ANOVA FOR DIFFUSIONS AND ITO PROCESSES. By Per Aslak Mykland and Lan Zhang

Transcription

1 Submed o he Annals of Sascs ANOVA FOR DIFFUSIONS AND ITO PROCESSES By Per Aslak Mykland and Lan Zhang The Unversy of Chcago, and Carnege Mellon Unversy and Unversy of Illnos a Chcago Io processes are he mos common form of connuous semmarngales, and nclude dffuson processes. The paper s concerned wh he nonparamerc regresson relaonshp beween wo such Io processes. We are neresed n he quadrac varaon (negraed volaly) of he resdual n hs regresson, over a un of me (such as a day). A man concepual fndng s ha hs quadrac varaon can be esmaed almos as f he resdual process were observed, he dfference beng ha here s also a bas whch s of he same asympoc order as he mxed normal error erm. The proposed mehodology, ANOVA for dffusons and Io processes, can be used o measure he sascal qualy of a paramerc model, and, nonparamercally, he appropraeness of a one-regressor model n general. On he oher hand, also helps quanfy and characerze he radng (hedgng) error n he case of fnancal applcaons. 1. Inroducon. We consder he regresson relaonshp beween wo sochasc processes Ξ and S, dξ = ρ ds + dz, T, (1.1) where Z s a resdual process. We suppose ha he processes S and Ξ are observed a dscree samplng pons = <... < k = T. Wh he adven of hgh frequency fnancal daa, hs ype of regresson has been a growng opc of neres n he leraure, cf. Secon 2.4. The processes Ξ and S wll be Io processes, whch s he mos commonly used ype of connuous semmarngale. Dffusons are a specal case Ths research was suppored n par by Naonal Scence Foundaon grans DMS (Mykland) and DMS (Mykland and Zhang). The auhors would lke o hank The Bendhem Cener for Fnance a Prnceon Unversy, where par of he research was carred ou. Keywords and phrases: ANOVA, connuous semmarngale, sascal uncerany, goodness of f, dscree samplng, paramerc and nonparamerc esmaon, small nerval asympocs, sable convergence, opon hedgng 1

2 2 P.A. MYKLAND AND L. ZHANG of Io processes. Defnons are made precse n Secon 2.1 below. The dfferenal n (1.1) s ha of an Io sochasc negral, as defned n Chaper I.4.d of Jacod and Shryaev (23) or Chaper 3.2 of Karazas and Shreve (1991). Our purpose s o assess nonparamercally wha s he smalles possble resdual sum of squares n hs regresson. Specfcally, for wo processes X and Y, denoe he quadrac covaraon beween X and Y on he nerval [, T ] by < X, Y > T = lm (X +1 X )(Y +1 Y ), (1.2) max +1 where = <... < k = T. (Ths objec exss by Defnon I.4.45 or Theorem I.4.47 n Jacod and Shryaev (23) (pp ), and smlar saemens n Karazas and Shreve (1991) and Proer (24)). In parcular, < Z, Z > T he quadrac varaon of Z s he sum of squares of he ncremens of he process Z under he dealzed condon of connuous observaon. We wsh o esmae, from medscree daa, mn < Z, Z > T, (1.3) ρ where he mnmum s over all adaped regresson processes ρ. An mporan movang applcaon for he sysem (1.1) s ha of sascal rsk managemen n fnancal markes. We suppose ha S and Ξ are he dscouned values of wo secures. A each me, a fnancal nsuon s shor one un of he secury represened by Ξ, and a he same me seeks o offse as much rsk as possble by holdng ρ uns of secury S. Z, as gven by (1.1), s hen he gan/loss, up o me, from followng hs rsk-neural procedure. In a complee (dealzed) fnancal marke, mn ρ < Z, Z > s zero; n a ncomplee marke, mn ρ < Z, Z > quanfes he unhedgeable par of he varaon n asse Ξ, when one adops he bes possble sraegy usng only asse S. And hs lower bound (1.3) s he arge ha a rsk managemen group wans o monor and conrol. The sascal mporance of (1.3) s hs. Once you know how o esmae (1.3), you know how o assess he goodness of f of any gven esmaon mehod for ρ. You also know more abou he appropraeness of a one regressor model of he form (1.1). We reurn o he goodness of f quesons n Secon 4. A model example of boh sascal and fnancal mporance s gven n Secon 2.2. To dscuss he problem of esmang (1.3), consder frs how one would fnd hs quany f he processes Ξ and S were connuously observed. Noe ha from (1.1),

3 ANOVA FOR DIFFUSIONS 3 one can wre < Z, Z > =< Ξ, Ξ > + ρ 2 u d < S, S > u 2 ρ u d < Ξ, S > u (1.4) (Jacod and Shryaev (23), I.4.54, p.55). Here, d < X, Y > s he dfferenal of he process < X, Y > wh respec o me. We shall ypcally assume ha < X, Y > s absoluely connuous as a funcon of me (for any realzaon). I s easy o see ha he soluon n ρ o he problem (1.3) s unquely gven by ρ = d < Ξ, S > = < Ξ, S > d < S, S > < S, S >, (1.5) where < Ξ, S > s he dervave of < Ξ, S > wh respec o me. Apar from s sascal sgnfcance, n fnancal erms ρ s he hedgng sraegy assocaed wh he mnmal marngale measure (see, for example, Föllmer and Sondermann (1986) and Schwezer (199)). The problem (1.3) hen connecs o an ANOVA, as follows. Le Z be he resdual n (1.1) for he opmal ρ, so ha he quany n (1.3) can be wren smply as < Z, Z > T. In analogy wh regular regresson, subsung (1.5) no (1.4), gves rse o an ANOVA decomposon of he form < Ξ, Ξ > }{{ T = } oal SS T ρ 2 ud < S, S > u + < Z, Z > T, (1.6) }{{}}{{} RSS SS explaned where SS s he abbrevaon for (connuous) sum of squares, and RSS sands for resdual sum of squares. Under connuous observaon, herefore, one solves he problem (1.3) by usng he ρ and Z defned above. Our arge of nference, < Z, Z >, would hen be observable. Dscreeness of observaon, however, creaes he need for nference. The man heorems n he curren paper are concerned wh he asympoc behavor of he esmaed RSS, as more dscree observaons are avalable whn a fxed me wndow. There wll be some choce n how o selec he esmaor < Z, Z >. We consder a class of such esmaors < Z, Z >. No maer whch of our esmaors s used, we ge he decomposon < Z, Z > < Z, Z > bas + ([Z, Z] < Z, Z > ), (1.7) o frs order asympocally, where [Z, Z] s he sum of squares of he ncremens of he (unseen) process Z a he samplng pons, [Z, Z] = (Z +1 Z ) 2 cf. he defnon (2.4) below.

4 4 P.A. MYKLAND AND L. ZHANG A prmary concepual fndng n (1.7) s he clear cu effec of he wo sources behnd he asympocs. The form of he bas depends only on he choce of he esmaor for he quadrac varaon. On he oher hand, he varaon componen s common for all he esmaors under sudy; comes only from he dscrezaon error n me dscree samplng. I s worhwhle o pon ou ha our problem s only angenally relaed o ha of esmang he regresson coeffcen ρ. Ths s n he sense ha he asympoc behavor of nonparamerc esmaors of ρ do no drecly mply anyhng abou he behavor of esmaors of < Z, Z > T. To llusrae hs pon, noe ha he convergence raes are no he same for he wo ypes of esmaors (O p (n 1/4 ) vs. O p (n 1/2 )), and ha he varance of he esmaor we use for ρ becomes he bas n one of our esmaors of < Z, Z > T (compare equaon (2.9) n Secon 2.4 wh Remark 1 n Secon 3). For furher dscusson, see Secon 2.4. Dependng on he goal of nference, sascal esmaes ˆρ of he regresson coeffcen can be obaned usng paramerc mehods, or nonparamerc ones ha are eher local n space or n me, as dscussed n Secons 2.4 and 4.1, and he references ced n hese secons. In addon, s also common n fnancal conexs o use calbraon ( mpled quanes, see Chapers 11 and 17 n Hull (1999)). The organzaon s as follows, n Secon 2, we esablsh he framework for ANOVA, and we nroduce a class of esmaors of he resdual quadrac varaon. Our man resuls, n Secon 3, provde he dsrbuonal properes of he esmaon errors for RSS. See Theorem 1-Theorem 2. In Secon 4, we dscuss he sascal applcaon of he man heorems. Paramerc and nonparamerc esmaon are compared n he conex of resdual analyss. The goodness of f of a model s addressed. Broad ssues, ncludng he analyss of varaon versus analyss of varance, he moderae level of aggregaon versus long run, he acual probably dsrbuon versus he rsk neural probably dsrbuon n he dervave valuaon seng, are dscussed n Secon Afer concludng n Secon 5, we gve proofs n Secons ANOVA for Io processes: framework Io processes, quadrac varaon, and dffusons. The assumpons and defnons n he followng wo subsecons are used hroughou he paper, somemes whou furher reference. Frs of all, we shall be workng wh a fxed flered probably space:

5 ANOVA FOR DIFFUSIONS 5 Sysem Assumpon I. We suppose ha here s an underlyng flered probably space (Ω, F, F, P ) T, sasfyng he usual condons (see, for example, Defnon 1.3 (p. 2) of Jacod and Shryaev (23), also n Karazas and Shreve (1991)). We shall hen be workng wh Io processes adaped o hs sysem, as follows. Noe ha Markov dffusons are a specal case. Defnon 1. (Io processes). By sayng ha X s an Io process, we mean ha X s connuous (a.s.), (F )-adaped, and ha can be represened as a smooh process plus a local marngale, X = X + X u du + σ X u dw X u, (2.1) where W s a sandard ((F ), P )-Brownan Moon, X s F measurable, and he coeffcens X and σ X are predcable, wh T X u du < + and T (σx u )2 du < +. We also wre X = X + X DR + X MG (2.2) as shorhand for he drf and local marngale pars of Doob-Meyer decomposon n (2.1). A more absrac way of pung hs defnon s ha X s an Io process f s a connuous semmarngale (Jacod and Shryaev (23), Defnon I.4.21, p. 42) whose fne varaon and local marngale pars, gven by (2.2), sasfy ha boh X DR and he quadrac varaon < X MG, X MG > are absoluely connuous. Obvously, an Io process s a specal semmarngale, also n he sense of he same defnon of Jacod and Shryaev (23). Dffusons are normally aken o be a specal case of Io processes, where one can wre (σ X )2 = a(x, ) and X = b(x, ), and smlarly n he muldmensonal seng. For a descrpon of he lnk, we refer o Chaper 5.1 of Karazas and Shreve (1991). The Io process defnon exends o a wo- or muldmensonal process, say (X, Y ), by requrng each componen X and Y ndvdually o be an Io process. Obvously, W X s ypcally dfferen for dfferen Io processes X. For wo processes X and Y, he relaonshp beween W X and W Y can be arbrary. The quadrac varaon < X, X > (formula (1.2)) can now be expressed n erms

6 6 P.A. MYKLAND AND L. ZHANG of he represenaon (2.1) by (Jacod and Shryaev (23), I.4.54, p.55) < X, X > = (σ X u )2 du. We denoe by < X, X > he dervave of < X, X > wh respec o me, hen < X, X > = (σx )2, and hs quany (or s square roo) s ofen known as volaly n he fnance leraure. Boh quadrac varaon and covaraon are absoluely connuous. Ths follows from he Io process assumpon and from he Kuna-Waanabe Inequaly (see, for example, p. 69 of Proer (24)). Defnon 2. (Volaly as an Io process). Denoe by < X, Y > he dervave of < X, Y > wh respec o me. We shall ofen suppose ha < X, Y > s self an Io process. For ease of noaon, we hen wre s Doob-Meyer decomposon as d < X, Y > = ddxy + dr XY XY = D d + dr XY. Noe ha he quadrac varaon of < X, Y > s he same as < R XY, R XY >, and ha n our noaon above, D XY = (< X, Y > ) DR and R XY = (< X, Y > ) MG 2.2. Model example: Adequacy of he one facor neres rae model. A common model for he rsk free shor erm neres rae s gven by he dffuson dr = µ(r )d + γ(r )dw, (2.3) where W s a Brownan moon. For example, n Vascek (1977), one akes µ(r) = κ(α r), and γ(r) = γ = consan, whle Cox e al. (1985) uses he same funcon µ, bu γ(r) = γr 1/2. For more such models, and a bref fnancal nroducon, see, for example, Hull (1999). A dscusson and revew of esmaon mehods s gven by Fan (25). One of he mplcaons of hs so-called one facor model s he followng. Suppose S and Ξ are he values of wo zero coupon governmen bonds wh dfferen maures. Fnancal heory hen predcs ha unl maury, S = f(r, ) and Ξ = g(r, ) for wo funcons f and g (see Hull (1999), Chaper 21, for deals and funconal forms). Under hs model, herefore, he relaonshp dξ = ρ ds holds from me zero unl he maury of he shorer erm bond. I s easy o see ha d < S, S > = f r(r, ) 2 γ(r ) 2 d and d < Ξ, S > = f r(r, )g r(r, )γ(r ) 2 d, wh

7 ANOVA FOR DIFFUSIONS 7 ρ = d < Ξ, S > /d < S, S > = g r(r, )/f r(r, ). Here, f r s he dervave of f wh respec o r, and smlarly for g r. The one facor model s only an approxmaon, and o assess he adequacy of he model, one would now wsh o esmae < Z, Z > T over dfferen me nervals. Ths provdes nsgh no wheher s worh whle o use a one facor model a all. If he concluson s sasfacory, one can esmae he quanes µ and γ (and hence f and g) wh paramerc or nonparamerc mehods (see Secon 2.4 and 4.1), and agan, use our mehods o assess he f of he specfc model, for example as dscussed n Secon Fnely many daa. We now suppose ha we observe processes, n parcular S and Ξ, on a fne se (paron, grd) G = { = < 1 < < k = T } of me pons n he nerval [, T ]. We ake he me pons o be nonrandom, bu possbly rregularly spaced. Noe ha hs also covers he case where he are random bu ndependen of he processes we seek o observe, so ha one can condon on G o ge back o rregular bu nonrandom spacng. Defnon 3. (Observed quadrac varaon). For wo Io processes X and Y observed on a grd G, [X, Y ] G = ( X )( Y ), (2.4) where X = X +1 X. When here s no ambguy, we use [X, Y ] for [X, Y ] G, or [X, Y ] (n) n case of a sequence G n. Noe ha hs s no he same as he usual defnon of [X, Y ] for Io processes. We use < X, Y > o refer o a connuous process, whle [X, Y ] refers o a (càdlàg) process whch only changes values a he paron pons. Snce he resuls of he paper rely on asympocs, we shall ake lms wh he help of a sequence of parons G n = { = (n) < (n) 1 < < (n) k n = T }. As n +, we le G n become dense n [, T ], n he sense ha he mesh δ (n) = max (n). (2.5) Here (n) = (n) +1 (n). In oher words, he mesh s he maxmum dsance beween he (n) s. On he oher hand, T remans fxed (excep brefly n Secon 4.5). In hs case, [X, Y ] = [X, Y ] Gn converges o < X, Y > unformly n probably, see Jacod and Shryaev (23), Theorem I.4.47 (p. 52), and Proer (24), Theorem

8 8 P.A. MYKLAND AND L. ZHANG II.23 (p. 68). More s rue, see Secon 5 of Jacod and Proer (1998), and (our) Secons 2.8 and 6 below. Noe ha, under (2.5), k n = G n. I s ofen convenen o consder he average dsance beween successve observaon pons, cf. Assumpon A() below. (n) = T k n, (2.6) 2.4. The regresson problem, and he esmaon of ρ. The processes n (1.1) wll be aken o sasfy he followng. Sysem Assumpon II. We le Ξ and S be Io processes. We assume ha nf < S, S [,T ] > > almos surely. (2.7) Ths assumpon assures ha ρ, gven by (1.5), s well defned under (2.7), by he Kuna-Waanabe Inequaly. As noed n he Inroducon, under connuous observaon of Ξ and S, one can drecly also observe he opmal ρ and Z. Our arge of nference, < Z, Z >, would hen be observable. Dscreeness of observaon, however, creaes he need for nference. In a non-connuous world, where Ξ and S can only be observed over grd mes, he mos sraghforward esmaor of ρ s, ˆρ = < Ξ, S > < S, S > = [Ξ, S] [Ξ, S] hn [S, S] [S, S] hn. (2.8) For smplcy, hs esmaor s he one we shall use n he followng. The resuls easly generalze when more general kernels are used. Noe ha we have o use a smoohng bandwdh h n. There wll naurally be a radeoff beween h n and (n). As we now argue, hs ypcally resuls n h n = O( (n)1/2 ). Asympocs for esmaors of he form < X, Y > = ([X, Y ] [X, Y ] hn )/h n and hence for ˆρ, are gven by Foser and Nelson (1996) and Zhang (21). Le (n) be he average observaon nerval, assumed o converge o zero. If < X, Y > s an Io process wh nonvanshng volaly, hen s opmal o ake h n = O(( (n) ) 1/2 ), and ( (n) ) 1/4 ( < X, Y > < X, Y > ) converges n law (for each fxed ) o a (condonal on he daa) normal dsrbuon wh mean zero and random varance. (The

9 ANOVA FOR DIFFUSIONS 9 mode of convergence s he same as n Proposon 1). The asympoc dsrbuons are (condonally) ndependen for dfferen mes. If < X, Y > s smooh, on he oher hand, he rae becomes ( (n) ) 1/3 raher han ( (n) ) 1/4, and he asympoc dsrbuon conans boh bas and varance. The same apples o he esmaor ˆρ. In he case when S and Ξ have a dffuson componen, he esmaor has (random) asympoc varance Vˆρ ρ () = < ρ, ρ > 3c + ch ()( < Ξ, Ξ > < S, S > ρ 2 ) (2.9) whenever h n /( (n) ) 1/2 c (, ), see Zhang (21). H() s defned n Secon 2.6 below. The scheme gven n (2.8) s only one of many for esmang < X, Y > by usng mehods ha are local n me. In parcular, Genon-Caalo e al. (1992) use waveles for hs purpose, and deermne raes of convergence and lm dsrbuons under he assumpon ha < X, Y > s deermnsc and has smoohness properes. Oher mporan leraure n hs area seeks o esmae < X, Y > as a funcon of he underlyng sae varables, by mehods ha are local n space, see n parcular Florens-Zmrou (1993), Hoffmann (1999) and Jacod (2). The ypcal seup s ha U = (X, Y,...) s a Markov process, so ha < X, Y > = f(u ) for some funcon f, and he problem s o esmae f. If all coeffcens n he Markov dffuson are smooh of order s, and subjec o regulary condons, he funcon f can be esmaed wh a rae of convergence of ( (n) ) s/(1+2s). The convergence obaned for he esmaor of f under Markov assumpons s consderably faser han wha can be obaned for (2.8). I does, however, rely on sronger (Markov) assumpons han he ones (Io processes) ha we shall be workng wh n hs paper. Snce we shall only be neresed n ρ as a (random) funcon of me, our developmen does no requre a Markov specfcaon, and n parcular does no requre full knowledge of wha poenal sae varables mgh be. Of course, hs s jus a subse of he leraure for esmaon of Markov dffusons. See Secon 4.1 for furher references. We emphasze ha he general ANOVA approach n hs paper can be carred ou wh oher schemes for esmang ρ han he one gven n (2.8). We have seen hs as a maer of fne unng and hence beyond he scope of hs paper. Ths s

10 1 P.A. MYKLAND AND L. ZHANG because Theorems 1 and 2 acheve he same rae of convergence as he one obaned n Proposon Esmaon schemes for he resdual quadrac varaon < Z, Z >. We now reurn o he esmaon of he quadrac varaon < Z, Z > of resduals. Gven he dscree daa of (Ξ, S), here are dfferen mehods o esmae he resdual varaon. One scheme s o sar wh model (1.1). For a fxed grd G, one frs esmaes Z hrough he relaon Ẑ = Ξ ˆρ ( S ), where all ncremens are from me o +1, and hen obans he quadrac varaon (q.v. hereafer) of Ẑ. Ths gves an esmaor of < Z, Z > as [Ẑ, Ẑ] = ( Ẑ ) 2 = [ Ξ ˆρ ( S )] 2 (2.1) where he noaon of square brackes (dscree me-scale q.v.) s nvoked, snce Ẑ s he ncremen over dscree mes. Alernavely, one can drecly analyze he ANOVA verson (1.6) of he model, where d < Z, Z > = d < Ξ, Ξ > ρ 2 d < S, S >. Ths yelds a second esmaor of < Z, Z >, < Z, Z > (1) = [( Ξ ) 2 ˆρ 2 ( S ) 2 ]. (2.11) In general, any convex combnaon of hese wo, < Z, Z > (α) = (1 α)[ẑ, Ẑ] + α< Z, Z > (1) (2.12) would seem lke a reasonable way o esmae < Z, Z >, and hs s he class of esmaors ha we shall consder. Parcular properes wll be seen o aach o < Z, Z > (1/2), whch we shall also denoe by < Z, Z >. For a sar, s easy o see ha < Z, Z > = [Ξ, Ẑ]. (2.13) Noe ha (2.13) also has a drec movaon from he connuous model. Snce < S, Z > =, (1.1) yelds ha < Ξ, Z > =< Z, Z >. We esablsh he sascal properes of he esmaor < Z, Z > (α), and n parcular hose of ([Ẑ, Ẑ], and < Z, Z > ) n Secon 3. Asympoc properes are naurally suded wh he help of small nerval asympocs Paradgm for asympoc operaons. The asympoc propery of he esmaon error s consdered under he followng paradgm.

11 ANOVA FOR DIFFUSIONS 11 Assumpon A (Quadrac varaon of me:) For each n N, we have a sequence of non-random parons G n = { (n) }, (n) = (n) +1 (n). Le max ( (n) ) = δ (n). Suppose ha () δ (n) as n, and δ (n) / (n) = O(1). () H (n) () = P ( (n) (n) ) 2 +1 (n) H() as n. () H() s connuously dfferenable. (v) The bandwdh h n sasfes (n) h n c, where < c <. (v) [H (n) () H (n) ( h n )]/h n H (),where he convergence s unform n. When he parons are evenly spaced, H() = and H () = 1. In he more general case, he lef hand sde of () s bounded by δ (n) / (n), whle he lef hand sde of (v) s bounded by δ (n)2 /( (n) h)+δ (n) / (n). In all our resuls, h s evenually bgger han (n), and hence boh he lef hand sdes are bounded because of (). The assumpons n () and (v) are, herefore, abou a unque lm pon, and abou nerchangng lms and dfferenaon. Noe ha we are no assumng ha he grds be nesed. Also as dscussed n Secon 2.4, how fas h n and (n) decay respecvely has a rade-off n erms of he asympoc varance of he esmaon error n ρ. I s opmal o ake h n = O( (n) ), whence A(v). From now on, we use h and h n nerchangeably Assumpons on he process srucure. The followng assumpons are frequenly mposed on he relevan Io processes. Assumpon B(X) (Smoohness): X s an Io process. Also, < X, X > and X are connuous, almos surely. The addon of Assumpon B o an Io process X, and smlar smoohness assumpons n resuls below, s parally due o he esmaon of ρ, whch requres sronger smoohness condons. In some nsances, Assumpon B s parally also a maer of convenence n a proof and can be dropped a he cos of more nvolved echncal argumens The lm for he dscrezaon error. The error < Z, Z > < Z, Z > can be decomposed no bas and pure dscrezaon error [Z, Z] < Z, Z >. We here

12 12 P.A. MYKLAND AND L. ZHANG dscuss he lm resul for he laer, followng Jacod and Proer (1998). We frs need he followng: Sysem Assumpon III (Descrpon of he flraon): There s a connuous muldmensonal P -local marngale X = (X (1),, X (p) ), for some fne p, so ha F s he smalles sgma-feld conanng σ(x s, s ) and N, where N conans all he null ses n σ(x s, s T ). The fnal saemen n he Assumpon assures ha he usual condons (Jacod and Shryaev (23), p. 2, Karazas and Shreve (1991), p. 1) are sasfed. The man mplcaon, however, s on our mode of convergence, as follows. Proposon 1. (Dscrezaon Theorem). Le Z be an Io process for whch T (< Z, Z > ) 2 d< a.s. and T Z 2 d < a.s. Subjec o assumpons A() () and Sysem Assumpons I and III, ( (n) ) 1 2 ( ) [Z, Z] (n) L.sable < Z, Z > 2H (u) < Z, Z > u dw u, where W s a sandard Brownan Moon, ndependen of he underlyng daa process X. The symbol L.sable denoes sable convergence of he process, as defned n Rény (1963) and Aldous and Eagleson (1978); see also Roozén (198) and Secon 2 of Jacod and Proer (1998). In he case of an equdsan grd, he resul concdes wh he applcable par of Theorems 5.1 and 5.5 n Jacod and Proer (1998), and he proof s essenally he same (see Secon 6). In absrac form, resuls of hs ype appear o go back o Roozén (198). The Jacod and Proer resul was used n fnancal applcaons by Zhang (21) and Barndorff-Nelsen and Shephard (22). The case where Z s observed dscreely and wh addve error s consdered n Zhang (24) and Zhang e al. (25). Noe ha he condons on < Z, Z > and Z are he same as n he equdsan case, due o he Lpschz connuy of H. Some furher dscusson and resuls are conaned n Secon ANOVA for dffusons: man dsrbuonal resuls Dsrbuon of [Ẑ, Ẑ] < Z, Z >. Recall ha he square bracke [Z, Z] and he angled bracke < Z, Z > represen he quadrac varaon of Z a dscree and

13 ANOVA FOR DIFFUSIONS 13 connuous me-scale, respecvely. Theorem 1. Under Sysem Assumpons I-II (and n parcular equaon (1.1)), assume ha Condon A holds. Suppose ha S, Ξ, ρ, < S, S >, < Ξ, S >, < R SS, R SS >, < R ΞS, R ΞS >, and < R ΞΞ, R ΞΞ > are Io processes, each sasfyng condon B. Le he esmaor [Ẑ, Ẑ] be defned as n (2.1). Then, as n unformly n, where D = 1 3c ( (n) ) 1 2 ([ Ẑ, Ẑ] < Z, Z > ) = D + ( (n) ) 1 2 ([Z, Z] (n) < Z, Z > ) + o p (1) (3.14) < ρ, ρ > u d < S, S > u +c H (u)d < Z, Z > u. (3.15) Remark 1. The consequence of Theorem 1 s ha he quany n (3.14) converges n law (sably) o D + 2H (u) < Z, Z > u dw u ; he o p (1) erm goes away by Lemma VI 3.31 (p. 352) n Jacod and Shryaev (23). Noe ha D n (3.15) can be expressed as D = Vˆρ ρ(u)d < S, S > u, where Vˆρ ρ () s he asympoc varance of ˆρ ρ, cf. (2.9), or Zhang (21). Hence he (random) varance erm for ˆρ becomes a bas erm for [Ẑ, Ẑ]. Ths s nuvely naural snce he ˆρ are asympocally ndependen for dfferen. Theorem 1, ogeher wh Proposon 1, says ha he esmaor [Ẑ, Ẑ] converges o < Z, Z > a he order of square roo of he average samplng nerval. In he lm, he error erm consss of a non-negave bas D, due o he esmaon uncerany [Ẑ, Ẑ] [Z, Z], and a mxure Gaussan, due o he dscrezaon [Z, Z] < Z, Z >. The non-negaveness of he asympoc bas s because he q.v. s (< ρ, ρ >, < S, S >, < Z, Z >) are non-decreasng processes. Furhermore, (3.15) dsplays a bas-bas radeoff, hus an opmal c for smoohng can be reached o mnmze he asympoc bas, hough we have no nvesgaed he effec of havng a random c. The dscrezaon erm s ndependen of he smoohng facor Dsrbuon of < Z, Z > < Z, Z >. Theorem 2. Under Sysem Assumpons I-II, assume ha Condon A holds. Also assume each of he followng processes exs, and are Io-processes sasfyng condon (B): Ξ, S, ρ, < Ξ, S >, < S, S >, < R SS, R SS >, < R ΞS, R SS >, and

14 14 P.A. MYKLAND AND L. ZHANG < R ΞS, R ΞS >. Also suppose ha he processes < Ξ, ρ > and < S, ρ > are connuous. Then, unformly n ( (n) ) 1 2 ( < Z, Z > < Z, Z > ) = 1 2c < Ξ, S > u dρ u (3.16) + ( (n) ) 1 2 ([Z, Z] (n) < Z, Z > ) + o p (1). Remark 1 apples smlarly. Unlke [Ẑ, Ẑ], he asympoc (condonal) bas assocaed wh < Z, Z > does no necessarly have a posve or negave sgn. Moreover, we are no longer faced wh a bas-bas radeoff due o he poson of c n (3.16). In hs case, he role of smoohng n he asympoc bas shall be dscussed n Secon General resuls for he < Z, Z > (α) class of esmaors. From (2.12), < Z, Z > (α) = (1 2α)[Ẑ, Ẑ] + 2α< Z, Z >, follows from he assumpons of Theorems 1-2 ha, f one ses hen, as n Remark 1, bas (α) = α c ( (n) ) 1 2 ( < Z, Z > (α) < Z, Z > ) = bas (α) L.sable bas (α) + < Ξ, S > u dρ u + (1 2α)D, (3.17) + ( (n) ) 1 (n) 2 ([Z, Z] < Z, Z > ) + o p (1) 2H (u) < Z, Z > u dw u, (3.18) As a summary for any lnear combnaon of he esmaors n Theorems 1-2, α [, 1], he convergence n (3.18) s n law as a process, and he lmng Brownan Moon W s ndependen of he enre daa process. For deals of sable convergence, see he dscusson and references n Secon 2.8 above. The varance erm ( (n) ) 1 2 ([Z, Z] < Z, Z > ) s he same for any esmaor n he lnear-combnaon class, hey are all asympocally perfecly correlaed. The common asympoc, condonal varance s ndependen of he smoohng bandwdh. I remans unclear wheher he common asympoc varance could, perhaps,

15 ANOVA FOR DIFFUSIONS 15 Table 1 The effec of consan ρ on he bas componens Esmaor Asympoc Bas [Ẑ, Ẑ] c R H (u)d < Z, Z > u < Z, Z > (1) c R H (u)d < Z, Z > u < Z, Z > be a lower bound under he nonparamerc seng (see Bckel e al. (1993) for a comprehensve dscusson). Ths needs furher nvesgaon. For he bas, on he oher hand, he esmaon procedure plays an mporan role, as he bas erm vares wh α. Also he smoohng effec eners he bas erms. From Theorem 1-2, excessve over-smoohng (smaller c) or under-smoohng (bgger c) can explode he bas of < Z, Z > (α), for α 1 2, hus (condonal) bas may be mnmzed opmally. When α = 1 2, s no obvous how o deal wh bas-bas radeoff. One mgh heorecally be able o reduce he bas for < Z, Z > (.e. < Z, Z > (1/2) ) by choosng he smalles possble bandwdh h. Ths hough should, however, be aken wh cauon. I s no obvous wheher he magnude of he hgher order erms n he earler resuls would reman neglgble f he esmaon wndow h were o decrease faser han he order. Table 1 shows ha assumng consan ρ, < Z, Z > wll be he bes choce among he hree. When ρ s random, none of he esmaon schemes n Secon 2.5 s obvously superor o he ohers Esmang he asympoc dsrbuon. For sascal nference concernng < Z, Z >, one needs, n vew of he above, o esmae he asympoc (random) bas and varance. The bas erm can be obaned by subsuon of esmaed quanes no he relevan expressons, mos generally (3.17). We shall here be concerned wh he varance erm n (3.18). In vew of he sable convergence n Proposon 1, we herefore seek o esmae 2H (u)(< Z, Z > u) 2 du. By a modfcaon over Barndorff-Nelsen and Shephard (22), and also Mykland (1995b), one can do hs by consderng he fourh order varaon. Defnon 4. (Observed fourh order varaon). For an Io processes X observed on a grd G, [X, X, X, X] = ( X ) 4 (3.19)

16 16 P.A. MYKLAND AND L. ZHANG Proposon 2. (Esmaon of Varance). Assume he regulary condons of Theorem 1, and le Ẑ be defned as n ha resul. Also assume Sysem Assumpon III. Then, as n, unformly n probably. 2 3 ( (n) ) 1 [Ẑ, Ẑ, Ẑ, Ẑ] 2H (u)(< Z, Z > u) 2 du (3.2) Ths esmae of varance can be used n connecon wh Secons below. The proof s gven n Secon 6. I can be noed from here ha he same saemen (3.2) would hold under weaker condons f Ẑ were replaced by Z, as follows. Remark 2. Assume he condons of Proposon 1, and also ha Z and < Z, Z > are bounded a.s. Then, as n, unformly n probably. 2 3 ( (n) ) 1 [Z, Z, Z, Z] 2H (u)(< Z, Z > u )2 du (3.21) Ths generalzes he correspondng resul a he end of page 27 n Barndorff- Nelsen and Shephard (22). The fndng n Mykland (1995b) s exac for small samples n he conex of explc embeddng, where follows from Barle denes. For anoher use of hs mehodology, see, for example, he proof of Lemma 1 n Mykland (1993). 4. Goodness of f. The purpose of ANOVA s o assess he goodness of f of a regresson model on he form (1.1). We here llusrae he use of Theorems 1 and 2 by consderng wo dfferen quesons of hs ype. In he frs secon, we dscuss how o assess he f of a paramerc esmaor for ρ. Aferwards, we focus on he ssue of how good s he one regressor model self, ndependenly of esmaon echnques. Ths s already measured by he quany < Z, Z > T, bu can be furher suded by consderng confdence bands for < Z, Z > as a process, and by an analogue o he coeffcen of deermnaon. Fnally, we dscuss he queson of he relaonshp beween hs ANOVA and he analyss of varance ha s used n he sandard regresson seng The assessmen of paramerc models. In he followng we suppose ha a paramerc model s f o he daa, and ρ s esmaed as a funcon of he parameer. Paramerc esmaon of dscreely observed dffusons has been suded by Genon- Caalo and Jacod (1994), Genon-Caalo e al. (1999, 2), Gloer (2), Gloer

17 ANOVA FOR DIFFUSIONS 17 and Jacod (21), Barndorff-Nelsen and Shephard (21), Bbby e al. (21), Eleran e al. (21), Jacobsen (21), Sørensen (21), and Hoffmann (22). Ths s, of course, only a small sample of he leraure avalable. Also, hese references only concern he ype of asympocs consdered n hs paper, where [, T ] s fxed, and, and here s also a subsanal leraure on he case where s fxed and T. In Secon 3, we have suded he nonparamerc esmae < Z, Z > (α) T for he resdual sum of squares. We here compare < Z, Z > (α) T o s paramerc counerpar o see how good he paramerc model s n capurng he rue regresson of Ξ on S. Specfcally, we suppose ha daa from he muldmensonal process X s observed a he grd pons. X has among s componens a leas S and Ξ, and possbly also oher processes. The paramerc model s of he form P θ,ψ, θ Θ, ψ Ψ, where he modelng s such ha dffuson coeffcens are funcons of θ, whle drf coeffcens can be funcons of boh θ and ψ. I s hus reasonable o suppose ha as, ˆθ converges n probably o a nonrandom parameer value θ, and ha ( ) 1/2 (ˆθ θ ) ηn(, 1) n law, sably, where η s a funcon of he daa and he N(, 1) erm s ndependen of he daa. (For condons under whch hs occurs, consul, for example, he references ced above). θ s he rue value of he parameer f he model does conan he rue probably, bu s oherwse also aken o be a defned parameer. Under P θ,ψ, he regresson coeffcen ρ s of he form β (θ). Mos commonly, β (θ) = b(x ; θ) for a nonrandom funconal b. We now ask wheher he rue regresson coeffcen can be correcly esmaed wh he model a hand. In oher words, we wsh o es he null hypohess H ha β (θ ) = ρ. For he ANOVA analyss, defne he heorecal resdual by and he observed one by dv = dξ β (θ )ds, V =, ˆV = Ξ β (ˆθ) S, ˆV =.

18 18 P.A. MYKLAND AND L. ZHANG Under he null hypohess, < V, V >=< Z, Z > and so a naural es sasc s of he form U = ( ) 1/2 ([ ˆV, ˆV ] T < Z, Z > (α) T ) We now derve he null dsrbuon for U, usng he resuls above. As an nermedae sep, defne he dscrezed heorecal resdual V d = Ξ β (θ ) S, V d = Subjec o obvous regulary condons, [ ˆV, ˆV ] T [V d, V d ] T = 2 (β (ˆθ) β (θ )) V d S + = 2(ˆθ θ ) = 2(ˆθ θ ) (β (ˆθ) β (θ )) 2 ( S ) 2 T β θ (θ ) V d S + O p ( ) β θ (θ )d < V, S > +O p ( ). Also, under he condons n Proposon 3 n Secon 6 (cf also he proof of Proposon 2 n he same secon), [V d, V d ] T = [V, V ] T + o p ( 1/2 ) as (snce < V d, V d > =< V, V > +o p ( 1/2 ), and < V d, V d > < V d, V > < V, V > ). Hence, under he condons of Theorem 1 or 2, where bas (α) T herefore, T U = 2( ) 1/2 (ˆθ θ ) +( ) 1/2 {([V, V ] T [Z, Z] T )} bas (α) T + o p (1) β θ (θ )d < V, S > has he same meanng as n Secon 3. If he null hypohess s sasfed, T U N(, 1) 2η β θ (θ )d < V, S > bas (α) T n law, sably. The varance and bas can be esmaed from he daa. Ths, hen, provdes he null dsrbuon for U.

19 ANOVA FOR DIFFUSIONS 19 Anoher approach s o use U o measure how close he paramerc resdual < V, V > s o he lower bound < Z, Z >. To frs order, ( ) 1/2 U P = < V, V > T < Z, Z > T T (β (θ ) ρ ) 2 d < S, S > The behavor of U ( ) 1/2 (< V, V > T < Z, Z > T ) depends on he jon lmng dsrbuon of ([V, V ] T < V, V > T ) ([Z, Z] T < Z, Z > T ) and ( ) 1/2 (ˆθ θ ). The former can be provded by Proposon 1 n Secon 2.8 (or Secon 5 of Jacod and Proer (1998)), bu furher assumpons are needed o oban he jon dsrbuon. A sudy of hs s beyond he scope of hs paper Confdence bands. In addon o provdng ponwse confdence nervals for (α) < Z, Z >, we can also consruc jon confdence band for he esmaed quadrac varaon < Z, Z > (α) of resduals. Ths s possble because < Z, Z > (α) converges as a process by Theorems 1-2. One proceeds as follows. As a process on [, T ], ( ) ( (n) ) 1 2 < Z, Z > (α) L < Z, Z > bas (α) + L, Under all esmaon schemes n he lnear combnaon class, we have, by Theorem 1- (α) 2 and subsequen resuls on < Z, Z >, L = 2H (u) < Z, Z > u dw u, where W s a sandard Brownan Moon ndependen of he complee daa flraon. Now condon on F T : by he sable convergence, L s hen a Gaussan process, wh < L, L > nonrandom. Use he change-of-me consrucon by Dambs (1965) and Dubns and Schwarz (1965) o oban L = W<L,L>, where W s a new Brownan Moon condonal on F T. I hen follows ha Now wre L n () = ( (n) ) 1 2 max L = max T 2 R W T H (u)(<z,z> u) 2 du mn T = 2 R T mn ( H (u)(<z,z> u )2 du ) (α) < Z, Z > < Z, Z > W bas (α), we have P ( L n () c, for all [, T ]) P ( L() c, for all [, T ]) ( ) = P mn W c, max W c τ τ

20 2 P.A. MYKLAND AND L. ZHANG Choose c = c τ such ha ( ) P mn W c τ, max W c τ τ = 1 α, τ τ wh τ = 2 T H (u)(< Z > u )2 du. To fnd c τ, one can refer o Karazas and Shreve (1991) Secon 2.8 for he dsrbuons of he runnng mnmum and maxmum of a Brownan Moon. τ self can be esmaed by usng Proposon 2. Ths complees our consrucon of a global confdence band The Coeffcen of Deermnaon R 2. In analogy wh sandard lnear regresson, one can defne he R 2 by R 2 = 1 < Z, Z > < Ξ, Ξ >. Ths quany would have been observed f he whole pah of he processes Ξ and S had been avalable. If observaons are on a grd, s naural o use ˆR 2 < = 1 Z, Z > (α). [Ξ, Ξ] Under he assumpons of Secon 2, he dsrbuon of ˆR 2 can be found by ( (n) ) 1/2 ( ˆR 2 R 2 ) = ( (n) ) 1/2 < Z, Z > (α) < Z, Z > [ (1 R 2 ) [Ξ, Ξ] < Ξ, Ξ > ]+o p (1) < Ξ, Ξ > < Ξ, Ξ > = ( (n) ) 1/2 1 < Ξ, Ξ > ( ([Z, Z] < Z, Z > ) (1 R 2 )([Ξ, Ξ] < Ξ, Ξ > ) ) bas(α) < Ξ, Ξ > + o p (1), where bas (α) s he bas correspondng o esmaor < Z, Z > (α). A sraghforward generalzaon of Proposon 1 yelds ha ( (n) ) 1/2 ([Z, Z] < Z, Z >, [Ξ, Ξ] < Ξ, Ξ > ) T converges (sably) o a process wh (bvarae) quadrac varaon g udu, where g = 2H () ( (< Z, Z > )2 (< Z, Ξ > )2 (< Z, Ξ > ) 2 (< Ξ, Ξ > ) 2 ),

21 ANOVA FOR DIFFUSIONS 21 and equaon (1.1) yelds ha < Z, Ξ > =< Z, Z >. I follows ha ( (n) ) 1/2 ( ˆR 2 R 2 ) L.sable R 2 < Ξ, Ξ > + 1 R2 < Ξ, Ξ > bas(α) < Ξ, Ξ >, 2H (u) < Z, Z > u dw u 2H (u)[(< Ξ, Ξ > u )2 (< Z, Z > u )2 ]dw u where W and W are ndependen Brownan moons. For fxed, he lm s condonally normal, wh mean bas (α) / < Ξ, Ξ > and varance + 1 < Ξ, Ξ > 2 R 4 2H (u)(< Z, Z > u) 2 du 1 < Ξ, Ξ > 2 (1 R 2 )2 2H (u)[(< Ξ, Ξ > u )2 (< Z, Z > u )2 ]du, whch can be readly esmaed usng he devce dscussed n Secon Varance versus varaon: Whch ANOVA?. The formulaon of (1.6) s n erms of quadrac varaon. Ths rases he queson of how our analyss relaes o he radonal meanng of ANOVA, namely a decomposon of varance. There are several answers o hs, some concernng he broad seng provded by model (1.1), and hey are dscussed presenly. More specfc srucure s provded by fnancal applcaons, and a dscusson s provded n Secon 4.5. In model (1.1), he varaon n Z can come from boh drf and marngale. As n (2.2), Z = Z + Z DR + Z MG. (4.1) Our analyss concerns mos drecly he varaon n Z MG, n ha var(z MG ) = E(< Z, Z > ), where should be noed ha < Z, Z > =< Z MG, Z MG >. Hence, f he Z DR erm s dencally zero, he analyss of varaon s an exac analyss of varance, n erms of expecaons. The quadrac varaon, however, s also a more relevan measure of varaon for he daa ha were acually colleced. The Dambs (1965)/Dubns and Schwarz (1965) represenaon yelds ha = V <Z,Z>, where V s a sandard Brownan moon on a dfferen me scale. Therefore, < Z, Z > = < Z MG, Z MG > conans nformaon abou he acual amoun of varaon ha has occurred n he process Z MG. Usng he quadrac varaon s, n hs sense, analogous o usng observed nformaon n a lkelhood Z MG

22 22 P.A. MYKLAND AND L. ZHANG seng (see, for example, Efron and Hnkley (1978)). The analogy s vald also on he echncal level: f one forms he dual lkelhood (Mykland (1995a)) from score funcon Z MG, he observed nformaon s, ndeed, < Z, Z >. If he drf Z DR n (4.1) s nonzero, he analyss apples drecly only o Z MG. So long as T s small or moderae, however, he varably n Z MG s he man par of he varably n Z. Specfcally, boh when T and T +, ZT MG = O p (T 1/2 ) and ZT DR = O p (T ). Thus he bas due o analyzng Z MG n leu of Z becomes a problem only for large T. A he same me he presen mehods provde esmaes for he varaon n Z MG for small and moderae T, whereas he varaon n Z DR can only be conssenly esmaed when T + (by Grsanov s Theorem). Thus, we recommend our curren mehods for moderae T, whle one should use oher approaches when dealng wh a me span T ha s long Fnancal applcaons: An nsance where varance and varaon relae exacly. I s que common n fnance o encouner he case from Secon 4.4, where Z self s a marngale, or where one s neresed n Z MG only. We here show a concepual example of hs, where one wans o es wheher he resdual Z s zero, or sudy he dsrbuon of he resdual under he so called Rsk Neural or Equvalen Marngale Measure P. If P s he rue, acual (physcal) probably dsrbuon under whch daa s colleced, P s, by conras, a probably measure equvalen o P n he sense of muual absolue connuy, and sasfes he condon ha he dscouned value of all raded secures mus be P -marngales. The value of fnancal asses, consequenly, are expecaons under P. For furher deals, refer o Harrson and Krebs (1979), Harrson and Plska (1981), Delbaen and Schachermayer (1995a,b), Duffe (1996), Hull (1999). If he resdual Z relaes o he value of a secury, one s ofen neresed n s behavor under P raher han under P. Specfcally we shall see ha one s neresed n Z MG, where hs s he marngale par n he Doob-Meyer decomposon (4.1), when aken under P ; Z = Z + Z DR + Z MG w.r.. P. (4.2) The quadrac varaon < Z, Z >=< Z MG, Z MG >=< Z MG, Z MG > s he same under P and P, bu under he laer dsrbuon, refers o he behavor of Z MG raher han Z MG. A smple example follows from he movang applcaon n he Inroducon.

23 ANOVA FOR DIFFUSIONS 23 Suppose ha Ξ and S are boh dscouned secures prces, and ha one seeks o offse rsk n Ξ by holdng ρ uns of S. The resdual s hen, self, he dscouned value of he unhedged par of Ξ. Under P, herefore, Z s a marngale, Z = Z + Z MG. A deeper example s encounered n Zhang (21), where we analyze mpled volales. In boh hese cases, n order o pu a value of he rsk nvolved n Z MG, one s neresed n bounds on he quadrac varaon < Z, Z >, under P. Ths wll help, for example, n prcng spread opons on Z. How do our resuls for probably P relae o P? They smply carry over, unchanged, o hs probably dsrbuon. Theorems 1 and 2 reman vald by absolue connuy of P under P. In he case of lmng resuls, such as hose n Proposons 1 and 3 (n Secon 6) and he developmen for goodness of f n Secon 4, we also nvoke he mode of sable convergence (Secon 2.8) ogeher wh he fac ha dp /dp s measurable wh respec o he underlyng σ-feld F T. Fnally, f one wans o es a null hypohess H ha Z MG s consan, hen H s equvalen o askng wheher < Z, Z > T s zero (wheher under P or P ). Ths can agan be answered wh our dsrbuonal resuls above. In he case of he example n hs Secon, he H of fully offseng he rsk n Ξ also ess wheher Z self s consan. 5. Concluson. Ths paper provdes a mehodology o analyze he assocaon beween Io processes. Under he framework of nonparamerc, one-facor regresson, we oban he dsrbuons of esmaors of resdual varaon < Z, Z >. We hen use hs n a varey of measures of goodness of f. We also show how he mehod yelds a procedure o es he appropraeness of a paramerc regresson model. The lmng dsrbuons denfy wo sources of uncerany, one from he dscree naure of he daa process, he oher from he esmaon procedure. Ineresngly, among he class of esmaors < Z, Z > (α) under consderaon, α [, 1], dscreeme samplng only mpacs he varance componen. On he oher hand, dfferen esmaon schemes lead o dfferen bases n he asympocs. ANOVA for dffusons perms nference over a me nerval. Ths s because he error erms n he quadrac varaon < Z, Z > (α) of resduals, and hence he error erms n he goodness of f measures, converge as a process, whereas he errors n he esmaed regresson parameers ˆρ are asympocally ndependen from one me pon o he nex. Ths feaure of me aggregaon makes ANOVA a naural procedure o deermne he adequacy of an adoped model. Also, he ANOVA s beer posed n ha he rae of he convergence s he square of he rae for ˆρ ρ.

24 24 P.A. MYKLAND AND L. ZHANG The ANOVA for dffusons approach s appealng also from he poson of applcaons. As long as one can collec he daa as a process, one can rely on he proposed ANOVA mehodology o draw nference whou mposng paramerc srucure on he underlyng process. In fnancal applcaons, as n Secon 4.5, can es wheher a fnancal dervave can be fully hedged n anoher asse. In he even of non-perfec hedgng, Theorems 1-2 ell us how o quanfy he amoun of hedgng error as well as s dsrbuon. 6. Convergence n law Proofs and furher resuls. In he followng, we deal wh processes ha are exemplfed by [Z, Z] < Z, Z >. We mosly follow Jacod and Proer (1998). Proof of Proposon 1. The applcable pars of he proof of he ced Theorems 5.1 and 5.5 of Jacod and Proer (1998) carry over drecly under Assumpon A()- (). When modfyng he proofs, as approprae; = max( (n) ) replaces [n]/n, δ n, replaces n 1, and so on. For example, he rgh hand sde of her equaon (5.1) (p. 29) becomes Kδn. 2 The man change due o he non-equdsan case occurs n par () of Jacod and Proer s Lemma 5.3 (p ), where n he defnon of α n, 2 Bk r should be replaced by (H( r + ) H( r ))Br k. Assumpon A()-() s clearly suffcen. Noe ha he resul exends n an obvous fashon o he case of muldmensonal Z = (Z (1),..., Z (p) ). Also, nsead of sudyng [Z, Z] < Z, Z >, one can, lke Jacod and Proer (1998), sae he (more general) resul for (Z() u Z () )dz u (j). In he sequel, we shall also be usng a rangular array form of Proposon 1, cf. he end of he proofs of boh Theorems 1 and 2. Proposon 3. (Trangular array verson of he dscrezaon heorem). Le Z be a vecor Io process for whch T < Z, Z > 2 d < and T Z 2 d < a.s. Also suppose ha Z (n), = 1, 2,... are Io processes sasfyng he same requremen, unformly. Suppose ha he (vecor) Brownan moon W s he same n he Io process represenaons of Z and of all he Z (n),.e., dz (n),mg u = σ (n) u dw and dz MG u = σ u dw. (6.3) Suppose ha T σ u (n) 4 σ u du = op (1). (6.4)

25 ANOVA FOR DIFFUSIONS 25 1 Then, subjec o Assumpon A()-(), he processes (n) 1 converge jonly wh he processes (n) (Z() u Z () (Z(,n) u )dz (j) u Z (,n) )dz u (j,n) o he same lm. If one requres sable convergence, one jus mposes Sysem Assumpon III, cf. Theorem 11.2 (pp. 338) and Theorem 15.2(c) (p. 496) of Jacod (1979). Proof of Proposon 3. Ths s manly a modfcaon of he developmen on p. 292 and he begnnng of p. 293 n Jacod and Proer (1998), and he furher developmen n (her) Theorem 5.5 s sraghforward. Agan we recollec ha H (from Assumpon A()-()) s Lpschz connuous. Noe ha o mach he end of he proof of Theorem 5.1, we really need o p (δ 4 n). Ths, of course, follows by approprae use of subsequences of subsequences. Fnally, we show he resul for Secon 3.4. Proof of Proposon 2. Le be he larges grd pon smaller or equal o. Then by Io s Lemma, d(z Z ) 4 = 4(Z Z ) 3 dz + 6(Z Z ) 2 d < Z, Z >. Hence, [Z, Z, Z, Z] = [ ] 4 (Z u Z ) 3 dz u + 6 (Z u Z ) 2 d < Z, Z > u = 6 usng Lemma 2 below. +1 (Z u Z ) 2 d < Z, Z > +O p (δ (n) 3 2 ɛ ), (6.5) Defne an nerpolaed verson of [Z, Z] = [Z, Z] G by leng [Z, Z] nerpol Z ) 2 + [Z, Z]. Then, agan by Io s Lemma, +1 (Z u Z ) 2 d < Z, Z > = (Z = 1 4 < ([Z, Z]nerpol < Z, Z >), ([Z, Z] nerpol < Z, Z >) > (6.6) Pung ogeher (6.5)-(6.6), Proposon 1, as well as Corollary VI.6.3 (p. 385) of Jacod and Shryaev (23), we oban (3.21). Replacng Z by Ẑ, and creang nerpolaed versons of Ẑ and [Ẑ, Ẑ] as a he begnnng of he proof of Theorem 1 below, (3.2) also follows.

26 26 P.A. MYKLAND AND L. ZHANG 7. Proofs of man resuls Noaon. In he followng proofs, always means (n), h always means h n, means (n), ec. Also, s he larges grd pon less han,.e., = max{ (n) }. We somemes wre < X, X > as < X >, and < X, X > as < X > for smplcy. Also, for convenence, we adop he followng shorhand for smoohness assumpons for Io processes: Assumpon B (Smoohness): B.(X): X s n C 1 [, T ]. B.1(X, Y ): < X, Y > s n C 1 [, T ]. B.2(X): he drf par of X (X DR ) s n C 1 [, T ]. Assumpon B(X) s equvalen o B.1(X, X) and B.2(X). We shall also be usng he followng noaons Υ X (h) = sup X u X s and Υ XY (h) = sup < X, Y > u < X, Y > s, u,s u,s where sup u,s means sup u,s [,T ]: u s h Prelmnary lemmas, and Proof of Theorem 1. Lemma 1. Le M,n (), T, = 1,, k n, k n = O( 1 ), be a collecon of connuous local marngales. Suppose ha sup 1 kn < M,n, M,n > T = O p ( β ). Then, for any ɛ >, sup 1 kn sup T M,n () = O p ( β 2 ɛ ). The above follows from Lenglar s nequaly. As a corollary, he followng s rue. Lemma 2. Le X and Y be Io processes. If X sasfes ha X and < X, X > are bounded a.s., hen for any ɛ >, Υ X (η) = O p ((η + ) 1 2 ɛ ). Smlarly, f X and Y sasfy ha D XY and < R XY, R XY > are bounded (a.s.), hen for any ɛ >, Υ XY (η) = O p ((η + ) 1 2 ɛ). Lemma 3. Suppose X, Y and Z are Io processes, and make assumpons A(), B.(Y ), B[(X), (Z)], and B.1(X, Z). Also, assume ha for any ɛ >, as n, ( ) 1/2 ɛ h 1/2 = o(1). Then sup 1 h 2 h < (X u X )(Z u Z )dy u ( 2 ) < X, Z > 2 Ỹ ] = o p( h ) +1 [ as n, where Ỹu = dy u /du.

27 ANOVA FOR DIFFUSIONS 27 Proof of Lemma 3. Whou loss of generaly, s enough o show he resul for X = Z. Ths s because one can prove he resuls for X, Z and X + Z, and hen proceed va he polarzaon deny. The condons mposed also means ha he assumpons of Lemma 3 are also sasfed for X + Z. Le I = 1 [+1 h 2 (< X > u < X > )dy u 1 ] 2 < X > Ỹ ( ) 2 II = 1 h 2 h < +1 h < [ (Xu X ) 2 (< X > u < X > ) ] dy u Obvously sup I =o p ( h 1 ). I s hen enough o show sup II =o p ( h 1 ), as follows. By Iô s Lemma, Recall ha dxv DR ges II = 2 h 2 h < +1 u (X v X )dx DR v dy u }{{} II,1 +1 u [ h < + 2 h 2 II,1 sup X u sup Ỹu 2 u u h 2 (X v X )dx MG v ]dy u. } {{ } II,2 = X v dv, hen by B.(Y ), B.2(X) and he connuy of X, one h < +1 u ( h < For II,2, usng negraon by pars, 1 +1 u h 2 [ (X v X )dxv MG = 1 h 2 h < +1 (7.1) has quadrac varaon bounded by 1 h 4 sup h < 1 h 4 sup +1 < X > sup ]du X v X dv)du = o p ( δ(n) h ) (X u X )[( ) (u )]dx MG u. (7.1) (X u X ) 2 ( +1 u) 2 d < X > u (7.2) h < +1 (Υ X (δ (n) )) 2 ( +1 u) 2 du=o p ( 3 ɛ h 3 )

28 28 P.A. MYKLAND AND L. ZHANG by Lemma 2 under Assumpon A(), B(X). Followng Lemma 1 and B.(Y ), sup II,2 = O p ( 3/2 ɛ h 3/2 ). The resul hen follows. Defnon 5. Suppose X and Y are connuous Io processes. Le and B XY { 1 1,, = h h (( h) u)d < X, Y > u h oherwse B XY { [2] j+1 2,, = h h j j+1 j (X s X j )dy s h oherwse (7.3) (7.4) where [2] ndcaes symmerc represenaon s.. [2] XdY = XdY + Y dx. Noe ha by negraon by pars va Io s Lemma, B1,, XY = 1 h (< X, Y > < X, Y > h) < X, Y >, and hence < X, Y > < X, Y > = B1,, XY + B2,, XY. Lemma 4. Under Assumpons A(), B(X, Y, < X, Y > ), and he order selecon of h 2 = O( ) for any ɛ >, In parcular, sup B XY 1,, = Op ( 1 4 ɛ ) and sup B XY 2,, = Op ( 1 4 ɛ ). T T sup < X, Y > < X, Y > = O p ( 1 4 ɛ ). T In addon, under condon B(< R XY, R ZV > ), sup < BXY 1,, B1, ZV > h 3 < RXY, R ZV > = O p (h 3/2 ɛ ), (7.5) sup < B2, XY, B2, ZV > G = o p ( 1 2 ), (7.6) where G = 1 h h 2 < (< X, Z > < Y, V > +< X, V > < Y, Z > )( ) 2, and < sup B ZV 1,, BXY 2, > = Op ( ) h (7.7)

29 ANOVA FOR DIFFUSIONS 29 Proof of Lemma 4. Usng Defnon 2, wre B XY 1,, = 1 (( h) u)dru XY + 1 (( h) u)ddu XY h h h }{{} h }{{} B XY,MG 1,, B XY,DR 1,, (7.8) Under assumpon B.2(< X, Y > ), sup B XY,DR 1,, = O p (h). Also, by assumpons A() and B.1(R XY, R XY ), we have ha sup u T < R XY > u = O p (1), whence sup < B1, XY, B1, XY > T = O p ( (n) ) 1 2 ). So sup sup B XY,MG = O p ( 1 4 ɛ ) by Lemma 1, and hence sup B XY 1, = Op ( 1 4 ɛ ). By smlar mehods, Lemmas 1-2 can be used o show ha sup B XY 2, = O p ( 1 4 ɛ ). The orders (7.5)-(7.7) follow from he represenaon (7.8), and dervaons smlar o hose of he proof of Lemma 3. 1,, The followng s now mmedae from Lemma 4, by Taylor expanson. Corollary 1. Suppose Ξ, S, and ρ are Io processes, where ρ and ˆρ are as defned n Secon 2. Then, under condons A(), B(Ξ, S, < Ξ, S >, < S, S > ), and (2.7), for any ɛ >, we have sup ˆρ ρ = O p ( 1 4 ɛ ) [,T ] In he res of hs subsecon, we shall se, by convenon, ˆρ o he value ˆρ, even f he defnon n Secon 2.4 perms oher values of ˆρ beween samplng pons. Ths s no conradcon snce we only use he ˆρ n our defnon of Ẑ and n he res of our developmen. Smlarly, we exend he defnon of Ẑ gven a he begnnng of Secon 2.5 o he case when s no a samplng pon. Specfcally Ẑ = Ẑ + Ξ Ξ ˆρ (S S ) = Ẑ + Ξ Ξ ˆρ u ds u by our convenon for ˆρ. We emphasze ha Ẑ s no more observable han S or Ξ. By hs defnon of Ẑ, n vew of (1.6) and (1.5), < Ẑ, Ẑ > =< Z, Z > + (ˆρ u ρ u ) 2 d < S, S > u. (7.9)

30 3 P.A. MYKLAND AND L. ZHANG We hen oban he followng prelmnary resul for Theorem 1. Proposon 4. Under he condons of Theorem 1, (< Ẑ, Ẑ > < Z, Z > ) / = D + o p (1), unformly n, where D s gven by (3.15). Proof of Proposon 4. () Le L,n () = 2 j=1 L j,,n(), where for j = 1, 2, L j,,n () = (Bj, ΞS ρ hbj, SS )/ < S, S > h for h, and zero oherwse. I now easly seen from Lemmas 2 and 4, and Corollary 1, and by he same mehods as n hese resuls, ha L,n () approxmaes ˆρ ρ, and n parcular ha (ˆρ u ρ u ) 2 d < S, S > u = unformly n, for any ɛ >. We now show ha L 2,n( ) < S, S > h = L 2,n( ) < S, S > h + O p ( 3 4 ɛ ) (7.1) To hs end, se Y () () = L 2,n () < L,n >, and Z n, = < L,n > < S, S > h + O p ( 3 4 ɛ ) (7.11) Y () ( ) < S, S > h + Y () ( ) < S, S > h. (7.12) By Lenglar s Inequaly (Jacod and Shryaev (23) Lemma I.3.3, p.35), (7.11) follows f one can show ha < Z n, Z n > T = O p ( 3 2 ɛ ), for any ɛ >. Ths s wha we do n he res of (). Snce < L,n, L j,n > = f ( h, ] and ( j h, j ] are dsjon for any T and j T, he quadrac varaon of he Z n s consdered n he overlappng me nerval. < Z n, Z n > T sup T(< S, S > )2 ( )( j ) j ( h, ] ( j h, j ] 2 sup S, S > T(< )2 ( )( j )I {<j: > j h}( j ( ( j h, ] d < Y (j) > u ) 1 2 ( j h, ] d < Y (), Y (j) > u d < Y () > u ) 1 2 (7.13) The above lne follows from Kuna-Waanabe Inequaly (p. 69 of Proer (24)).