Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models

Transcription

1 Optimally Thresholded Realized Power Variatios for Lévy Jump Diffusio Models José E. Figueroa-López Jeffrey Nise March 8, 13 Abstract: Thresholded Realized Power Variatios (TPV) are oe of the most popular oparametric estimators for geeral cotiuous-time processes with a wide rage of applicatios. I spite of their popularity, a commo drawback lies i the ecessity of choosig a suitable threshold for the estimator, a issue which so far has mostly bee addressed by heuristic selectio methods. To address this importat issue, we propose a objective selectio method based o desirable optimality properties of the estimators. Cocretely, we develop a well-posed optimizatio problem which, for a fixed sample size ad time horizo, selects a threshold that miimizes the expected total umber of jump misclassificatios committed by the thresholdig mechaism associated with these estimators. We aalytically solve the optimizatio problem uder mild regularity coditios o the desity of the uderlyig jump distributio, allowig us to provide a explicit ifill asymptotic characterizatio of the resultig optimal thresholdig sequece at a fixed time horizo. The leadig term of the optimal threshold sequece is show to be proportioal to the Lévy s modulus of cotiuity of the uderlyig Browia motio, hece theoretically justifyig ad sharpeig selectio methods previously proposed i the literature based o power fuctios or multiple testig procedures. Furthermore, buildig o the aforemetioed asymptotic characterizatio, we develop a estimatio algorithm, which allows for a feasible implemetatio of the ewfoud optimal sequece. Simulatios demostrate the improved fiite sample performace offered by optimal TPV estimators i compariso to other popular o-optimal alteratives. Keywords: volatility estimatio, jump detectio, Lévy processes, additive processes, oparametric estimatio, thresholded estimators, power variatios. 1 Itroductio A Lévy jump-diffusio, X t := γt + σw t + J t, is costructed via the superpositio of a Browia motio with drift γt + σw t ad a idepedet compoud Poisso process J t. This is oe of the first ad simplest extesios to the classical geometric Browia motio uderlyig the famous Black-Scholes-Merto framework for optio pricig. The key motivatio behid jump-diffusio models is the icorporatio of market shocks, which result i large ad sudde chages i the price of a risky security ad which ca hardly be modeled by the large deviatio of a diffusive compoet. Jump-diffusios costitute a semiparametric subclass of the fully oparametric class of Itô semimartigales X t := t γ sds + t σ udb u + J t. Over the last decade, several estimatio methods for the itegrated variace IV t := t σ udu have bee proposed while comparatively less attetio has bee paid to the problem of jump estimatio. I this paper, we put forward a ovel approach, which produces a optimal solutio for both estimatio problems, withi the class of Lévy jump-diffusios ad a type of Itô semimartigales with determiistic local characteristics. Our method builds o the well kow class of Thresholded Realized Power Variatios (TPV) estimators, which were itroduced by Macii (1, 4). TPV estimators s strategy is based o the atural idea that if the observed icremet i X := X it/ X (i 1)t/ of the process durig a small time period is large the this likely happes due to the presece of oe or more jumps durig that time period. I that case, the icremet is to be discarded for estimatig the diffusio coefficiet or kept for estimatig the jump features of the process. Departmet of Statistics, Purdue Uiversity, West Lafayette, IN, 4797, USA (figueroa@purdue.edu). Departmet of Statistics, Purdue Uiversity, West Lafayette, IN, 4797, USA (jise@purdue.edu). 1

2 Sice the semial work of Macii, several authors have leveraged or exteded the thresholdig cocept to deal with more complex stochastic models. For istace, i Macii (9) ad Jacod (7, 8), the problem of jump detectio ad itegrated volatility estimatio was studied withi the class of Itô semimartigales with fiite ad ifiite jump activity Lévy compoets. More recetly, Gegler ad Stadtmüller (1) also studied the problem of oparametrically estimatig the characteristic triplet of a geeral Lévy process by way of thresholdig type estimators. Other variatios of jumps detectio methods via thresholdig ca be foud i Lee ad Haig (1) ad Jig et al. (1). We also refer the reader to Sectio. for more details about other applicatios of threshold estimators i the literature. Let us also remark that, eve though the oparametric estimatio of the Lévy triplet has a log history (cf. Rubi ad Tucker (1959)), the problem has received some reewed attetio durig the last decade (cf. Woerer (6), Figueroa-López (4, 9), Neuma ad Reiss (7), Comte ad Geo-Catalot (9, 11)). The statistical performace of TPV estimators critically depeds upo the choice of a good thresholdig sequece. The mai theme of the preset work is twofold: first, we precisely determie what costitutes a good threshold sequece ad, secod, we propose a objective method for selectig such a sequece based o judiciously chose criteria of statistical optimality. To address the first poit, we provide a complete picture of the rates of covergece of the three mai statistical measures of performace (bias, variace, ad mea-squared error) for the TPV estimators of the diffusio coefficiet σ, the uderlyig Poisso process, ad the whole compoud Poisso compoet. As a cosequece of our results, we obtai explicit ecessary ad sufficiet coditios o the class of threshold sequeces, which allow for mea-squared cosistecy of the TRV ad jump compoet estimators. Sufficiet coditios have bee kow i the literature for some time, however, here we are able to tighte the results up to simple equivalece coditios. To address the issue of selectig a suitable threshold, we propose a optimal threshold sequece, which, o oe had, is capable of rederig mea-squared cosistet estimates of the volatility coefficiet ad the jump times simultaeously, ad, o the other, miimizes a atural statistical measure of performace. I order to achieve this, we itroduce a loss fuctio which pealizes threshold estimators for jump-misclassificatios ad, furthermore, we are able to show that such a loss fuctio is asymptotically equivalet to oe which directly pealizes the mea-squared errors of the TPV estimators i questio. We the proceed to show that proposed optimizatio problem is well-posed ad admits a uique solutio. We also derive a explicit ifill asymptotic characterizatio of the optimal threshold sequece. The leadig term of the optimal threshold sequece is show to be proportioal to the Lévy s modulus of cotiuity of the uderlyig Browia motio. The latter fidig furishes a theoretical justificatio to the fuctioal form of the Boferroi type threshold estimator proposed i Aderse et al. (7) ad Gegler ad Stadtmüller (1), ad also a objective method to choose the expoet of the power-based threshold sequece proposed by Macii (4), which is further improved via the iclusio of a logarithmic compoet. Furthermore, buildig o the aforemetioed asymptotic characterizatio, we develop a estimatio algorithm which allows for a feasible implemetatio of the ewfoud optimal sequece. I the big picture, our results ope a ew lie of research to be exteded i future works to larger classes of stochastic processes with jumps. Furthermore, the approach put forward ca be combied with other popular oparametric methods such as multi-power variatio estimators (cf. Bardorff-Nielse ad Shephard (4, 6); Bardorff-Nielse et al. (6)). I a far reachig applicatio, optimality ca potetially be modified to hadle microstructure oise compoets as i the semial two-scale method of Zhag et al. (5), the pre-averagig approach of Jacod et al. (9), or the blocked multipower variatio method of Myklad et al. (1). The rest of the paper is orgaized as follows. Sectio itroduces the framework, estimatio problem, ad a review of the TPV estimators. Sectio 3 surveys the asymptotics of the bias, variace, ad mea-squared error of the TPV estimators. I Sectio 4 we itroduce ad aalyze the optimizatio problem for both Lévy jump diffusios ad additive processes with absolutely cotiuous characteristics. Sectio 5 presets our estimatio algorithms ad the results of Mote Carlo studies desiged to ivestigate the fiite sample performace of optimally thresholded TPV estimators. Sectio 6 cotais some cocludig remarks, while all proofs are deferred to two fial appedices. Estimatio Problem I this sectio we itroduce the model framework ad stadig assumptios. We the proceed to state the estimatio problem ad itroduce the estimators which will be examied throughout the remaider of the paper.

3 .1 The Framework Throughout, let W = (W t ) t ad J = (J t ) t respectively deote a Browia motio ad a idepedet compoud Poisso process defied o a complete filtered probability space (Ω, F, (F t ) t, P). Give model parameters (γ, σ, λ) R R + R + ad a absolutely cotiuous probability measure F o (R, B(R)) with desity f, we cosider a Lévy process X = (X t ) t of the form N t X t := γt + σw t + J t = γt + σw t + ζ j, (.1) where {ζ j } j 1 deote the successive jumps of the compoud Poisso process J, which are by defiitio i.i.d. with distributio F, ad N = (N t ) t deotes the coutig process associated with the jumps of J, which is ecessarily a Poisso process with itesity λ. The desity f of the jump distributio is take to be of the form j=1 f(x) = pf + (x)1 x ] + qf (x)1 x<], (.) with p, 1] ad q := 1 p, ad where f + ad f are bouded probability desity fuctios o, ) ad (, ], respectively, such that mi{f + (), f ()} >. We also assume both f + ad f have bouded ad cotiuous derivatives f (1) + ad f (1) o (, ) ad (, ), respectively, such that f (1) + () := lim x + f (1) + (x) ad f (1) () := lim x f (1) (x) exist ad are fiite. The followig costat will also be eeded C(f) := pf + () + qf (). For estimatio purposes, we assume our data cosists of a discrete record of observatios from the process X o a give fixed time iterval, t]. Cocretely, we observe ( i X) t, where i X := X t i X ti 1 deotes the i th icremet, ad t i := t i := i, for i = 1,..., t, gives rise to the samplig grid. Let us fiish with some otatio that will be eeded i what follows. Throughout, Φ( ) ad φ( ) deote the stadard Gaussia distributio ad desity fuctios, respectively, while Φ 1 (y) := if{x : Φ(x) y} ad Φ(x) = φ(z)dz deote the correspodig quatile ad survival x fuctios. Also, for brevity we will ofte refer to the costats γ := γ/ ad σ := σ /.. Threshold Style Estimators The class of threshold style estimators bega with the work of Macii (1, 4), uder the jump-diffusio model (.1). Sice the, TRV estimators have bee applied to a larger class of models ragig from Lévy processes to geeral semimartigales with time-varyig stochastic volatility (cf. Macii (4, 9), Jacod (7, 8), Corsi, Pirio ad Reó (1), ad Gegler ad Stadtmüller (1)). Besides estimatio, these statistics ca also be used to costruct oparametric tests for path-wise properties of the uderlyig process such as the presece of jumps or of a diffusive compoet (see, e.g., Aït-Sahalia ad Jacod (9a, 1), Cot ad Macii (11)). See also Aït-Sahalia ad Jacod (9b), Figueroa-López (1), Jig et al. (1), ad refereces therei for further applicatios of the TPV statistics i estimatig the degree of small-jump activity i Lévy ad semimartigale models. Throughout the rest of the paper, we will take a thresholdig sequece to be a determiistic sequece B = (B ) 1 of o-egative real umbers such that lim B =. Give a thresholdig sequece B = (B ) 1, we defie the correspodig TRV estimator as t T RV (X)B] t := i X 1 i X B ]. (.3) Similarly, with certai abuse of otatio, we will also refer to the followig TRV estimator associated to a idividual thresholdig level B, ]: t T RV (X)B] t := i X 1 i X B]. (.4) As explaied i the itroductio, the basic idea of thresholdig estimators is to filter out those icremets which, due to their large magitude, may cotai jumps. Disceribly, the accuracy of threshold style estimators critically depeds upo a thresholdig sequece B = (B ) 1. Several explicit fuctioal forms for thresholdig sequeces have bee proposed i the literature. Macii (4) proposed a power threshold, which takes the form B P ow(α, ω)] := α ω, (.5) 3

4 for some α > ad ω (, 1/). The previous domai for ω esures that the sufficiet cosistecy coditio (3.9) below is satisfied. Both Aderse et al. (7) ad Gegler ad Stadtmüller (1) idepedetly proposed the threshold B BF ( σ, C)] = σ ( Φ 1 1 C ), (.6) for some costat C, which may deped o the samplig frequecy or expected jump misclassificatio rate, ad some prior estimate σ of σ. We shall refer to the resultig TRV estimator as a Boferoi threshold estimator, sice this threshold is ispired by the Boferoi Type I error cotrol procedure. The i-fill asymptotic properties of TRV estimators are well uderstood for large classes of threshold sequeces satisfyig covergece coditios which are easy to check i practice (see e.g. Macii (4, 9) ad Jacod (8) for precise details). It is importat to realize that TRV estimators were desiged with jump detectio i mid ad, thus, it is ot surprisig they ca be adapted to develop estimators for the jump compoet parameters λ ad F ( ) i a highfrequecy/log-horizo samplig scheme (see Macii (1) ad Gegler ad Stadtmüller (1) for more details). The idea is simple: if it is believed that at least oe jump has occurred durig a specified subiterval t i 1, t i ], the the icremet i X = X t i X ti 1 ca be used as a proxy of the jump size itself. I that case, the followig are atural statistics to estimate the uderlyig compoud Poisso process (J t ) t of the jump-diffusio (.1) ad its correspodig jump coutig process (N t ) t : t ĴB] t := ( i X)1 i X >B ], NB] t := 1 i X >B ]. (.7) Aalogously to (.3)-(.4), we will also defie the estimators ĴB] t ad NB] t correspodig to a sigle threshold level B as i (.7) but replacig B by B i the idicator fuctios. The statistics (.7) aturally lead to the followig estimators for the successive {(τ j, ζ j )} j 1 arrival times ad sizes of the jumps of J: τ j+1 := mi {t i = i } : t i > τ j & i X > B, ζ j := τ j X, (j 1), (.8) where ˆτ := ad mi :=. Uder certai coditios o the threshold sequece B = (B ) 1 (see Theorem 3.4 below for the detailed coditios), Macii (9) s Theorem 1 implies the strog cosistecy of the previous estimators to their correspodig true values: t ĴB] t a.s. a.s. J t, NB] t N t, τ j 1 τ j t a.s. τ j 1 τj t, ζ j 1 τ j t a.s. ζ j 1 τj t, as, for each fixed t >. Although the previous estimators have may useful attributes, the choice of the thresholdig sequece is otrivial ad ca critically affect the fiite-sample performace of the resultig estimators. The latter issue is the mai motivatio behid the preset work. I Sectio 4, we itroduce the cocept of optimal thresholdig ad, more importatly, we explicitly characterize the leadig order term of the optimal threshold sequece. Before this, we shall preset some results related to the first ad secod order statistical properties of the proposed TRV estimators. 3 Properties of the Estimators As oe would expect, ot all sequeces B = (B ) of positive umbers covergig to zero will yield useful threshold type estimators. Ituitively, i order for a threshold estimator to perform well, the threshold sequece should ot coverge to zero too fast, otherwise there would be a over filterig of icremets that actually do ot cotai ay jumps. O the other had, if the sequece coverges very slowly, the estimator is expected to exhibit poor efficiecy. I this sectio we compare the rates of covergece of the bias, variace, ad mea-squared error (MSE) for the estimators of both the cotiuous ad jump compoets. As a cosequece, we will also be able to characterize simple ecessary ad sufficiet coditios for the estimators to be mea-squared cosistet. 4

5 Withi the cotext of the jump compoet estimatio, the accurate determiatio of the uderlyig jump times plays a key role. For ay give observed icremet, two types of jump detectio errors ca occur: false positive or false egative misclassificatios. Due to their close coectio with hypothesis testig, we will refer to these mistakes as Type I ad Type II jump misclassificatios errors, respectively. Cocretely, we say that a Type I error occurs durig a iterval ((i 1)/, i/] whe the correspodig icremet i X cotais o jump ad still exceeds the commesurate threshold i magitude; i.e., the idicator 1 i X >B, i N=] is oe. O the other had, we say that a Type II error occurs durig ((i 1)/, i/] whe i X cotais at least oe jump but the correspodig icremet i X fails to exceed the threshold i absolute value so that the idicator 1 i X B, i N ] is oe. The previous aalogy with hypothesis testig will help us to iterpret some of our results below. Our first result gives a decompositio of the mea-squared estimatio error (MSE) of the estimator NB] t defied i (.7) ad characterizes the rate of covergece of each of their compoets. As a cosequece, we obtai a simple ecessary ad sufficiet coditio o the threshold sequece i order to achieve mea-squared cosistet estimatio of the drivig Poisso process. The resultig coditio is iterestig i that it gives a implicit lower boud o how fast the threshold sequece should coverge to zero. The proof follows from the decompositio NB] t N t = ( t ) t t 1 i N ] N t + 1 i X >B, i N=] 1 i X B, i N ], (3.1) ad Lemma B.1 below together with simple tail asymptotics for the Gaussia distributio (see Nise ad Figueroa- López (13) for the details). Propositio 3.1 (Cosistet estimatio of the Poisso process). Suppose the jump desity f is of the form (.) ad B = (B ) is a thresholdig sequece. The, the followig assertios hold: (1) The mea-squared error of the estimator NB] t itroduced i (.7), relative to the uderlyig Poisso process N t, coverges to if ad oly if ( ) B lim φ =. (3.) B σ () Furthermore, uder (3.), the mea-squared error of NB] t admits the followig asymptotic decompositio as : ] MSEN; B] t := E NB] t N t σt ( ) B φ + λtb C(f). (3.3) B σ The followig result is the aalog of Propositio 3.1 for the estimator ĴB] t of the compoud Poisso process J t. Similarly to Propositio 3.1, the result is a cosequece of the atural decompositio ĴB] t J t = ( t ) ( i X)1 i N ] J t + ( i X)1 i X >B, i N=] ( i X)1 i X B, i N ] (3.4) as well as Lemma B.1 i Appedix B ad the well-kow asymptotic property t Γ(a, b) b a 1 e b, (a >, b ), (3.5) for the upper icomplete Gamma fuctio Γ(a, b) := x a 1 e x dx (see Nise ad Figueroa-López (13) for the b details). Propositio 3.. Uder the assumptios ad otatio of Propositio 3.1, the followig assertios hold: (1) The mea-squared error of the estimator ĴB] t itroduced i (.7), relative to the uderlyig compoud Poisso process J t, coverges to if ad oly if the followig coditio is satisfied: lim B =. (3.6) t 5

6 () Uder (3.6), the mea-squared error of the estimator ĴB] t admits the followig decompositio whe : ] MSEJ; B] t := E ĴB] t J t tσ ( ) B B φ + tλb3 σ 3 C(f ζ) + 4tλσ. (3.7) I comparig the ecessary ad sufficiet coditios give by (3.) ad (3.6), it traspires that the coditios required to recover the jump compoet J t are weaker tha those required to recover the Poisso process N t drivig J t. A heuristic explaatio of this pheomeo ca be draw from the decompositios (3.1) ad (3.4), which suggest that the estimator NB] t is highly sesitive to jump misclassificatios (regardless of the type but especially Type I errors) ad, hece, it is ecessary that the expected umber of Type I ad Type II errors coverges to zero i order to achieve mea-squared cosistecy. By cotrast, the estimator ĴB] t is sesitive to Type II errors, but it is far less sesitive to Type I errors tha what its couterpart NB] t is. The above MSE decompositios also idicate that, i order to cotrol Type II errors, it suffices that the threshold sequece coverge to zero, irrespective of its rate. By cotrast, i order to cotrol Type I errors, or spurious jump detectios, the threshold sequece caot ted to zero too quickly, which is precisely why the ecessary coditios impose lower bouds o the rate of covergece. We ow proceed to cosider the estimatio of the cotiuous compoet of the process. Cocretely, the followig result compares the rates of covergece of the bias, variace, ad mea-squared error for the TRV estimator (.4). These follow from the atural decompositio t T RV (X)B] t tσ = t i X 1 i N=] tσ + t i X 1 i X B, i N ] i X 1 i X >B, i N=], (3.8) together with Lemma B.1 ad the asymptotic property (3.5). We refer the reader to Nise ad Figueroa-López (13) for the details. Theorem 3.3. Suppose the jump distributio has a desity f of the form (.) ad B = (B ) is a give thresholdig sequece. The, the TRV estimator i (.4) will coverge i mea-square to tσ if ad oly if B =. lim Furthermore, i this case, the rates of covergece for the bias, variace, ad mea squared error, as, are give by: E T RV (X)B] t ] tσ = λt σ + tλc(f) B 3 σt ( ) B B φ + o( 1 R ), Var (T RV (X)B] t ) tσ4 3 σ, MSET RV ; B] t := E T RV (X)B] t tσ ] = tσ4 + λ t ( ) 9 B6 C(f) + 4t σ Bφ B + o( 1 R σ ), where above R := B 3 B φ(σ 1 B ). Iterestigly eough, the ecessary ad sufficiet coditio idetified i Theorem 3.3 is idetical to that i Propositio 3.. As see from the decompositios (3.4) ad (3.8), this equivalece lies i the fact that both estimators are affected by jump misclassificatios i a similar way. That is, while both estimators are sesitive to Type II errors, they are sigificatly less susceptible to Type I errors whe compared to the estimator NB] t. For compariso we ow preset two related results from Macii (9) (Theorem 1 ad Corollary therei). Theorem 3.4. Let B = (B ) be a threshold sequece such that lim l() B =. (3.9) The, for P-almost every ω Ω, there exists a N(ω) > large eough such that 1 i N=](ω) = 1 i X B ](ω), for all i = 1,..., t ad all N(ω). I particular, it follows that T RV (X)B] t P tσ as. Remark 3.1. It ca be show that every sequece satisfyig (3.9) also satisfies (3.), which i tur implies (3.6); however, either coverse holds. For the class of jump-diffusio processes that we are cosiderig here, Theorem 3.3 above sharpes Theorem 1 from Macii (9) i that the former characterizes ecessary ad sufficiet coditios for mea-squared cosistecy. 6

7 4 Optimally Thresholded TPV Estimators 4.1 Lévy Jump-Diffusio models I light of Propositios 3.1 ad 3. ad Theorem 3.3, it is clear that differet choices of the thresholdig sequece will give rise to differig rates of covergece for the three mai statistical measures of performace: bias, variace, ad mea-squared error. For a give estimator this observatio aturally leads to the questio of how to select a optimal threshold, where optimality is measured by how fast the aforemetioed statistical measures coverge to zero. The formulatio of a rigorous ad feasible optimizatio problem, givig rise to a uique optimal threshold sequece will serve as the focal poit for this sectio. A importat questio related to the otio of optimal thresholdig is whether or ot it ca be feasibly implemeted i practice. To address this poit, we will characterize explicitly the asymptotic behavior (as ) of the optimal threshold sequece B = (B). As it turs out, the leadig term of B depeds oly o the volatility parameter σ, a fact which will allow us to develop a iterative method to estimate the optimal threshold sequece (Sectio 5). I order to simplify the proof of the quasi-covexity property of the loss fuctio (Theorem 4. below), we assume throughout what follows that the drift parameter γ is zero, eve though this assumptio ca actually be relaxed as show by the more geeral Theorem 4.5 below. The proofs of all the results i this sectio ad their supportig lemmas ca be foud i Appedices A ad B. A essetial igrediet to ay well-posed optimizatio problem is the selectio of a objective fuctio, which imposes proper pealties o the feasible set of iterest, ad which possesses good covexity properties. For our purposes, it seems reasoable that a well chose threshold sequece should lead to a accurate estimatio of both the cotiuous ad jump compoets of the model. Specifically, for a fixed time horizo t > ad a fixed samplig grid mesh 1/, we seek a threshold sequece which simultaeously miimizes the mea-squared errors of both estimators T RV (X)B] t ad NB] t, so that we may optimally recover the diffusio parameter σ ad the jump times (τ i ) Nt. Obviously, i the light of the well-kow variace-bias decompositio of the MSE, i miimizig the MSEs of the two estimators T RV (X)B] t ad NB] t, we are also cotrollig their variaces ad biases as well. With these ideas i place, we propose the followig two cadidate objective fuctios for our optimizatio framework which is to follow. Defiitio 4.1. (TPV mea-squared error based loss fuctio) Loss (1) (B) := E T RV (X)B] t tσ ] ] + E NB] t N t. (4.1) Defiitio 4.. (Total jump misclassificiatio based loss fuctio) t Loss () ( ) (B) := E 1 i X >B, i N=] + 1 i X B, i N ]. (4.) As before, we ca similarly defie the loss fuctios Loss (1) (B) ad Loss () (B) correspodig to a sigle threshold level B, ) by replacig B by B o the right-had side of (4.1) ad B by B o the right-had side of (4.). For a fixed N, whe optimized over all B >, the loss fuctio Loss (1) (B) will favor those threshold levels which simultaeously miimize the mea-squared errors of both estimators T RV (X)B] t ad NB] t. O the other had, whe the same procedure is cosidered for the loss fuctio Loss () (B), threshold levels which miimize the expected total umber of jump misclassificatios will be favored. O the surface, the two proposed objective fuctios appear to differ by a cosiderable degree, however, as the followig result shows, they are actually asymptotically equivalet. Theorem 4.1. Give a arbitrary threshold sequece B = (B ) satisfyig (3.6), there exists a positive sequece (C (B)), with lim C (B) =, such that the followig relatioship holds: where, as, Loss () (B) + R (B) Loss (1) (B) (1 + C (B))Loss () (B) + R (B) + R (B), (4.3) R (B) tλ + t ( ( σ B φ σ B ) ] 6tσ 4 λb C(f)), R (B) + 3B6 t λ C(f). 7

8 Furthermore, lim if B> Loss(1) (B) = 1. (4.4) if B> Loss() (B) Theorem 4.1 suggests that the objective fuctio Loss () ( ) ca act as a asymptotic lower ad upper approximatio to the objective fuctio Loss (1) ( ). As it will be show below, the former objective fuctio gives rise to a tractable well-posed optimizatio problem, while the latter is aalytically more difficult to deal with. I light of Theorem 4.1, we will work hereafter with the more tractable objective fuctio Loss () ( ) ad, for simplicity, we will write Loss ( ) istead of Loss () ( ). Havig laid out the key ideas, we ca ow state the formal optimizatio problem. For each N, cosider the followig miimizatio problem, wheever it is well-posed, B := arg if B> Loss (B). (4.5) We will ow proceed to study the aalytical properties of the proposed objective fuctio, ultimately leadig to a pair of key results which first shows that the optimizatio problem (4.5) is well-posed ad secodly provides a explicit asymptotic characterizatio of the resultig optimal solutio. Throughout what follows, it will be coveiet to decompose the loss fuctio as ( Loss (B) = t e λ/ Φ ( where F k (x) := P σ 1 W + ) k j=1 ζ j x. B σ ) + t k=1 ( ) k λ e λ/ k! F k (B) F k ( B) ], (4.6) I order to establish the existece ad uiqueess of the optimal threshold sequece, we first show that the objective fuctio Loss ( ) is differetiable ad, for sufficietly large N, quasi-covex. Together these imply that if a critical poit exists, it must correspod to the global miimum of the fuctio. Recall that a mappig g : X R defied o a covex domai X R is said to be quasi-covex if for ay λ, 1] ad x, y X, g(xλ+y(1 λ)) max{g(x), g(y)}. Theorem 4.. (Quasi-covexity of the Loss Fuctio) Suppose the jump distributio has a desity f of the form (.). The, there exists a N N such that for all N, the loss fuctio Loss (B) is quasi-covex ad possesses a uique global miimum B. Remark 4.1. As a cosequece of the proof, oe ca deduce the lower boud N C(f) πσλ ] /3 o the critical sample size N of the previous result. This boud mostly serves a qualitative role as it depeds o the ukow parameters. Still, this boud suggests that the higher the volatility, the greater the jump itesity, ad the more cocetrated the jumps are aroud the origi, the larger the sample size must be i order to obtai a well-posed optimizatio problem. We ow derive a asymptotic characterizatio of the optimal threshold sequece. Although the above optimizatio problem (4.5) caot, i geeral, be solved explicitly, the first order term below ca serve as a blueprit to fid a suitable threshold sequece whe the samplig frequecy is sufficietly large ad a prior estimate of the volatility is available. I Sectio 5 below, we further develop this idea. Theorem 4.3. (Asymptotic Characterizatio of the Optimal Threshold Sequece) Let B,1 The, uder the coditios of Theorem 4., the optimal threshold sequece (B) is such that B = B,1 + o := 3σ l()/. ( l()/ ), ( ). (4.7) Remark 4.. Iterestigly eough, the leadig term of (4.7) is exactly proportioal to the Lévy s modulus of cotiuity of a Browia motio (see, e.g., (Karatzas ad Shreve, 1991, Chapter )): lim sup h 1 h l(1/h) sup W t W s = 1, a.s. t s <h,s,t,1] 8

9 Theorem 4.3 provides a alterative theoretical justificatio for the Boferroi threshold sequece (.6) based o statistical optimality criteria. Ideed, usig the asymptotic for the quatile fuctio Φ 1 provided i Domiici (3), it turs out that the leadig term of (.6) is give by B BF ( σ, C)] = σ ( Φ 1 1 C ) l() l() σ, which is agai proportioal to the leadig term of the optimal threshold sequece (4.7). As a side ote, both B ad B,1 do t meet the secod coditio i (3.9) ad, thus, they are outside of the class of estimators cosidered by Macii (9). Both the optimal sequece B ad its leadig term B,1 satisfy (3.) ad, thus, (3.6), which, i light of Theorem 3.3, implies that the MSE of their correspodig TRV estimators will exhibit the rate of covergece described therei. For completeess we ow summarize the i-fill asymptotic properties of the differet thresholded estimators based o the optimal threshold sequece B = (B) 1 as well as the rate of covergece of the objective fuctio. Let us also remark that the same asymptotics are satisfied by the threshold estimators correspodig to B,1 := (B,1 ) 1. Corollary 4.4. (Optimally Thresholded Objective Fuctio, Jump Compoet ad TRV Estimators) Suppose the assumptios of Theorem 4.3 are eforced. The followig assertios hold for the optimal sequece B := (B ) 1, as : Loss (B ) λtc(f) ] 1/ l() 3σ, MSEN; B ] t λtc(f) 3σ l() ] 1/, MSEJ; B ] t 4λtσ. (4.8) E T RV (X)B ] t ] tσ λtσ 1, V ar (T RV (X)B ] t ) tσ4, MSET RV ; B ] t tσ4. (4.9) 4. Additive Processes with Absolutely Cotiuous Characteristics I this subsectio, we will preset some prelimiary results which demostrate that the idea of optimal thresholdig ca be exteded beyod the fiite jump activity Lévy settig. Throughout, we suppose the uderlyig process is a Itô semimartigale with determiistic local characteristics ad takes the form X s := s γ(u)du + s N s σ(u)dw u + ζ j =: Xs c + J s, (4.1) where (N s ) s is a o-homogeeous Poisso process with rate fuctio {λ(s)} s ad, as before, we suppose the cotiuous ad jump compoets are idepedet. Here, γ :, ) R, λ :, ) R +, ad σ :, ) R + are assumed to be cotiuous determiistic fuctios such that, for ay give fixed t >, j=1 if λ(s) > ad if σ(s) >. s t s t Let us also relax the samplig scheme of Sectio ad assume a o-uiform samplig grid of the fiite time horizo, t]; cocretely, it is assumed that we observe the process at the sample times π : = t < t 1 < < t m = t. The followig is a atural geeralizatio of the optimal thresholdig problem (4.5) to the curret cotext: m ( if E 1 Xti (B 1,B,...,B m) R m X ti 1 >B i,n ti N ti 1 =] + 1 Xti X ti 1 B i,n ti N ti 1 ]) ]. (4.11) + That is, we are ow lookig for a fiite vector of thresholdig levels B = ( ) Bt 1, Bt,..., Bt m R m + (oe for each subiterval of the partitio) at which the miimal value (4.11) is reached. As before, the loss fuctio behid the 9

10 problem (4.11) ca be iterpreted as the overall expected umber of jump misclassificatios. additivity of this loss fuctio, each Bt i will solve a optimal thresholdig problem of the form I tur, due to the if B> L (B) = L ti,h i (B t i ), where we had set h i := t i t i 1 ad L s,h (B) := P ( X s+h X s > B, N s+h N s = ) + P ( X s+h X s B, N s+h N s ), which is aalogous to the loss fuctio give by (4.6) but assumig that we oly have at our had the observatio of the icremet X s+h X s. As it turs out, the theoretical framework we have developed i the previous sectio ca be exteded to the curret cotext. Cocretely, for each fixed samplig scheme π = (t,..., t m ), there exists a uique optimal threshold scheme B = ( B t 1, B t,..., B t m ), provided that the samplig mesh max1 i m {t i t i 1 } is small eough. Furthermore, a asymptotic characterizatio similar to that of Theorem 4.3 is also satisfied. The followig theorem summarizes these two results. Note that, i the absece of the drift γ, Theorem 4.5 below would ideed be a direct cosequece of our earlier Theorems 4. ad 4.3. However, the iclusio of the drift γ causes some techical subtleties that ca evertheless be hadled. Theorem 4.5 (Quasi-covexity ad Asymptotic Characterizatio). For ay fixed t >, there exists h := h (t) > such that, for all s, t] ad h (, h ], the followig two assertios hold true: 1. The fuctio L s,h (B) is strictly quasi-covex i B ad possesses a uique global miimum B s,h.. The optimal threshold B s,h is such that, B s,h = 3σ (s)h l(1/h) ] 1/ + o( h l(1/h)), (h ). (4.1) As with our earlier Lévy jump-diffusio framework, despite the fact that the optimizatio problem (4.11) caot be solved explicitly, the characterizatio (4.1) provides a basis o which the optimal threshold ca be implemeted i practice. I the sequel, we propose a kerel based iterative threshold estimatio scheme desiged to approximate the spot volatility ad optimal threshold i tadem. 5 Mote Carlo Study I this sectio, we address the feasibility of the optimal TRV estimators itroduced i the previous two sub-sectios by developig a iterative algorithm aimed at estimatig the first-order approximatios of the optimal thresholdig sequece. By Mote Carlo simulatios, we show that the proposed estimator is i some sese adaptive to a wide rage of sigal-to-oise data scearios. 5.1 Lévy Jump Diffusios As previously explaied, the optimal threshold B does ot admit a explicit form. A remedy to this issue is provided by the first-order approximatio (4.7), which possesses a explicit ad parsimoious form. Ufortuately, (4.7) itself depeds o the ukow parameter σ, which is precisely what we would be able to estimate if we had the threshold. A atural approach is the to implemet a sort of fixed poit method where we start with a prelimiary guess for σ ad proceed to iteratively improve this guess by feedig it back ito the TRV estimator. We the propose the followig method: B,1 Optimal TRV Based Iterative Algorithm for Lévy jump diffusios: t Set σ, := 1 t i X ad B, := while σ,k 1 > σ,k do σ,k+1 1 t T RV (X) B,k ] t 3 σ ad B,k+1, l() ] 1/ 3 σ ] 1/,k+1 l() 1

11 ed while { } Let k := if k 1 : σ,k+1 = σ,k ad take σ,k as the fial estimate for σ ad the correspodig B,k as a estimate for B. Note that, by desig, σ,k σ,k 1 σ, for ay k ad, sice each σ,k takes values i the fiite set {t 1 t i X δ i : δ i {, 1}}, these two facts esure that the algorithm fiishes i fiite time ad the quatity σ,k is well defied. It is iterestig to study the behavior of lim σ,k, however, this will be the subject of future research. We ow proceed to compare the statistical performace of our proposed estimator σ,k to the followig three alteratives: (1) A power threshold estimator as defied i (.5), with the same parameter values, α = 1 ad ω =.495, as give i the Mote Carlo study of Macii (9). () A Boferroi type threshold estimators as defied i (.6) usig a data-depedet algorithm proposed by Gegler ad Stadtmüller (1), wherei they take C = 1 ad use a two-step procedure to calibrate 1 the threshold. (3) Fially, we cosider a TRV estimator of the form B := βσ/, where β 3, 6]. I order to implemet this estimator i practice, we would first eed to fix the value of β ad, the, estimate a suitable value for σ usig, e.g., a two-step procedure similar to that of Gegler ad Stadtmüller (1) (see previous poit ()). Still, β will have to be chose i a kid of ad hoc fashio. For compariso purposes, we shall use below the true value of σ ad, due to this fact, we called it a oracle TRV estimator ad use the shorthad otatio Or(β) to refer to it. Below, we take β {3, 4.5, 6}. I order to examie the estimatio accuracy of the cosidered TRV estimators, we geerate sample paths from both the Merto ad Kou models (see Merto (1976) ad Kou () for details), where the jump desity fuctios take the followig forms: ( ) x µ 1 f Merto (x) = φ δ δ, f Kou(x) = p (1 p) e x/α+ 1 x ] + e x /α 1 x<], (5.1) α + α for some costats δ, α +, α R +, µ R, ad p, 1]. We cosider three differet scearios: (S1) For the Merto model we simulate icremets at a daily frequecy (1/ = 1/5) for N = 1, days, which correspods to a time horizo of t 4 years. We set the model parameters to σ =.3, λ = 5, µ =, ad δ =.6. This samplig desig ad parameterizatio coicides with that give i Aït-Sahalia (4), which was subsequetly used by both Macii (9) ad Gegler ad Stadtmüller (1). Due to the low samplig frequecy ad high sigal-tooise ratio, as measured by V ar( i J)/V ar( i W ) =, the Merto model simulatio study tests the ability of the threshold estimators to detect a sparse high-amplitude sigal via a coarse samplig, over a log time horizo. (S) For the Kou model, we simulate icremets at a 5-miute frequecy (1/ = 1/( )) for N = 39 time periods, which equates to a time horizo of t = 1 week. This samplig desig is cosistet with observig 5-miute returs for 6.5 hours per day over a 5 day tradig week. For this study we set the model parameters to σ =.5, λ = 5, p =.45, α + =.5, ad α =.1. As a cosequece of the high samplig frequecy ad low sigal-to-oise ratio, V ar( i J)/V ar( i W ) =.65, the first Kou model simulatio study ivestigates the capability of the threshold estimators to idetify a semi-sparse low-amplitude sigal by meas of fie samplig over short time horizos. (S3) Agai, usig the Kou model, we simulate icremets at a 5-miute frequecy, however, this time for N = 19, 656 time periods, which is cosistet with a t = 1-year time horizo. We set the jump distributio parameters to p =.5 ad α + = α =.1, ad chaged the other model parameters to σ =.4 ad λ = 1. This samplig desig together with the high sigal-to-oise-ratio, as measured by V ar( i J)/V ar( i W ) = 15, provides a test case for the ability of threshold estimators to detect a low-amplitude abudat sigal via frequet samplig over log time horizos. Sice the parameters of the jump distributio are such that E i J] =, P( ζ 1) 1, ad the Poisso process rate parameter λ is take to be large, this third simulatio also offers a approximate experimet i optimal thresholdig for Lévy processes possessig a ifiite activity jump compoet. 1 Gegler ad Stadtmüller (1) calibratio s method has two steps. (1) Use the sample stadard deviatio, σ, as a iitial estimate of the true σ, which yields the iitial threshold B BF ( σ, 1)]. () Compute a updated TRV estimate for σ 1, ad a updated threshold B 1 BF ( σ 1, 1)]. This type of estimator, which was proposed to us by a aoymous referee, is commoly used i practice. 11

12 (S1) Merto Model: 4-year / 1-day (S) Kou Model: 1-week / 5-miute (S3) Kou Model: 1-year / 5-miute σ =.3 λ = 5 σ =.5 λ = 5 σ =.4 λ = 1 µ =, δ =.6 p =.45, α + =.5, α =.1 p =.5, α + = α =.1 Method T RV S T RV Loss S Loss T RV S T RV Loss S Loss T RV S T RV Loss S Loss B,k P ow BF Or(3) Or(4.5) Or(6) Table 1: Fiite-sample performace of the threshold realized variatio (TRV) estimators for the three simulatio scearios (S1)-S(3) based o K = 5, sample paths. Here, Loss represets the total umber of Jump Misclassificatio Errors, while T RV, Loss, S T RV, ad S Loss deote the correspodig sample meas ad stadard deviatios, respectively. B,k deotes the estimate of the optimal threshold produced by the iterative algorithm, P ow deotes the power thresold with parameters α = 1 ad ω =.495. BF deotes the Boferroi threshold ad Or(β) deotes the oracle threshold with parameter β. A summary of the simulatio results are give i Table 1. The umbers give i bold represet the best estimates for both the mea TRV ad Loss statistics i each simulatio study for each of the two categories of estimators, amely the o-oracle ad oracle varieties. I all cases the TPV estimators correspodig to the optimal threshold ad oracle threshold with β = 4.5 rak first i their respective classes. Furthermore, both of these thresholds give rise to very comparable results across all three simulatio studies. For the Merto model study, all six estimators are able to recover the diffusio volatility parameter withi decimal ad exhibit early idetical variability. O the other had, i the first Kou model study the optimal threshold ad Boferroi based TRV estimators give the least biased estimates withi the o-oracle class while the power threshold estimator exhibits high egative bias. This is due to the power threshold s excessive screeig of icremets which do ot cotai jumps, as idicated by the large average Loss reported i the table. I the secod Kou model study, the disparities betwee the o-oracle estimators become eve more apparet. I this circumstace it is clear that although all of the estimators commit, o average, a large umber of jump idetificatio errors, the optimal threshold based estimator commits fewer tha either the power or Boferroi threshold estimators. The power threshold s low estimate for σ idicates that it commits a large umber of Type I errors, while o the other had the Boferroi threshold s high estimate for σ suggests it commits a large umber of Type II errors. By keepig tighter cotrol of both type of errors the optimal threshold estimator is able to achieve the sharpest estimate of σ amogst these three methods. I summary, the simulatio results demostrate the robust adaptive ature of the optimal thresholdig based estimatio procedure. The fact that it gives very comparable results to the oracle estimators, which caot be used i practice, is quite promisig. The self-adaptive ature of this ew estimator is appealig sice ulike may other threshold estimators foud i the literature it does ot require the ad hoc selectio of tuig parameters. I essece, the optimal threshold based estimator is self-calibratig. 5. Additive processes with absolutely cotiuous characteristics We ow preset a algorithm which is desiged to approximate, to first-order, the optimal thresholds for the Itô semimartigale model as stated i (4.1). The idea for this algorithm is similar to that give for Lévy jump diffusios, however, sice the characteristics of the uderlyig model are time varyig, the previous algorithm must be modified accordigly. For this we propose a atural kerel based iterative estimatio method. We assume we are give a symmetric o-egative kerel K, such that K(x)dx = 1, xk(x)dx = ad x K(x)dx <, alog with a badwidth parameter θ >, ad averagig widow size l N. The idea is to take 1

13 a local average of the squared thresholded icremets ad by iteratio, successively refie estimates for both the spot volatility ad optimal threshold, simultaeously. Below we use the otatio K θ to deote the ormalized kerel weights. This is the proposed algorithm: Optimal Threshold Spot Volatility Estimatio Algorithm: For each i {1,,..., m}, set σ () := l j= l while There exists i {1,,..., m} such that σ (k 1) σ (k+1) 1 ] 1/ i+j X K θ (t i t i+j ) ad h B() := 3 σ () h i l(1/h i ) i+j l 1 j= l h i+j i+j X 1 (k) i+jx B > σ (k) do t i+j,h i+j ] K θ (t i t i+j ) ad ed while { } Let km := if k 1 : σ (k+1) = σ (k) ; for all i = 1,,..., m correspodig B (k m ) as a estimate for Bt i,h i. ad take σ (k m ) ] 1/ (k+1) B := 3 σ (k+1) h i l(1/h i ) Notice that, similar to the costat volatility based algorithm, for each i = 1,,..., m, we have ad sice each σ (k) σ (k) σ (k 1) σ (1) σ (), as the fial estimate for σ ad the { l j= l h 1 i+j i+j X K θ (t i t i+j )δ i+j : δ i+j {, 1}}, it is guarateed that the algorithm termiates i fiite time ad the elemets of the collectio ( σ (k m ) t 1,h 1, σ (k m ) t,h,..., σ (k m ) t m,h m ) are well defied. I order to umerically assess the performace of the optimal threshold spot volatility estimatio algorithm, we preset results based o a simulatio of the model give i (4.1) where the parameter fuctios ad jump distributio are take to be of the form: γ(t) =.1t σ(t) = 4.5t si(πe t ) +. λ(t) = 5(e 3t 1) ζ i.i.d. = D N (µ =.5, δ =.5). For this experimet we used a stadard Euler discretizatio scheme to geerate the icremets of the cotiuous compoet ad a thiig algorithm (cf. Ross ()) to simulate the icremets of the o-homogeeous compoud Poisso process jump compoet over the time horizo, 1]. Each sample path was discretized over a uiform samplig grid at the 5-mi frequecy (i.e., h i 1/( )). I order to implemet the iterative estimatio algorithm we employed two types of kerel: (i) the quadratic kerel K(x) := (3/4)(1 x )1 x 1] ad (ii) the uiform kerel K(x) = (1/)1 { x 1}. The averagig widow parameter ad badwidth were set to l = 7 78 ad h = l h i, respectively. This choice of widow size ad badwidth correspod to a 7-day kerel weighted thresholded movig average of the realized volatility. Although we will omit the details here, it is importat to metio that a boudary kerel type adjustmet was used for those observatios ear the begiig ad ed of time horizo uder cosideratio. A illustratio of the adaptive ature ad iheret variability of these estimators is give i Figure 1. From Figure 1 pael (A), it is clear that the first pass made by each estimator is quite rough ad sigificatly off the mark from the actual smooth spot volatility. This is obviously due to the icreasig umber of observed jumps across the time horizo ad the fact that the iitial estimatio passes do ot threshold sample icremets, but merely serve to obtai crude estimates of the actual spot volatility. However, durig the secod ad subsequet estimatio passes each type of estimator uses previous estimates to successively refie the spot volatility ad optimal threshold approximatios. After oly oe iteratio the estimates sigificatly improve, as show i pael (B). From the figure it is clear that, by desig, each estimator produces a decreasig sequece of upperbouds for the true spot volatility. Each iteratio of the algorithm successively refies previous estimates of the spot volatility, i tur workig to properly idetify those icremets cotaiig jumps. Iterestigly, i this experimet, the algorithms termiate after 4 iteratios, providig satisfactory fial estimates for the spot volatility as see i pael (C). Lastly, i order to depict the path-to-path variability associated with this estimatio procedure, we geerate 5 sample paths ad plot the termial spot volatility estimates agaist the true spot volatility via a scatter diagram, as see i pael (D). As poited out to us by oe of the editors, the performace ca be sped up if the the first pass is doe usig a TPV estimator with a reasoable threshold. 13

14 (A) Iitial Estimates (B) Itermediate Estimates.6 Actual Spot Volatility Est. Spot Vol. (Uiform) Est. Spot Vol. (Quad) Sample Icremets Actual Spot Volatility Est. Spot Vol. (Uiform) Est. Spot Vol. (Quad) Sample Icremets Time Horizo Time Horizo (C) Termial Estimates (D) Estimatio Variability Actual Spot Volatility Est. Spot Vol. (Uiform) Est. Spot Vol. (Quad) Sample Icremets 1. Actual Spot Volatility Time Horizo Time Horizo Figure 1: Estimatio of Spot Volatility usig Adaptive Kerel Weighted Realized Volatility. (A) The iitial estimates. (B) Itermediate estimates. (C) The termial estimates. (D) Estimatio variability, based o 5 geerated sample paths, for the Quadratic Kerel based estimator. 14

15 6 Coclusios We have obtaied explicit ifill asymptotic decompositios of the three statistical measures of bias, variace, ad mea-squared error for the class of thresholded realized variatio ad jump compoet estimators uder a Lévy jump diffusio model. I particular, our aalysis reveled simple ecessary ad sufficiet coditios o the threshold sequece for achievig mea-squared cosistecy, ad ucovered a itimate coectio betwee the performace of these estimators ad their iheret jump detectio capabilities. Motivated by the previously metioed coectio, we proposed a ovel approach for threshold selectio based o miimizig the expected total umber of jump misclassificatios. We showed that the proposed optimizatio problem is well-posed ad gives rise to a uique optimal thresholdig sequece, for which we provide a asymptotic characterizatio up to first-degree. We also exted the otio of optimal thresholdig to the class of Itô semimartigales with determiistic local characteristics. I light of these ew results, we propose two iovative algorithms to estimate the leadig order term of the optimal thresholds based o a type of fixed poit argumet. Based o extesive Mote Carlo experimetatio, the ew procedures are show to produce accurate ad stable estimates of the uderlyig spot volatility, ad exhibit iheret robustess agaist a wide rage of sigal-to-oise scearios. Ackowledgemets José E. Figueroa-López s research has bee partially supported by the NSF grat DMS Jeffrey Nise has bee supported by a Purdue Research Foudatio (PRF) Research Grat. We also wish to express our gratitude to a aoymous referee ad the editors, especially Professor Jacod, for their costructive suggestios, which sigificatly cotributed to improve the quality of the mauscript. A Proofs of the Mai Results Before we prove the mai results of Sectio 4.1, we shall give some supportig Lemmas. Lemma A.1. Let W := max 1 i t i W be the maximal icremet of the Browia motio over the samplig grid. The, for every p >, ( ) p/ lim E (W ) p ] = 1 (A.1) l(t) Proof of Lemma A.1. The estimate (A.1) is a direct cosequece of the followig asymptotic result for i.i.d. stadard Normal variables {Z i }, obtaied i Fischer ad Nappo (1): ( ) p ] lim ( l()) p/ E max Z i = 1. 1 i Lemma A.. Give ay threshold sequece B = (B ), there exists a positive sequece C (B), such that, the mea-squared error of the correspodig TRV estimator ca be bouded from above as follows: MSET RV ; B] t C (B)Loss () (B) + R (B), (A.) where ] 6tσ 4 R (B) +3B6 t λ C(f), ( ). (A.3) Proof of Lemma A.. Let DB] t := T RV (X)B] t tσ, which from (3.8) ca be decomposed as the sum of three terms that we deote T (i), i = 1,, 3. Note that t t T () B 1 i X B, i N ] ad T (3) σ (W) 1 i X >B, i N=], 15