Statistics and high frequency data

Transcription

1 Statistics ad high frequecy data Jea Jacod Itroductio This short course is devoted to a few statistical problems related to the observatio of a give process o a fixed time iterval, whe the observatios occur at regularly spaced discrete times. This kid of observatios may occur i may differet cotexts, but they are particularly relevat i fiace: we do have ow huge amouts of data o the prices of various assets, exchage rates, ad so o, typically tick data which are recorded at every trasactio time. So we are maily cocered with the problems which arise i this cotext, ad the cocrete applicatios we will give are all pertaiig to fiace. I some sese they are ot stadard statistical problems, for which we wat to estimate some ukow parameter. We are rather cocered with the estimatio of some radom quatities. This meas that we would like to have procedures that are as model-free as possible, ad also that they are i some sese more aki to oparametric statistics. Let us describe the geeral settig i some more details. We have a uderlyig process X = X t t, which may be multi-dimesioal its compoets are the deoted by X, X 2,. This process is defied o some probability space Ω, F,. We observe this process at discrete times, equally spaced, over some fixed fiite iterval [, T ], ad we are cocered with asymptotic properties as the time lag, deoted by, goes to. I practice, this meas that we are i the cotext of high frequecy data. The objects of iterest are various quatities related to the particular outcome ω which is partially observed. The mai object is the volatility, but other quatities or features are also of much iterest for modelig purposes, for example whether the observed path has jumps ad, whe this is the case, whether several compoets may jump at the same times or ot. All these quatities are related i some way to the probabilistic model which is assumed for X: we do ideed eed some model assumptio, otherwise othig ca be said. I fact, ay give set of observed values X, X,, X i,, with fixed, is of course compatible with may differet models for the cotiuous time process X: for example we ca suppose that X is piecewise costat betwee the observatio times, or that it Istitut de mathématiques de Jussieu, Uiversité ierre et Marie Curie aris-6 ad CNRS, UMR 7586, 4 place Jussieu, aris, Frace

2 is piecewise liear betwee these times. Of course either oe of these two models is i geeral compatible with the observatios if we modify the frequecy of the observatios. So i the sequel we will always assume that X is a Itô semimartigale, that is a semimartigale whose characteristics are absolutely cotiuous with respect to Lebesgue measure. This is compatible with virtually all semimartigale models used for modelig quatities like asset prices or log-prices, although it rules out some o-semimartigale models sometimes used i this cotext, like the fractioal Browia motio. Before statig more precisely the questios which we will cosider, ad i order to be able to formulate them i precise terms, we recall the structure of Itô semimartigales. We refer to [3], Chapter I, for more details. Semimartigales: We start with a basic filtered probability space Ω, F, F t t,, the family of sub-σ-fields F t of F beig icreasig ad right-cotiuous i t. A semimartigale is simply the sum of a local martigale o this space, plus a adapted process of fiite variatio meaig, its paths are right-cotiuous, with fiite variatio o ay fiite iterval. I the multidimesioal case it meas that each compoet is a real-valued semimartigale. Ay multidimesioal semimartigale ca be writte as t X t = X + B t + Xt c + κxµ νds, dx + R d I this formula we use the followig otatio: t R d κ xµds, dx.. - µ is the jump measure of X: if we deote by X t = X t X t the size of the jump of X at time t recall that X is right-cotiuous with left limits, the the set {t : X t ω } is at most coutable for each ω, ad µ is the radom measure o, R d defied by µω; dt, dx = ε s, Xs ωdt, dx, ε a = the Dirac measure sittig at a. s>: X s ω - ν is the compesator or, predictable compesator of µ. This is the uique radom measure o, R d such that, for ay Borel subset A of R d at a positive distace of, the process ν, t] A is predictable ad the differece µ, t] A ν, t] A is a local martigale. - κ is a trucatio fuctio, that is a fuctio: R d R d, bouded with compact support, such that κx = x for all x i a eighborhood of. This fuctio is fixed throughout, ad we choose it to be cotiuous for coveiece. - κ is the fuctio κ x = x κx. - B is a predictable process of fiite variatio, with B =. - X c is a cotiuous local martigale with X c =, called the cotiuous martigale part of X. With this otatio, the decompositio. is uique up to ull sets, but the process B depeds o the choice of the trucatio fuctio κ. The cotiuous martigale part 2

3 does ot deped o the choice of κ. Note that the first itegral i. is a stochastic itegral i geeral, whereas the secod oe is a pathwise itegral i fact for ay t is is simply the fiite sum s t κ X s. Of course. should be read compoetwise i the multidimesioal settig. I the sequel we use the shorthad otatio to deote the possibly stochastic itegral w.r.t. a radom measure, ad also for the possibly stochastic itegral of a process w.r.t. a semimartigale. For example,. may be writte more shortly as X = X + B + X c + κ µ ν + κ µ..2 The * symbol will also be used, as a superscript, to deote the traspose of a vector or matrix o cofusio may arise. Aother process is of great iterest, amely the quadratic variatio of the cotiuous martigale part X c, which is the followig R d R d -valued process: C = X c, X c, that is, compoetwise, C ij = X i,c, X j,c..3 This is a cotiuous adapted process with C =, which further is icreasig i the set M + d of symmetric oegative matrices, that is C t C s belogs to M + d for all t > s. The triple B, C, ν is called the triple of characteristics of X, this ame comig from the fact that i good cases it completely determies the law of X. The fudametal example of semimartigales is the case of Lévy processes. We say that X is a Lévy process if it is adapted to the filtratio, with right-cotiuous ad leftlimited paths ad X =, ad such that X t+s X t is idepedet of F t ad has the same law as X s for all s, t. Such a process is always a semimartigale, ad its characteristics B, C, ν are of the form B t ω = bt, C t = ct, νω; dt, dx = dt F dx..4 Here b R d ad c M + d ad F is a measure o Rd which does ot charge ad itegrates the fuctio x x 2. The triple b, c, F is coected with the law of the variables X t by the formula for all u R d Ee i u,xt = exp t i u, b 2 u, cu + F dx e i u,x i u, κx,.5 called Lévy-Khitchie s formula. So we sometimes call b, c, F the characteristics of X as well, ad it is the Lévy-Khitchie characteristics of the law of X i the cotext of ifiitely divisible distributios. b is called the drift, c is the covariace matrix of the Gaussia part, ad F is called the Lévy measure. As see above, for a Lévy process the characteristics B, C, ν are determiistic, ad they do characterize the law of the process. Coversely, if the characteristics of a semimartigale X are determiistic oe ca show that X has idepedet icremets, ad if they are of the form.4 the X is a Lévy process. Itô semimartigales. By defiitio, a Itô semimartigale is a semimartigale whose characteristics B, C, ν are absolutely cotiuous with respect to Lebesgue measure, i 3

4 the followig sese: B t ω = t b s ωds, C t ω = t c s ωds, νω; dt, dx = dt F ω,t dx..6 here we ca always choose a versio of the processes b or c which is optioal, or eve predictable, ad likewise choose F i such a way that F t A is optioal, or eve predictable, for all Borel subsets A of R d. It turs out that Itô semimartigales have a ice represetatio i terms of a Wieer process ad a oisso radom measure, ad this represetatio will be very useful for us. Namely, it ca be writte as follows where for example κ δ µ t deotes the value at time t of the itegral process κ δ µ: X t = X + t b s ds + t σ s dw s + κδ µ ν t + κ δ µ t..7 I this formula W is a stadard d -dimesioal Wieer process ad µ is a oisso radom measure o, E with itesity measure νdt, dx = dt λdx, where λ is a σ-fiite ad ifiite measure without atom o a auxiliary measurable set E, E. Of course the process b t is the same i.6 ad i.7, ad σ = σ ij i d, j d is a R d R d -valued optioal or predictable, as oe wishes to process such that c = σσ, ad δ = δω, t, x is a predictable fuctio o Ω [, E that is, measurable with respect to E, where is the predictable σ-field of Ω [,. The coectio betwee δ above ad F i.6 is that F t,ω is the image of the measure λ by the map x δω, t, x, ad restricted to R d \{}. Remark. Oe should be a bit more precise i characterizig W ad µ: W is a F t -Wieer process, meaig it is F t adapted ad W t+s W t is idepedet of F t o top of beig Wieer, of course. Likewise, µ is a F t -oisso measure, meaig that µ, t] A is F t -measurable ad µt, t + s] A is idepedet of F t, for all A E. Remark.2 The origial space Ω, F, o which X is defied may be too small to accommodate a Wieer process ad a oisso measure, so we may have to elarge the space. Such a elargemet is always possible. Remark.3 Whe the matrix c t ω is of full rak for all ω, t ad d = d, the it has a uique square-root σ t ω, which further is ivertible. I this case we have W = σ X c. Otherwise, there are may ways of choosig σ such that σσ = c, hece may ways of choosig W ad its dimesio d which ca always be take such that d d. I a similar way, we have a lot of freedom for the choice of µ. I particular we ca choose at will the space E, E ad the measure λ, subject to the above coditios, ad for example we ca always take E = R with λ the Lebesgue measure, although i the d-dimesioal case it is somewhat more ituitive to take E = R d. Of course a Lévy process is a Itô semimartigale compare.2 ad.6. I this case the two represetatios.2 ad.7 coicide if we take E = R d ad λ = F the 4

5 Lévy measure ad µ = µ the jump measure of a Lévy process is a oisso measure ad δω, t, x = x, ad also if we recall that i this case the cotiuous martigale or Gaussia part of X is always of the form X c = σw, with σσ = c. The settig of Itô semimartigales ecompasses most processes used for modelig purposes, at least i mathematical fiace. For example, solutios of stochastic differetial equatios drive by a Wieer process, or a by a Lévy process, or by a Wieer process plus a oisso radom measure, are all Itô semimartigales. Such solutios are obtaied directly i the form.7, which of course implies that X is a Itô semimartigale. The volatility. I a fiacial cotext, the process c t is called the volatility sometimes it is σ t which is thus called. This is by far the most importat quatity which eeds to be estimated, ad there are may ways to do so. A very widely spread way of doig so cosists i usig the so-called implied volatility, ad it is performed by usig the observed curret prices of optios draw o the stock uder cosideratio, by somehow ivertig the Black-Scholes equatio or extesios of it. However, this way usually assumes a give type of models, for example that the stock prices is a diffusio process of a certai type, with ukow coefficiets. Amog the coefficiets there is the volatility, which further may be stochastic, meaig that it depeds o some radom iputs other tha the Wieer process which drives the price itself. But the it is of primary importace to have a soud model, ad this ca be checked oly by statistical meas. That is, we have to make a statistical aalysis, based o series of ecessarily discrete observatios of the prices. I other words, there is a large body of work, essetially i the ecoometrical literature, about the statistical estimatio of the volatility. This meas fidig good methods for estimatig the path t c t ω for t [, T ], o the basis of the observatio of X i ω for all i =,,, [T/ ]. I a sese this is very similar to the o-parametric estimatio of a fuctio ct, say i the -dimesioal case, whe oe observes the Gaussia process Y t = t cs dws here W is a stadard -dimesioal Wieer process at the time i, ad whe is small that is, we cosider the asymptotic. As is well kow, this is possible oly uder some regularity assumptios o the fuctio ct, whereas the itegrated value t csds ca be estimated as i parametric statistics, sice it is just a umber. O the other had, if we kow t csds for all t, the we also kow the fuctio ct, up to a Lebesgue-ull set, of course: it should be emphasized that if we modify c o such a ull set, we do ot chage the process Y itself; the same commet applies to the volatility process c t i.6. This is why we maily cosider, as i most of the literature, the problem of estimatig the itegrated volatility, which with our otatio is the process C t. Oe has to be aware of the fact that i the case of a geeral Itô semimartigale, this meas estimatig the radom umber or matrix C t ω, for the observed ω, although of course ω is ideed ot fully observed. 5

6 Let us cosider for simplicity the -dimesioal case, whe further X is cotiuous, that is X t = X + t b s ds + t σ s dw s,.8 ad σ t equivaletly, c t = σ 2 t is radom. It may be of the form σ t ω = σx t ω, it ca also be by itself the solutio of aother stochastic differetial equatio, drive by W ad perhaps aother Wieer process W, ad perhaps also some oisso measures if it has jumps eve though X itself does ot jump. By far, the simplest thig to do is to cosider the realized itegrated volatility, or approximate quadratic variatio, that is the process B2, t = [t/ ] i X 2, where i X = X i X i..9 The if.8 holds, well kow results o the quadratic variatio goig back to Itô i this case, we kow that B2, t C t. covergece i probability, ad this covergece is eve uiform i t over fiite itervals. Further, as we will see later, we have a rate of covergece amely / uder some appropriate assumptios. Now what happes whe X is discotiuous? We o loger have., but rather B2, t C t + X s 2. s t the right side above is always fiite, ad is the quadratic variatio of the semimartigale X, also deoted [X, X] t. Nevertheless we do wat to estimate C t : a good part of these otes is devoted to this problem. For example, we will show that both quatities B,, t = [t/ ] i X i+x, B2, ϖ, α t = [t/ ] i X 2 { i X α ϖ }.2 coverge i probability to 2 π C t ad C t respectively, ad as soo as ϖ, /2 ad α > for the secod oe. Iferece for jumps. Now, whe X is discotiuous, there is also a lot of iterest about jumps ad, to begi with, are the observatios compatible with a model without jumps, or should we use a model with jumps? More complex questios may be posed: for a 2-dimesioal process, do the jumps occur at the same times for the two compoets or ot? Is there ifiitely may small jumps? I this case, what is the cocetratio of the jumps ear? Here agai, the aalysis is based o the asymptotic behavior of quatities ivolvig sums of fuctios of the icremets i X of the observed process. So, before goig to the mai results i a geeral situatio, we cosider first two very simple cases: whe X = σw 6

7 for a costat σ >, ad whe X = σw +Y whe Y is a compoud oisso process. It is also of primary importace to determie which quatities ca be cosistetly estimated whe, ad which oes caot be. We begi with the latter questio. 2 What ca be estimated? Recall that our uderlyig process X is observed at discrete times,, 2,, up to some fixed time T. Obviously, we caot have cosistet estimators, as, for quatities which caot be retrieved whe we observe the whole path t X t ω for t [, T ], a situatio referred to below as the complete observatio scheme. We begi with two simple observatios: The drift b t ca ever be idetified i the complete observatio scheme, except i some very special cases, like whe X t = X + t b sds. 2 The quadratic variatio of the process is fully kow i the complete observatio scheme, up to time T of course. This implies i particular that the itegrated volatility C t is kow for all t T, hece also the process c t this is of course up to a -ull set for C t, ad a dω dt-ull set for c t ω. 3 The jumps are fully kow i the complete observatio scheme, up to time T agai. Now, the jumps are ot so iterestig by themselves. More importat is the law of the jumps i some sese. For Lévy processes the law of jumps is i fact determied by the Lévy measure. I a similar way, for a semimartigale the law of jumps ca be cosidered as kow if we kow the measures F t,ω, sice these measures specify the jump coefficiet δ i.7. Warig: this specificatio is i a weak sese, exactly as c specifies σ; we may have several square-root of c, as well as several δ such that F t is the image of λ, but all choices of σ t ad δ which are compatible with a give c t ad F t give rise to equatios that have exactly the same weak solutios. Cosider Lévy processes first. Basically, the restrictio of F to the complemet of ay eighborhood of, after ormalizatio, is the law of the jumps of X lyig outside this eighborhood. Hece to cosistetly estimate F we eed potetially ifiitely may jumps far from, ad this possible oly if T. I our situatio with T fixed there is o way of cosistetly estimatig F. We ca still say somethig i the Lévy case: for the complete observatio scheme, if there is a jump the F is ot the zero measure; if we have ifiitely may jumps i [, T ] the F is a ifiite measure; i this case, we ca also determie for which r > the sum s T X s r is fiite, ad this is also the set of r s such that { x < } x r F dx <. The same statemets also hold for more geeral semimartigales: we ca decide for which r s the sum s T X s r is fiite, ad also if we have zero, or fiitely may, or ifiitely may jumps. Those are characteristics of the model which are of much iterest for modellig purposes. 7

8 Hece we will be iterested, whe comig back to the actual discrete observatio scheme, i estimatig C t for t T, ad whether there are zero or fiitely may or ifiitely may jumps i [, T ]. 3 Some simple limit theorems for Wieer plus compoud oisso processes This sectio is about a very particular case: the uderlyig process is X = σx + Y for some σ >, ad Y a compoud oisso process idepedet of W. Ad i the first subsectio we eve cosider the most elemetary case of X = σw. I these two cases we state all limit theorems that are available about sums of a fuctio of the icremets. We do ot give the full proofs, but heuristic reasos for the results to be true. The reaso for devotig a special sectio to this simple case is to show the variety of results that ca be obtaied, whereas the full proofs ca be easily recostructed without aoyig techical details. Before gettig started, we itroduce some otatio, to be used also for a geeral d- dimesioal semimartigale X later o. Recall the icremets i X i.9. First for ay p > ad j d we set Bp, j, t = [t/ ] i X j p. 3. I the -dimesioal case this is writte simply Bp, t. Next if f is a fuctio o R d, the state space of X i geeral, we set V f, t = [t/ ] f i X, V f, t = [t/ ] f i X/. 3.2 The reaso for itroducig the ormalizatio / will be clear below. These fuctioals are related oe of the other by the trivial idetity V f, = V f, with f x = fx/. Moreover, with the otatio y R h p y = y p, x = x j R d h j px = x j p, 3.3 we also have Bp, j, = V h j p, = p/2 V h j p,. Fially if we eed to emphasize the depedecy o the process X, we write these fuctioals as BX; p, j, or V X; f, or V X; f,. 3. The Wieer case. Here we suppose that X = σw for some costat σ >, so d =. Amog all the previous fuctioals, the simplest oes to study are the fuctioals V f, with f a fixed fuctio o R. We eed f to be Borel, of course, ad ot too big, for example with polyomial growth, or eve with expoetial growth. I this case, the results are 8

9 straightforward cosequeces of the usual law of large umbers LNN ad cetral limit theorem CLT. Ideed, for ay the variables i X/ : i are i.i.d. with law N, σ 2. I the formulas below we write ρ σ for the law N, σ 2 ad also ρ σ g the itegral of a fuctio g with respect to it. Therefore, with f as above, the variables f i X/ whe i varies are i.i.d. with momets of all orders, ad their first ad secod momets equal ρ σ f ad ρ σ f 2 respectively. The the classical LLN ad CLT give us that V f, t tρ σ f V f, t tρ σ g L N, tρ σ f 2 ρ σ f We clearly see here why we have put the ormalizig factor / iside the fuctio f. The reader will observe that, cotrary to the usual LNN, we get covergece i probability but ot almost surely i the first part of 3.4. The reaso is as follows: let ζ i be a sequece of i.i.d. variables with the same law tha fx. The LLN implies that t [t/ ] [t/ ] ζ i coverges a.s. to tρ σ f. Sice V f, t has the same law as Z = Z we deduce the covergece i probability i 3.4 because, for a determiistic limit, covergece i probability ad covergece i law are equivalet. However the variables V f, t are coected oe with the others i a way we do ot really cotrol whe varies, so we caot coclude to V f; t tρ σ f a.s..9 gives us the covergece for ay time t, but we also have fuctioal covergece: First, recall that a sequece g of oegative icreasig fuctios o R + covergig poitwise to a cotiuous fuctio g also coverges locally uiformly; the, from the first part of.9 applied separately for the positive ad egative parts f + ad f of f ad usig a subsequece priciple for the covergece i probability, we obtai V f, t u.c.p. tρ σ f 3.5 u.c.p. where Zt Z t meas covergece i probability, locally uiformly i time : that is, sup s t Zs Z s for all t fiite. 2 Next, if istead of the -dimesioal CLT we use the fuctioal CLT, or Dosker s Theorem, we obtai V f, t tρ σ f t L = ρ σ f 2 ρ s f 2 W 3.6 where W is aother stadard Wieer process, ad = L stads for the covergece i law of processes for the Skorokhod topology. Here we see a ew Wieer process W appear. What is its coectio with the basic uderlyig Wieer process W? To study that, oe ca try to prove the joit covergece of the processes o the left side of 3.6 together with W or equivaletly X itself. This is a easy task: cosider the 2-dimesioal process Z whose first compoet is the left side of 3.6 ad secod compoet is X [t/ ] the discretized versio of 9

10 X, which coverges poitwise to X. The Z takes the form Zt = [t/ ] ζi, where the ζi are 2-dimesioal i.i.d. variables as i varies, with the same distributio as g X, g 2 X, where g x = fx ρ σ f ad g 2 x = x. The the 2-dimesioal versio of Dosker s Theorem gives us that V L f; t tρ σ f, X t = B, X 3.7 t ad the pair B, X is a 2-dimesioal correlated Wieer process, characterized by its variace-covariace at time, which is the followig matrix: ρσ f 2 ρ s f 2 ρ σ fg 2 ρ σ fg 2 σ ote that σ 2 = ρ σ g 2 2 ad also ρ σg 2 =, so the above matrix is semi-defiite positive. Equivaletly, we ca write B as B = ρ σ f 2 ρ s f 2 W with W a stadard Browia motio as i 3.7 which is correlated with W, the correlatio coefficiet beig ρ σ fg 2 /σ ρ σ f 2 ρ s f 2. Now we tur to the processes Bp,. Sice Bp, = p/2 V h p, this is just a particular case of 3.5 ad 3.7, which we reformulate below m p deotes the pth absolute momet of the ormal law N, : p/2 Bp, p/2 Bp, t tσ p m p, X t t u.c.p. tσ p m p, 3.9 } L = B, X, 3. with B a Wieer process uit variace σ 2p m 2p m 2 p, idepedet of X the idepedece comes from that fact that ρ σ g =, where gx = x x p. Fially for the fuctioals V f,, the importat thig is the behavior of f ear, sice the icremets i X are all goig to as. I fact, sup i [t/ ] i X poitwise, so whe the fuctio f vaishes o a eighborhood of, for all bigger tha some radom fiite umber N depedig also o t we have V f, s = s t. 3. For a geeral fuctio f we ca combie 3.9 with 3.: we easily obtai that 3.9 holds with V f, istead of Bp, as soo as fx x p as x, ad the same holds for 3. if we further have fx = x p o a eighborhood of. Of course these results do ot exhaust all possibilities for the covergece of V f;. For example o may prove the followig: fx = x p log x p/2 log/ V f, u.c.p. 2 tσp m p, 3.2 ad a CLT is also available i this situatio. Or, we could cosider fuctios f which behave like x p as x ad like x p as x, with p p. However, we essetially restrict our attetio to fuctios behavig like h p : for simplicity first, ad sice more geeral fuctios do ot really occur i the applicatios we have i mid, ad also because the extesio to processes X more geeral tha the Browia motio is ot easy for other fuctios.

11 3.2 The Wieer plus compoud oisso case. Our secod example is whe the uderlyig process X has the form X = σw + Y, where as before σ > ad W is a Browia motio, ad Y is a compoud oisso process idepedet of W. We will write X = σw. Recall that Y has the form Y t = p Φ p {Tp t}, 3.3 where the T p s are the successive arrival times of a oisso process, say with parameter they are fiite stoppig times, positive, strictly icreasig with p ad goig to, ad the Φ p s are i.i.d. variables, idepedet of the T p s, ad with some law G. Note that i 3.3 the sum, for ay give t, is actually a fiite sum. The processes V f,, which were particularly easy to study whe X was a Wieer process, are ot so simple to aalyze ow. This is easy to uderstad: let us fix t; at stage, we have i X = i X for all i [t/ ], except for those fiitely may i s correspodig to a iterval i, i ] cotaiig at least oe of the T p s. Furthermore, all those exceptioal itervals cotai exactly oe T p, as soo as is large eough depedig o ω, t. Therefore for large we have V f, t = V X ; f, t + A t, where A t = [t/ ] p {i <T p i } fφ p + i X f i X /. 3.4 The double sum i A t is ideed a fiite sum, with as may o-zero etries as the umber of T p s less tha [t/ ]. Therefore the behavior of V f, depeds i a essetial way o the behavior of f ear ifiity. There are essetially two possibilities: The fuctio f is bouded, or more geerally satisfies fx K + x p for some p < 2. The A t above is essetially smaller tha K q:t + Φ q t p r p/2 for some costat K, ad thus A t. So obviously the covergece 3.5 holds. If further p < we eve have A t. Therefore 3.7 holds. Observe that i this situatio, the presece of the jumps does ot modify the results that held for the Browia case; this will be the rule for more geeral processes X as well. 2 The fuctio f is equivalet to x p at ifiity, for some p > 2. The i 3.4 the leadig term becomes A t, which is approximately equal to p/2 s t X s p. So V f, t coverges i probability to the variable p/2 Bp t = s t X s p 3.5 we have just proved the covergece for ay give t, but it is also a fuctioal covergece, for the Skorokhod topology, i probability. Agai, these cases do ot exhaust the possible behaviors of f, ad further we have ot give a CLT i the secod situatio above. But, whe f is ot bouded it looks a

12 bit strage to impose a specific behavior at ifiity, ad without this there is simply o covergece result for V f, t, ot to speak about CLTs. Now we tur to the processes V f,. To begi with, we observe that, similar to 3.4, we have V f, t = V X, f, t + A t, where A t = [t/ ] p 3.6 {i <T p i } fφ p + i X f i X. The first - fudametal - differece with the cotiuous case is that 3. fails ow whe f vaishes o a eighborhood of. I this case, though, for each give t ad all bigger tha some umber depedig o ω, t, we have V X ; f, s = for all s t by 3., hece V f, s = [s/ ] {i <T p i }fφ p + i X, s t. 3.7 p The, as soo as f is cotiuous ad vaishes o a eighborhood of, we get V f, t Sk V f t := f X s. 3.8 s t Here Sk meas covergece for the Skorokhod topology, poitwise i ω the reaso for which we have covergece i the Skorokhod sese will be explaied later; what is clear at this poit is that we have the - poitwise i ω - covergece for all t such that X is cotiuous at t; we also have for each t a almost sure covergece above. Next, we cosider the case where f is still cotiuous ad, say, coicides with h p for some p > o a eighborhood of. For ay give ε > we ca write f = f ε + f ε with f ε ad f ε cotiuous, ad f ε x = h p x if x ε/2 ad f ε x = if x ε ad f ε h p everywhere. Sice f ε vaishes aroud, we have V f ε, t V f ε t by 3.8, ad V f ε t coverges to V f t as ε. O the other had the process A associated with f ε by 3.6 is the sum of summads smaller tha 2ε p, the umber of them beig bouded for each ω, t by a umber idepedet of ε: hece A t is egligible ad V f ε, ad V X ; f ε, behave essetially i the same way. This meas heuristically that, with the symbol meaig approximately equal to, we have V f ε, t V f t, V f ε, t p/2 t σ p m p. 3.9 Addig these two expressios, we get V f, t V f, t Sk V f t if p > 2 Sk V f t + tσ 2 if p = 2 r/2 V f, t u.c.p. tσ p/2 m p if p < This type of LLN, which shows a variety of behaviors accordig to how f behaves ear, will be foud for much more geeral processes later, i almost exactly the same terms. 2

13 Now we tur to the CLT. Here agai we sigle out first the case where f vaishes i a eighborhood of. We eed to fid out what happes to the differece V f, V f. It is easier to evaluate is the differece V f, t V f [t/ ], sice by 3.7 we have V f, s V f [s/ ] = [s/ ] p {i <T p i } fφ p + i X fφ p 3.2 for all s t, as soo as is large eough. rovided f is C, with derivative f, the pth summad above is approximately f Φ p i X. Now the ormalized icremet i X /, for the value of i such that i < T p i, has the law N, σ 2 because X ad Y are idepedet, ad it is asymptotically idepedet of the process X more details are to be foud later. Thus if U p p deotes a sequece of i.i.d. N, variables, idepedet of X, it is ot difficult to see that V f, t V f [t/ ] L = Bf t := p:t p t f Φ p σu p, 3.22 ad i fact, this covergece i law for the Skorokhod topology is eve stable deoted L s =, a stroger property tha the mere covergece i law, which will be defied later oly but evertheless is used i the statemets below. Whe ow f coicide with h p for some p > o a eighborhood of ad is still C outside, exactly as for 3.9 we obtai heuristically that V f ε, t V f [t/ ] + U t, V f ε, t p/2 tσ p m p + p/2 /2 U t, where U ad U coverge stably i law to the right side of 3.22 ad to the process B of 3., respectively. We the have two coflictig rates, ad we ca ideed prove that, with Bf as i 3.22 ad B as i 3. thus depedig o r: V f, t V f [t/ ] L s = Bf t if p > 3 V f, t V f [t/ ] p/2 p/2 V f, t V f [t/ ] L s = tσ 3 m 3 + Bf t if p = 3 u.c.p. tσ p m p if 2 < p < 3 V f, t V f [t/ ] tσ 2 L s = B t + Bf t if p = 2 V f; t tσ p Sk m p V f t if < p < 2 p/2 V f, t tσm L s = V f t + B t if p = p/2 V f, t tσ p L s m p = B t if p < Hece we obtai a geuie CLT, relative to the LLN 3.2, i the cases p > 3, p = 2 ad p <. Whe p = 3 ad p = we still have a CLT, with a bias. Whe 2 < p < 3 or < p < 2 we have a secod order LNN, ad the associated geuie CLTs ru as follows: p/2 /2 V f, t V f [t/ ] p/2 tσ p L s m p = Bf t if 2 < p < 3 V f, t V f [t/ ] p/2 tσ p L s m p = B t if < p <

14 We see that these results exhibit agai a large variety of behavior. This will be ecoutered also for more geeral uderlyig processes X, with of course more complicated statemets ad proofs i the preset situatio we have ot really give the complete proof, of course, but it is relatively easy alog the lies outlied above. However, i the geeral situatio we will ot give such a complete picture, which is useless for practical applicatios. Oly 3.2 ad the cases r > 2 i 3.23 will be give. 4 Auxiliary limit theorems The aims of this sectio are twofold: first we defie the stable covergece i law, already metioed i the previous sectio. Secod, we recall a umber of limit theorems for partial sums of triagular arrays of radom variables. Stable covergece i law. This otio has bee itroduced by Réyi i [22], for the very same reasos as we eed it here. We refer to [4] for a very simple expositio ad to [3] for more details. It ofte happes that a sequece of statistics Z coverges i law to a limit Z which has, say, a mixed cetered ormal distributio: that is, Z = ΣU where U is a N, variable ad Σ is a positive variable idepedet of U. This poses o problem other tha computatioal whe the law of Σ is kow. However, i may istaces the law of Σ is ukow, but we ca fid a sequece of statistics Σ such that the pair Z, Σ coverges i law to Z, Σ; so although the law of the pair Z, Σ is ukow, the variable Z /Σ coverges i law to N, ad we ca base estimatio or testig procedures o this ew statistics Z /Σ. This is where the stable covergece i law comes ito play. The formal defiitio is a bit ivolved. It applies to a sequece of radom variables Z, all defied o the same probability space Ω, F,, ad takig their values i the same state space E, E, assumed to be olish = metric complete ad separable. We say that Z stably coverges i law if there is a probability measure η o the product Ω E, F E, such that ηa E = A for all A F ad EY fz Y ωfxηdω, dx 4. for all bouded cotiuous fuctios f o E ad bouded radom variables Y o Ω, F. This is a abstract defiitio, similar to the defiitio of the covergece i law which says that EfZ fxρdx for some probability measure ρ. Now for the L covergece i law we usually wat a limit, that is we say Z Z, ad the variable Z is ay variable with law ρ, of course. I a similar way it is coveiet to realize the limit Z for the stable covergece i law. We ca always realize Z i the followig way: take Ω = Ω E ad F = F E ad edow Ω, F with the probability η, ad put Zω, x = x. But, as for the simple covergece i law, we ca also cosider other extesios of Ω, F, : that is, we have a probability space Ω, F,, where Ω = Ω Ω ad F = F F for some auxiliary measurable space Ω, F ad is a probability measure o Ω, F whose first margial 4

15 is, ad we also have a radom variable Z o this extesio. The i this settig, 4. is equivalet to sayig with Ẽ deotig the expectatio w.r.t. EY fz ẼY fz 4.2 for all f ad Y as above, as soo as A {Z B} = ηa B for all A F ad B E. We the say that Z coverges stably to Z, ad this covergece is deoted by. L s Clearly, whe η is give, the property A {Z B} = ηa B for all A F ad B E simply amouts to specifyig the law of Z, coditioally o the σ-field F. L s Therefore, sayig Z Z amouts to sayig that we have the stable covergece i law towards a variable Z, defied o ay extesio Ω, F, of Ω, F,, ad with a specified coditioal law kowig F. Obviously, the stable covergece i law implies the covergece i law. But it implies L s much more, ad i particular the followig crucial result: if Z Z ad if Y ad Y are variables defied o Ω, F, ad with values i the same olish space F, the Y Y, Z L s Y, Z. 4.3 Y O the other had, there are criteria for stable covergece i law of a give sequece Z. The σ-field geerated by all Z is ecessarily separable, that is geerated by a coutable algebra, say G. The if for ay fiite family A p : p q i G, the sequece Z, Ap p q of E R q -valued variables coverges i law as, the ecessarily Z coverges stably i law. 2 Covergece of triagular arrays. Our aim is to prove the covergece of fuctioals like i 3. ad 3.2, which appear i a atural way as partial sums of triagular arrays. We really eed the covergece for the termial time T, but i most cases the available covergece criteria also give the covergece as processes, for the Skorokhod topology. So ow we provide a set of coditios implyig the covergece of partial sums of triagular arrays, all results beig i [3]. We are ot lookig for the most geeral situatio here, ad we restrict our attetio to the case where the filtered probability space Ω, F, F t t, is fixed. For each we have a sequece of R d -valued variables ζi : i, the compoets beig deoted by ζ,j i for j =,, d. The key assumptio is that for all, i the variable ζi is F i -measurable, ad this assumptio is i force i the remaider of this sectio. Coditioal expectatios w.r.t. F i will play a crucial role, ad to simplify otatio we write it E i istead of E. F i, ad likewise i is the coditioal probability. Lemma 4. If we have [t/ ] E i ζ i t >, 4.4 5

16 the [t/ ] ζ i u.c.p.. The same coclusio holds uder the followig two coditios: I particular whe ζi to imply [t/ ] ζi [t/ ] [t/ ] E i ζ i E i ζ i 2 u.c.p., 4.5 t >. 4.6 is a martigale differece, that is E i ζ i =, the 4.6 is eough u.c.p.. Lemma 4.2 If we have [t/ ] E i ζ i u.c.p. A t 4.7 for some cotiuous adapted R d -valued process of fiite variatio A, ad if further 4.6 holds, the we have [t/ ] ζi u.c.p. A t. Lemma 4.3 If we have 4.7 for some determiistic cotiuous R d -valued fuctio of fiite variatio A, ad also the followig two coditios: [t/ ] E i ζ,j i ζ,k i E i ζ,j i E i ζ,k i C jk t t >, j, k =,, d, 4.8 [t/ ] E i ζ i 4 t >, 4.9 where C = C jk is a determiistic fuctio, cotiuous ad icreasig i M + d, the the processes [t/ ] ζi coverge i law to A + B, where B is a cotiuous cetered Gaussia R d -valued process with idepedet icremets with EB j t Bk t = C jk 4.9 is a coditioal Lideberg coditio, whose aims is to esure that the limitig process is cotiuous; other, weaker, coditios of the same type are available, but ot eeded here. The coditios give above completely characterize, of course, the law of the process B. Equivaletly we could say that B is a Gaussia martigale relative to the filtratio it geerates, startig from, ad with quadratic variatio process C. t. 3 Stable covergece of triagular arrays. The reader will have observed that the coditios 4.7 ad 4.8 i Lemma 4.3 are very restrictive, because the limits are oradom. I the sequel, such a situatio rarely occurs, ad typically these coditios are satisfied with A ad C radom. But the we eed a additioal coditio, uder which it turs out that the covergece holds ot oly i law, but eve stably i law. Note that the stable covergece i law has bee defied for variables takig values i a olish space, so it also applies to right-cotiuous ad left limited d-dimesioal 6

17 processes: such a process ca be viewed as a variable takig its values i the Skorokhod space DR d of all fuctios from R + ito R d which are right-cotiuous with left limits, provided we edow this space with the Skorokhod topology which makes it a olish space. See [] or Chapter VI of [3] for details o this topology. I fact, i Lemma 4.3 the covergece i law is also relative to this Skorokhod topology. The stable covergece i law for processes is deoted as = L s below. I the previous results the fact that all variables were defied o the same space Ω, F, F t t, ad the ζ i s were F i -measurable was essetially irrelevat. This is o loger the case for the ext result, for which this settig is fudametal. Below we sigle out, amog all martigales o Ω, F, F t t,, a possibly multidimesioal Wieer process W. The followig lemma holds for ay choice of W, ad eve with o W at all i which case a martigale orthogoal to W below meas ay martigale but we will use it maily with the process W showig i.7. The followig is a particular case of Theorem IX.7.28 of [3]. Lemma 4.4 Assume 4.7 for some cotiuous adapted R d -valued process of fiite variatio A, ad 4.8 with some cotiuous adapted process C = C jk with values i M + d ad icreasig i this set, ad also 4.9. Assume also [t/ ] E i ζ i i N t > 4. wheever N is oe of the compoets of W or is a bouded martigale orthogoal to W. The the processes [t/ ] ζi coverge stably i law to A + B, where B is a cotiuous process defied o a extesio Ω, F, of the space Ω, F, ad which, coditioally o the σ-field F, is a cetered Gaussia R d -valued process with idepedet icremets satisfyig ẼBj t Bk t F = C jk t. The coditios stated above completely specify the coditioal law of B, kowig F, so we are exactly i the settig explaied i above ad the stable covergece i law is well defied. However oe ca say eve more: lettig F t be the smallest filtratio o Ω which make B adapted ad which cotais F t that is, A Ω F t wheever A F t, the B is a cotiuous local martigale o Ω, F, F t t, which is orthogoal i the martigale sese to ay martigale o the space Ω, F, F t t,, ad whose quadratic variatio process is C. Of course, o the exteded space B is o loger Gaussia. The coditio 4. could be substituted with weaker oes. For example if it holds whe N is orthogoal to W, whereas [t/ ] E i ζ i i W j coverges i probability to a cotiuous process for all idices j, we still have the stable covergece i law of [t/ ] ζi, but the limit has the form A + B + M, where the process M is a stochastic itegral with respect to W. Se [3] for more details. 7

18 5 A first LNN Law of Large Numbers At this stage we start givig the basic limit theorems which are used later for statistical applicatios. erhaps givig first all limit theorems i a purely probabilistic settig is ot the most pedagogical way of proceedig, but it is the most ecoomical i terms of space... We are i fact goig to provide a versio of the results of Sectio 3, ad other coected results, whe the basic process X is a Itô semimartigale. There are two kids of results: first some LNNs similar to 3.5, 3.9, 3.8 or 3.2; secod, some cetral limit theorems CLT similar to 3. or We will ot give a complete picture, ad rather restrict ourselves to those results which are used i the statistical applicatios. Warig: Below, ad i all these otes, the proofs are ofte sketchy ad sometimes abset; for the full proofs, which are sometimes a bit complicated, we refer essetially to [5] which is restricted to the -dimesioal case for X, but the multidimesioal extesio is straightforward. I this sectio, we provide some geeral results, valid for ay d-dimesioal semimartigale X = X j j d, ot ecessarily Itô. We also use the otatio 3. ad 3.2. We start by recallig the fudametal result about quadratic variatio, which says that for ay idices j, k, ad as recall : [t/ ] i X j i X k Sk [X j, X k ] t = C jk t + s t X j s X k s. 5. This is the covergece i probability, for the Skorokhod topology, ad we eve have the joit covergece for the Skorokhod topology for the d 2 -dimesioal processes, whe j, k d. Whe further X has o fixed times of discotiuity, for example whe it is a Itô semimartigale, we also have the covergece i probability for ay fixed t. Theorem 5. Let f be a cotiuous fuctio from R d ito R d. a If fx = o x 2 as x, the V f, t Sk f µ t = f X s. 5.2 s t b If f coicide o a eighborhood of with the fuctio gx = d j,k= γ jkx j x k here each γ jk is a vector i R d, the V f, t Sk d j,k= γ jk C jk t + f µ t. 5.3 Moreover both covergeces above also hold i probability for ay fixed t such that X t = = hece for all t whe X is a Itô semimartigale. roof. Suppose first that fx = whe x ε, for some ε >. Deote by S, S 2, the successive jump times of X correspodig to jumps of orm bigger tha 8

19 ε/2, so S p. Fix T >. For each ω Ω there are two itegers Q = QT, ω ad N = NT, ω such that S Q ω T < S Q+ ω ad for all N ad for ay iterval i, i ] i [, T ] the either there is o S q i this iterval ad i X ε, or there is exactly oe S q i it ad the we set αq = i X X S q. Sice fx = whe x ε we clearly have for all t T ad N: V f, t q: S q [t/ ] f X Sq The the cotiuity of f yields 5.2, because α q for all q. Q f X Sq + αq f X Sq. 2 We ow tur to the geeral case i a. For ay η > there is ε > such that we ca write f = f ε + f ε, where f ε is cotiuous ad vaishes for x ε, ad where f εx η x 2. By virtue of 5. ad the first part of the proof, we have q= V f ε, η [t/ ] i X 2 Sk η d j= [Xj, X j ], V f ε, Sk f ε µ Moreover, f ε µ u.c.p. f µ as ε follows easily from Lebesgue covergece theorem ad the property fx =o x 2 as x, because x 2 µ t < for all t. Sice η > ad ε > are arbitrarily small, we deduce 5.2 from V f, = V f ε, + V f ε,. 3 Now we prove b. Let f = f g, which vaishes o a eighborhood of. The if we combie 5. ad 5.2, plus a classical property of the Skorokhod covergece, we obtai that the pair V g,, V f, coverges for the 2d -dimesioal Skorokhod d topology, i probability to the pair j,k= γ jkc jk + g µ, f µ, ad by addig the two compoets we obtai 5.3. Fially the last claim comes from a classical property of the Skorokhod covergece, plus the fact that a Itô semimartigale has o fixed time of discotiuity. I particular, i the -dimesioal case we obtai recall 3.: p > 2 Bpr, Sk Bp t := X s p. 5.4 s t This result is due to Lépigle [8], who eve proved the almost sure covergece. completely fails whe r 2 except uder some special circumstaces. It 6 Some other LNNs 6. Hypotheses. So far we have geeralized 3.8 to ay semimartigale, uder appropriate coditios o f. If we wat to geeralize 3.5 or 3.4 we eed X to be a Itô semimartigale, plus the fact that the processes b t ad σ t ad the fuctio δ i.7 are locally bouded ad σ t is either right-cotiuous or left-cotiuous. 9

20 Whe it comes to the CLTs we eed eve more. So for a clearer expositio we gather all hypotheses eeded i the sequel, either for LNNs or CLTs, i a sigle assumptio. Assumptio H: The process X has the form.7, ad the volatility process σ t is also a Itô semimartigale of the form σ t = σ + t bs ds + t σ dw s + κ δ µ ν t + κ δ µ t. 6. I this formula, σ t a d d matrix is cosidered as a R dd -valued process; b t ω ad σ t ω are optioal processes, respectively dd ad dd 2 -dimesioal, ad δω, t, x id a dd - dimesioal predictable fuctio o Ω R + E; fially κ is a trucatio fuctio o R dd ad κ x = x κx. Moreover, we have: a The processes b t ω ad sup x E δω,t,x γx ad sup x E e δω,t,x eγx are locally bouded, where γ ad γ are o-radom oegative fuctios satisfyig E γx2 λdx < ad E γx2 λdx <. b All paths t b t ω, t σ t ω, t δω, t, x ad t δω, t, x are left-cotiuous with right limits. Recall that b t is locally bouded, for example, meas that there exists a icreasig sequece T of stoppig times, with T, ad such that each stopped process bt t = b t T is bouded by a costat depedig o, but ot o ω, t. Remark 6. For the LNNs, ad also for the CLTs i which there is a discotiuous limit below, we eed a weaker form of this assumptio, amely Assumptio H : this is as H, except that we do ot require σ t to be a Itô semimartigale but oly to be càdlàg the of course b, σ, δ are ot preset, ad b t is oly locally bouded. As a rule, we will state the results with the metio of this assumptio H, whe the full force of H is ot eeded. However, all proofs will be made assumig H, because it simplifies the expositio, ad because the most useful results eed it ayway. Apart from the regularity ad growth coditios a ad b, this assumptio amouts to sayig that both X ad the process σ i.7 are Itô semimartigales: sice the dimesio d is arbitrary large ad i particular may be bigger tha d, this accommodates the case where i.7 oly the first d compoets of W occur by takig σ ij t = whe j > d, whereas i 6. other compoets of W come i, thus allowig σ t to be drive by the same Wieer process tha X, plus a additioal multidimesioal process. I the same way, it is o restrictio to assume that both X ad σ are drive by the same oisso measure µ. So i fact this hypothesis accommodates virtually all models of stock prices or exchage rates or iterest rates, with stochastic volatility, icludig those with jumps, ad allows for correlatio betwee the volatility ad the asset price processes. For example if we cosider a q-dimesioal equatio dy t = fy t dz t 6.2 2

21 where Z is a multi-dimesioal Lévy process, ad f is a C 2 fuctio with at most liear growth, the if X cosists i a subset of the compoets of Y, it satisfies Assumptio H. The same holds for more geeral equatios drive by a Wieer process ad a oisso radom measure. 6.2 The results. Now we tur to the results. The first, ad most essetial, result is the followig; recall that we use the otatio ρ σ for the law N, σσ, ad ρ k σ deotes the k-fold tesor product. We also write ρ k σ f = fxρ k σ dx if f is a Borel fuctio o R d k. With such a fuctio f we also associate the followig processes V f, k, t = [t/ ] f i X/,, i+k X/. 6.3 Of course whe f is a fuctio o R d, the V f,, = V f,, as defied by 3.2. Theorem 6.2 Assume H or H oly, see Remark 6., ad let f be a cotiuous fuctio o R d k for some k, which satisfies fx,, x k K k + x j p 6.4 for some p ad K. If either X is cotiuous, or if p < 2, we have V f, k, t u.c.p. t ρ k σ u fdu. I particular, if X is cotiuous ad the fuctio f o R d satisfies fλx = λ p fx for all x R d ad λ, the p/2 V f, t u.c.p. j= t ρ σu fdu. 6.5 The last claim above may be viewed as a extesio of Theorem 5. to the case whe the limit i 5.2 vaishes. The cotiuity of f ca be somehow relaxed. The proof will be give later, after we state some other LLNs, of two kids, to be proved later also. Recallig that oe of our mai objective is to estimate the itegrated volatility C jk t, we observe that Theorem 5. does ot provide cosistet estimators for C t whe X is discotiuous. There are two ways to solve this problem, ad the first oe is as follows: whe X has jumps, 5. does ot give iformatio o C t because of the jumps, essetially the big oes. However a big jump gives rise to a big icremet i X. So a idea, followig Macii [9], [2], cosists i throwig away the big icremets. The cutoff level has to be chose carefully, so as to elimiate the jumps but keepig the icremets which are maily due to the cotiuous martigale part X c, ad those are of order. So we choose two umbers ϖ, /2 ad α >, ad for all idices j, k d we set V jk ϖ, α, t = [t/ ] i X j i X k { i X α ϖ }

22 More geerally oe ca cosider the trucated aalogue of V f, k, of 6.3. With ϖ ad α as above, ad if f is a fuctio o R d k, we set V ϖ, α; f, k, t = [t/ ] f i X/,, i+k X/ j=,,k { i+j X α ϖ }. 6.7 Theorem 6.3 Assume H or H oly, ad let f be a cotiuous fuctio o R d k for some k, which satisfies 6.4 for some p ad some K >. Let also ϖ, 2 ad α >. If either X is cotiuous, or X is discotiuous ad p 2 we have V u.c.p. ϖ, α; f, k, t t ρ k σ u fdu. I particular, V jk ϖ, α, u.c.p. C jk t. This result has o real iterest whe X is cotiuous. Whe X jumps, ad at the expese of a more complicated proof, oe could show that the result holds whe p 4, ad also whe p > 4 ad ϖ p 4 2p 2r 4 whe additioally we have γx r λdz < for some r [, 2 where γ is the fuctio occurrig i H. The slight improvemet o the coditio o p, upo the previous theorem, allows to easily estimate ot oly C t, but also the itegral t gc sds for ay polyomial g o the set of d d matrices. For example if we take fx,, x k = k x m j j x j j, 6.8 j= for arbitrary idices m j ad j i {,, d}, the we get V ϖ, α; f, k, t u.c.p. t k j= c m j j s ds. 6.9 The problem with this method is that we do ot really kow how to choose ϖ ad α a priori: empirical evidece from simulatio studies leads to choose ϖ to be very close to /2, like ϖ =.47 or.48, whereas α for estimatig C jj t, say, should be chose betwee 2 ad 5 times the average c jj recall c = σσ. So this requires a prelimiary rough estimate of the order of magitude of c jj : of course for fiacial data this order of magitude is usually pretty much well kow. Aother way, iitiated by Bardorff-Nielse ad Shephard see [6] ad [7] cosists i usig the so-called bipower, or more geerally multipower, variatios. This is i fact a particular case of the Theorem 6.2. Ideed, recallig that m r is the rth absolute momet of N,, we set for ay r,, r l, 2 with r + + r l = 2 hece l 2: V jk r,, r l, t = 4m r m rl [t/ ] l i+v X j + X k r v v= 22 l i+v X j X k r v 6. v=