supports of two ∑jump measures. Hence T4 →P d d s≤t C(θ, g, s)∆s X1 ∆s X2 = 0, where C(θ, g, s) is abounded functional depending only on g, θ, and s. ∆s Z = Zs − Zs− denotes jumpsize of Z at s. Proof of Theorem 2 When Yr , (r = 1, 2) have continuous path, the proofsof un-truncated version have been already finished. Therefore it suffices to show that ∆1/2 (V2n (Y) − V1n (Yc )) →P 0. n (6.11) Rewrite the left hand side as ∆1/2 (V2n (Y) − V1n (Yc )) n 14
  • 15. ∑ n−kn +1 = ∆1/2 (∆n n Y1c )(∆n n Y2c )1{|∆n (1) (2) − (∆n n Y1c )(∆n n Y2c ) i,kn Y1 |≤un ,|∆i,kn Y2 |≤un } n i,k i,k n i,k i,k i=0 ∑ n−kn +1 +∆1/2 (∆n n Y1c )(∆n n Y2d )1{|∆n (1) (2) i,kn Y1 |≤un ,|∆i,kn Y2 |≤un } n i,k i,k n i=0 ∑ n−kn +1 +∆1/2 (∆n n Y1d )(∆n n Y2d )1{|∆n (1) (2) i,kn Y1 |≤un ,|∆i,kn Y2 |≤un } n i,k i,k n i=0 ∑ n−kn +1 +∆1/2 (∆n n Y1d )(∆n n Y2c )1{|∆n (1) (2) i,kn Y1 |≤un ,|∆i,kn Y2 |≤un } n i,k i,k n i=0 =: A1 + A2 + A3 + A4 . Since the continuous and discontinuous components are independent, it is easy toshow that A2 →P 0 and A4 →P 0. Next, we show A3 →P 0. For any arbitrarily small ϵ > 0, there exists an integerN1 , as long as n > N1 , we have ∑ n−kn +1 |A3 | ≤ ∆1/2 n |(∆n n Y1d )(∆n n Y2d )|1{|∆n n Y1 |≤ϵ/2,|∆n n Y2 |≤ϵ/2} . i,k i,k i,k i,k i=0By the Levy law for modulus of continuity of Brownian motion’s paths (see, Theorem9.25, Karatzas and Shreve, 1999) and time changed Brownian motion (Theorems1.9-1.10, Revuz and Yor, 2001), there exists an integer N2 , when n > N2 , we have ∆n n Xr i,k c sup √ ≤ Λ(ω), i ∆1/4 1 2 log ∆n nfor r = 1, 2, and some constant Λ only depending on ω. On the other hand, thecentral limit theorem implies that √ 1 1 √ P (|∆n n ϵr | ≤ 1/kn log i,k ) = P (|∆n n ϵr | ≤ ∆1/4 log i,k n ) = 1 − o(1/ kn ). ∆n ∆nWe take N3 = max(N1 , N2 ) and let n > N3 , we get ∑ n−kn +1 |A3 | ≤ ∆1/2 n |(∆n n Y1d )(∆n n Y2d )|1{|∆n n Y1 |≤ϵ,|∆n n Y2 |≤ϵ} + oP (1) i,k i,k i,k i,k i=0 ∑ → P |h(g, θ, s)(∆s Y1 )(∆s Y2 )|1{|∆s Y1 |≤ϵ,|∆s Y2 |≤ϵ} , s≤t 15
  • 16. where h(g, θ, s) only depends on g, θ and s. Now, taking ϵ → 0 yields A3 →P 0. 1/2 ∑n−kn +1 i Finally, we show that A1 =: ∆n i=0 A1 →P 0. We consider the followingdisjoint cases. Note that l1 , l2 , r1 , r2 denote any positive real numbers. (r) • If |∆n n Yrc | ≥ un /2, for r = 1, 2, we have for some appropriate constant K, i,k K|∆n n Y1c |1+l1 |∆n n Y2c |1+l2 i,k i,k |Ai | 1 ≤ (1) (2) . (6.12) (un )l1 (un )l2 (r) • If |∆n n Yrc | ≤ un /2, for r = 1, 2, similarly, we have, i,k K|∆n n Y1c ||∆n n Y2c ||∆n n Y1d |r1 |∆n n Y2d |r2 i,k i,k i,k i,k |Ai | 1 ≤ (1) (2) . (6.13) (un )r1 (un )r2 (1) (2) • If |∆n n Y1c | ≥ un /2, |∆n n Y2c | ≤ un /2, we have, i,k i,k  K|∆n n Y1 |1+l1 |∆n n Y2 ||∆n n Y2 |r2 c c d   i,k i,k i,k , if |∆i,kn Y2 | ≥ un (2)   (1) (2) (un )l1 (un )r2 |Ai | ≤ 0, (2) if |∆i,kn Y2 | ≤ un and |∆n n Y1c | ≤ un , (1) 1   i,k   K|∆n n Y1 |1+l1 |∆n n Y2 | c c (2) (1) i,k i,k (1) , if |∆i,kn Y2 | ≤ un and |∆n n Y1c | ≥ un . i,k (un )l1 (6.14) (1) (2) • The case of |∆n n Y1c | ≤ un /2, |∆n n Y2c | ≥ un /2 is similar to (6.14). i,k i,kNext, we estimate |∆n n Y c | and |∆n n Y d |. By Holder’s and Birkholder’s inequalities, i,k i,kwe have   E(|∆n Y d |2 ) = ∫ (i+kn )∆n ∫ g 2 ( 1 ⌊ s−i∆n ⌋)x2 F (dx)ds ≤ K(k ∆ ), i,kn i∆n R kn ∆n t n n (6.15)  E(|∆n Y c |l ) ≤ K (k ∆ )l/2 for l > 0. i,kn l n nWithout loss of generality, we let l1 = l2 = r1 = r2 = 1. Then we deduce from above 3/2 1/2inequalities and estimations that ∆n E(|Ai |) ≤ K( 1 ∆n (1) (2) ). In view of (3.8), we get un unA1 →P 0. 16
  • 17. Proof of Theorem 3. By Theorem 4.1 of Jacod et al. (2010), it suffices to show ∑ n−kn +1 ∆1/4 (V2n (Y) n − V1n (Y )) =: c B1 →P 0. i (6.16) i=0 d dBy assumption, X1 and X2 are of finite variation, i.e., β1 < 1 and β2 < 1. Therefore c dwe have the following decomposition: Xr (t) = Xr (t) + Xr (t) for r = 1, 2, where, ∫t ∑Xr (t) = brs − 0 xFrs (dx) and Xr (t) = s≤t ∆s Xr . c d We have the following estimation in this case:   E(|∆n X d |) = ∫ (i+kn )∆n ∫ g( 1 ⌊ s−i∆n ⌋)|x|F (dx)ds ≤ K(k ∆ ), i,kn i∆n R kn ∆n t n n (6.17)  E(|∆n Y c |l ) ≤ K (k ∆ )l/2 for l > 0. i,kn l n nThen we have for some constant K, ( l +l −2 (l +l )ϖ r1 +r2 (r +r )ϖ ) 1 2 + 1 2 2 2 + 4 − 1 22 2 1 E|B1 | ≤ K∆n ∆n i 8 + ∆n 8 =: K∆n (∆c1 + ∆c2 ). n nThen we can choose l1 , l2 , r1 , r2 and ϖ2 such that, c1 > 0, c2 > 0; see A¨ ıt-Sahalia andJacod (2010) or Jing, Liu and Kong (2010) on how to select these values. This proves(6.16), hence Theorem 3. Proof of Corollary 1 The corollary is a consequence of Theorem 3 andSlutsky Theorem for stable convergence.Acknowledgements The authors would like to thank the Editor, an Associate Edition and tworeferees for their constructive suggestions that improved this paper considerably.JING’s research is supported by Hong Kong RGC grants HKUST6011/07P andHKUST6015/08P, and in part by Fundamental Research Funds for the Central Uni-versities, and Research Funds of Renmin University of China (Grant No. 10XNL007)and by NSFC (Grant No. 71071155). LIU’s research is supported in part by Fu-jian Key Laboratory of Statistical Sciences, China, and in part by NSFC (Grant No.71171103). 17
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