Sochasic Processes ad heir Alicaios 119 29 1845 1865 www.elsevier.co/locae/sa Power variaio for Gaussia rocesses wih saioary icrees Ole E. Bardorff-Nielse a, José Mauel Corcuera b, Mark Podolskij c, a Deare of Maheaical Scieces, Uiversiy of Aarhus, Ny Mukegade, DK-8 Aarhus C, Deark b Uiversia de Barceloa, Gra Via de les Cors Caalaes 585, 87 Barceloa, Sai c CREATES, School of Ecooics ad Maagee, Uiversiy of Aarhus, Buildig 1322, DK-8 Aarhus C, Deark Received 23 Noveber 27; received i revised for 15 July 28; acceed 8 Seeber 28 Available olie 16 Seeber 28 Absrac We develo he asyoic heory for he realised ower variaio of he rocesses X = φ G, where G is a Gaussia rocess wih saioary icrees. More secifically, uder soe ild assuios o he variace fucio of he icrees of G ad cerai regulariy codiios o he ah of he rocess φ we rove he covergece i robabiliy for he roerly oralised realised ower variaio. Moreover, uder a furher assuio o he Hölder idex of he ah of φ, we show a associaed sable ceral lii heore. The ai ool is a geeral ceral lii heore, due esseially o Hu ad Nualar [Y. Hu, D. Nualar, Reoralized self-iersecio local ie for fracioal Browia oio, A. Probab. 33 25 948 983], Nualar ad Peccai [D. Nualar, G. Peccai, Ceral lii heores for sequeces of ulile sochasic iegrals, A. Probab. 33 25 177 193] ad Peccai ad Tudor [G. Peccai, C.A. Tudor, Gaussia liis for vecor-valued ulile sochasic iegrals, i: M. Eery, M. Ledoux, M. Yor Eds., Seiaire de Probabilies XXXVIII, i: Lecure Noes i Mah, vol. 1857, Sriger-Verlag, Berli, 25,. 247 262], for sequeces of rado variables which adi a chaos rereseaio. c 28 Elsevier B.V. All righs reserved. MSC: riary 6G15; 6F17; secodary 62G2 Keywords: Ceral lii heore; Chaos exasio; Gaussia rocesses; High-frequecy daa; Mulile Wieer Iô iegrals; Power variaio We hak Giovai Peccai ad Sved-Erik Graverse for helful coes ad suggesios. Ole E. Bardorff- Nielse ad Mark Podolskij ackowledge fiacial suor fro CREATES fuded by he Daish Naioal Research Foudaio, ad fro Thiele Cere. The work of José Mauel Corcuera is suored by he MEC Gra No. MTM26-3211. Corresodig auhor. Fax: +45 492343214559. E-ail addresses: oeb@if.au.dk O.E. Bardorff-Nielse, jcorcuera@ub.edu J.M. Corcuera, odolskij@creaes.au.dk M. Podolskij. 34-4149/$ - see fro aer c 28 Elsevier B.V. All righs reserved. doi:1.116/j.sa.28.9.4
1846 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1. Iroducio This aer esablishes resuls o covergece i robabiliy ad i law sably for roerly oralised realised ower variaios of rocesses of he for X = φ G. Here G is a Gaussia rocess wih saioary icrees whose icrees have a variace fucio ha saisfies cerai regulariy codiios, ad G ad he rocess φ are defied o oe ad he sae filered robabiliy sace. The secial case of φ G where φ is a cosa ad G iself is saioary was reaed i early aers [17,23]. I geeral, rocesses of ye φ G are o seiarigales ad he roofs of he lii resuls use he heory of isooral rocesses ad echiques develoed i [15] for derivig siilar lii resuls for rocesses φ B H where B H deoes fracioal Browia oio. For Iô seiarigales, boh oe- ad uli-diesioal, a exesive heory of realised ower ad uliower variaios is available. For discussios of his heory ad is alicaios, see [2,3,5 1,19,21,32,33]. Geeral ad shar crieria whe a rocess of he for φ G is a seiarigale are available i [11,12]. Secio 2 ses u he roble ad exelifies he kids of rocesses G o which he heory alies, ad he covergece i robabiliy ad ceral lii resuls for rocesses φ G are give i Secios 3 ad 5, resecively. Secio 4 derives a ulivariae ceral lii heore via chaos exasios which should be of wide geeral ieres. I aricular i covers i-fill asyoics or riagular array schees. The ai buildig blocks i he heore are coaied i he rece aers [18,27,29]. The cocludig Secio 6 idicaes lies for furher research. Mos of he roofs are relegaed o a Aedix. 2. The seig We sar wih a Gaussia rocess G defied o a filered colee robabiliy sace Ω, F, F, P, which has ceered ad saioary icrees. We defie R as he variace fucio of he icrees of G, i.e. R = E[ G s+ G s 2 ],. 2.1 I his aer we cosider a rocess of he for X = X + φ s dg s, defied o he sae robabiliy sace as G, which is assued o be observed a ie ois i/, i =, 1,..., []. We are ieresed i he asyoic behaviour of he roerly oralised realised ower variaio [] i X, wih i X = X i X i 1, for >. Before we roceed wih he asyoic resuls for he fucioals defied i 2.3 we eed o esure ha he iegral i 2.2 is well-defied i a suiable sese. For his urose we use he coce of a ahwise Riea Sieljes iegral. Recall ha for a real-valued fucio f : [, ] R he r-variaio is defied as var r f ; [, ] = su π 2.2 2.3 1/r f i f i 1 r, 2.4
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1847 where he sureu is ake over all ariios π = { = < 1 < < = }. Trivially, whe f is α-hölder coiuous i has fiie 1/α-variaio o ay coac ierval. I his case we se f α = f s f u su s<u s u α. 2.5 I [34] i is show ha he Riea Sieljes iegral f sdgs exiss if f ad g have fiie q-variaio ad r-variaio, resecively, i he ierval [, ], where 1/r + 1/q > 1, ad hese fucios have o coo discoiuiies. I order o give a saee abou r-variaio of he Gaussia rocess G we require he followig assuio o he behaviour of he fucio R defied i 2.1. A1 R = β L for soe β, 2 ad soe osiive slowly varyig a fucio L, which is coiuous o,. Recall ha a fucio L :, R is called slowly varyig a whe he ideiy Lx li x Lx = 1 2.6 holds for ay fixed >. Provided L is coiuous o,, we have Lx Cx α, x, ] 2.7 for ay α > ad ay > where he cosa C > deeds o α ad. See [14] Page 16 for siilar roeries of slowly varyig fucios a. Assuio A1 ilies he ideiy E[ G G s 2 ] = s β L s, 2.8 fro which we deduce by 2.7 ha he rajecories of G are β/2 ɛ-hölder coiuous alos surely for ay ɛ, β/2. Clearly, G has fiie r-variaio for ay r > 2/β ad var r G; [a, b] C b a 1/r a.s. 2.9 for ay a < b ad for soe cosa C which deeds o ad r. Cosequely, he iegral i 2.2 is well-defied as a ahwise Riea Sieljes iegral for ay sochasic rocess φ of fiie q-variaio wih q < 1 β/2 1. I he followig we sudy he asyoic roeries of he rocess V X, = 1 τ [] i X, 2.1 where τ 2 = R 1 = E[ i G 2 ] ad >. 3. Covergece i robabiliy I his secio we rove he covergece i robabiliy for he quaiy V X,. For his urose we require he followig addiioal assuios o he variace fucio R: A2 R = β 2 L 2 for soe slowly varyig fucio L 2, which is coiuous o,. A3 There exiss b, 1 wih K = li su su L 2 y x L x <. y [x,x b ]
1848 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 We sar wih rovig he weak law of large ubers for he sequece V G,. Throughou his aer we wrie Y uc Y whe su [,T ] Y P Y for ay T >. Proosiio 1. Assue ha codiios A1 A3 are saisfied. The we have V G, uc µ, where µ = E[ U ], U N, 1. 3.1 Proof. See he Aedix. Reark 1. The raher echical codiio A3 ca be relaced by he followig weaker assuio: R j+1 j 1 + R 2R j 2R 1 r j, 1 r 2 j, 3.2 for soe sequece r j see Lea 1 ad he roof of Proosiio 1 i he Aedix. The ai resul of his secio is he followig heore. Theore 2. Assue ha codiios A1 A3 are saisfied ad he rocess φ has fiie q-variaio wih q < 1 β/2 1. The we have V X, uc µ φ s ds. 3.3 Proof. See he Aedix. Exale 3 Cauchy Class. For odellig uroses, a ieresig ad flexible class of rocesses is give by G α,γ, where he G α,γ s are saioary ceered Gaussia rocesses wih variace 1 ad auocorrelaio fucio h = 1 + α γ /α. Here he araeers have o saisfy α, 2] ad γ > see [16]. Wih h = 1 h we fid, for >, wih Furher, ad h = α L L = 1 + α γ /α 1 α 1 + α γ /α. h = γ α 1 1 + α γ /α 1 h = α 2 L 2, L 2 = γ γ + 1 α α 11 + α γ /α 2.
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1849 Boh L ad L 2 are slowly varyig. Now, L 2 = αγ α 1 1 + α γ /α 3 [ γ + 1 + α 1 γ /α + 2 γ + 1γ /α + 1 α] showig ha L 2 is decreasig or icreasig i a eighbourhood of deedig o wheher α is greaer or saller ha c γ, where c γ deoes he osiive roo of he equaio γ + 1 + α 1 γ /α + 2 =. I ay case, su L 2 y L x α 1 α <, y [x,x b ] as x, for ay b, 1. Thus codiios A1 A3 are fulfilled for ay α, 2 ad γ >, ad Proosiio 1 ad Theore 2 aly o he class G α,γ α,2,γ >. 4. A geeral ulivariae ceral lii heore via chaos exasio I his secio we rese a ulivariae ceral lii heore for a sequece of rado variables which adi a chaos rereseaio. This resul is based o he heory for ulile sochasic iegrals develoed i [27,29,18] ad i aears ilicily i [15]. The ceral lii heore will be used o show he weak covergece of he rocess V G,. However, he lii resuls of his secio igh be of ieres for ay oher alicaios. Le us recall he basic oios of he heory of ulile sochasic iegrals. Cosider a searable Hilber sace H. For ay 1, we defie H o be he h esor roduc of H ad we wrie H for he h syeric esor roduc of H, which is edowed wih he odified or! H. A ceered Gaussia faily B = {Bh h H}, defied o he robabiliy sace Ω, F, P, is called a isooral rocess o H whe E[BhBg] = h, g H, h, g H. I his secio we assue ha F is geeraed by B. For 1, we deoe by H he h Wieer chaos associaed wih B, i.e. he closed subsace of L 2 Ω, F, P geeraed by he rado variables H Bh, where h H wih h H = 1 ad H is he h Herie olyoial. Recall ha he Herie olyoials H are defied as follows: H x = 1, H x = 1 e x2 2 d x2 e 2 dx, 1. The firs hree Herie olyoials are H 1 x = x, H 2 x = x 2 1 ad H 3 x = x 3 3x. By I we deoe he liear isoery bewee he syeric esor roduc H, equied wih he or! H, ad he h Wieer chaos ha is defied by I h = H Bh see, for isace, Chaer 1 i [26] for ore deails. For ay h = h 1 h ad g = g 1 g H, we defie he h coracio of h ad g, deoed by h g, as he elee of H 2 give by h g = h +1, g 1 H h, g H h 1 h g +1 g. This ca be exeded by lieariy o ay elee of H. Noe ha if h ad g belog o H, h g does o ecessarily belog o H 2. For ay h = h 1 h H, we deoe
185 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 by h H he syerizaio of h, i.e. h = 1 h ζ1 h ζ,! ζ S where S is he grou of eruaios of {1,..., }. Moreover, we wrie h g for he syerizaio of h g. Now, we rese a ulivariae ceral lii heore which is a sraighforward cosequece of Theore 1 ad Proosiio 2 i [29] ad he roofs herei. Theore 4. Cosider a collecio of aural ubers 1 2 d ad a collecio of elees { f 1,..., f d 1} such ha f k H k ad he followig codiios are saisfied: 1 For ay k, l = 1,..., d we have cosas C kl such ha li k! f k 2 H k = C kk, li E[I k f k I l f l ] = C kl, k l, ad he arix C = C kl 1 k,l d is osiive defiie. 2 For every k = 1,..., d we have li f k f k 2 H 2 k = for ay = 1,..., k 1. The we obai he ceral lii heore T I 1 f 1,..., I d f d D Nd, C. 4.1 Noice ha C kl i 1 of Theore 4 is equal o whe k l, because I k ad I l are orhogoal by cosrucio. Fially, we cosider a d-diesioal rocess Y = Y 1,..., Y dt, defied o Ω, F, P, which has a chaos rereseaio Y k = =1 I f, k, k = 1,..., d, 4.2 wih f k, H. Noice ha EY =. The followig resul rovides a ceral lii heore for he sequece Y. Theore 5. Suose ha he followig codiios hold: i For ay k = 1,..., d we have li li su! f, k N 2 H =. =N+1
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1851 ii For ay 1, k, l = 1,..., d we have cosas Ckl such ha li! f k, 2 H = Ckk, li E[I f k, I f, l ] = C kl, k l, ad he arix C = C kl 1 k,l d is osiive defiie for all. iii =1 C = C R d d. iv For ay 1, k = 1,..., d ad = 1,..., 1 The we have Y li f k, f, k 2 H 2 =. D Nd, C. 4.3 Proof. Defie he rucaed rado variable Y,N = Y 1,N,..., Y d,n T by Y k,n = N =1 I f, k, k = 1,..., d. Sice I 1 ad I 2 are orhogoal whe 1 2, Theore 4 ilies uder codiios ii ad iv of Theore 5 ha D N ξn N d, C Y,N =1 for a fixed N. By assuio iii we obai he covergece i disribuio ξ N D ξ Nd, C as N. Fially, codiio i ad he Markov iequaliy ily li li su P Y,N Y δ = N for ay δ > here deoes he axiu or. By sadard argues we obai he desired resul. 5. A sable ceral lii heore for ower variaio Firs, we rese a fucioal ceral lii heore for he sequece V G,. I he followig discussio we use he oaio Hx = x µ. Noice ha he fucio H has he rereseaio Hx = a j H j x, j=2 where a 2 > ad H j j are Herie olyoials. Uder a resricio o he araeer β we obai he followig resul. 5.1 5.2
1852 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 Theore 6. Assue ha codiios A1 A3 hold ad < β < 2 3. The we obai he weak covergece i he sace D[, T ] 2 equied wih he Skorohod oology G, V G, µ G, τ W, 5.3 where W is a Browia oio ha is defied o a exesio of he filered robabiliy sace Ω, F, F, P ad is ideede of F, ad τ 2 is give by l 1 τ 2 = j!a 2 β j λ2 j, λ2 j = 1 + 2 2l β + l + 1 β j 2 j. 5.4 j=2 l=1 Proof. See he Aedix. Reark 2. Theore 6 alies o he Cauchy class G α,γ wih γ > ad α, 3/2 of Gaussia rocesses ha has bee iroduced i Exale 3. The roof of Theore 6 relies o he ceral lii heore reseed i Theore 5. I [15] he resul of Theore 6 is show wih he sae lii for he case of fracioal Browia oio wih Hurs araeer H, 1; see also [24] he araeer β corresods o 2H. Their derivaio relies o he self-siilariy of he fracioal Browia oio. The asyoic heory reseed i his secio rovides a aural exesio of heir work o geeral Gaussia rocesses wih saioary icrees. Reark 3. i A ceral lii heore for he quaiy V G, 1 i.e. for = 1 was origially roved i [17] uder assuios A1 A3. For Theore 6 he echical codiio A3 ca be relaced by he weaker assuio: R j+1 j 1 + R 2R j 2R 1 r j, r 2 j <, 5.5 for soe sequece r j. Noice ha 5.5 ilies codiio 3.2 i Reark 1 wih r j = r j for all j 1. See Lea 1 ad he roof of Theore 6 i he Aedix for ore deails. ii Furherore, i [17] i is show ha he lii of he secod cooe i 5.3 is a elee of he secod Wieer chaos whe G is a saioary Gaussia rocess wih EG =, EG 2 = 1, ad 3 2 < β < 2. For he covariace fucio R = E[G s G s+ ] he auhors assued he followig codiios: 1 R saisfies A1, R saisfies A2 wih L 2 x = β1 βl x1 + o1 ear, R is decreasig ear ad A3 holds. Uder hese assuios hey have roved he covergece 2 β L 1 V G, 1 µ D µ 4 I 2, where I 2 is he Wieer Iô iegral e ix1+x2 1 I 2 = f 1 2 x1 f 1 2 x2 W dx 1 W dx 2, R 2 ix 1 + x 2 W is a Browia oio ad f is give by f x = e ix R d. R
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1853 See [23] for a fucioal ceral lii heore for he case 3 2 < β < 2. If β = 3 2 boh liis ca aear: whe R is iegrable ear we obai a elee of he secod Wieer chaos i he lii, whereas he lii is oral whe R is o iegrable ear alhough he covergece rae chages. Noice ha he weak covergece i 5.3 is equivale o he sable covergece i D[, T ] 2 V G, µ F G s τ W, 5.6 where F G deoes he σ -algebra geeraed by he rocess G see [1,2] or [3] for ore deails o sable covergece. The laer resul is crucial for rovig a fucioal ceral lii heore for he sequece V X, for F G -easurable rocesses φ. Theore 7. Suose ha φ is F G -easurable ad has Hölder coiuous rajecories of order a > 1/2 1. Whe < β < 3 2 ad assuios A1 A3 hold we obai he sable covergece V X, µ φ s F ds G s τ φ s dw s 5.7 i he sace D[, T ] 2. Proof. See he Aedix. Reark 4. Noice ha if φ = f G for soe sooh fucio f, he codiios of Theore 7 ily ha > 1/β ad β 1, 3 2. This leads o a serious resricio o he araeers ad β. O he oher had, Theore 7 reais valid whe he rocess φ is ideede of G his follows fro Theore 6 if we relace he rocess G by φ. I his case we oly require he codiio a > 1/2 1. Alyig he roeries of sable covergece we ca obai a feasible versio of Theore 7. Sice V X, 2 P µ 2 s ds, we deduce he followig resul. φ2 Corollary 1. For ay fixed >, we have V X, µ φ s ds F G s U, µ 1 2 τ 2 V X, 2 where U is ideede of F ad U N, 1. 6. Coclusio The resuls derived i he rese aer cosiue a aural exesio of earlier work o ower variaio, as idicaed i he Iroducio. The ossibiliy of furher exesio o biower, ad ore geerally uliower, variaios is uder cosideraio. Fro aoher oi of view, he resuls rovide a se i a larger rojec ha ai o develo robabilisic ad ifereial rocedures for he sudy of volailiy odulaed Volerra rocesses, as defied i [4]. Moreover, Theore 7 ca be alied o esiae he araeer β, 3/2 see e.g. [22] for he
1854 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 cosrucio of iiax esiaors usig ower variaios. We hik ha sable ceral lii heores ca be obaied for all β, 2 by cosiderig ower fucioals of he secod order icrees of X see [24] for he case of he fracioal Browia oio. Fially, closer liks o Malliavi calculus, cf. [25] or [28], offer exciig rosecs. Aedix I he followig we deoe all cosas which do o deed o by C. Throughou his secio we use he oaio r j = Cov 1 G, 1+ j G, j. A.1 τ τ By he riagular ideiy we kow ha r j = R j+1 + R j 1 2R j 2R 1, j 1, A.2 where he fucio R is give by 2.1. Firs, le us rove he followig echical lea which exeds Leas 2 ad 3 i [17]. Lea 1. Suose ha codiios A1 A3 hold. Le ɛ > wih ɛ < 2 β. Defie he sequece r j by r j = j 1 β+ɛ 2, j 2, A.3 ad r = r1 = 1. The we obai he followig asserios: i I holds ha 1 r 2 j. If, oreover, β + ɛ 2 < 2 1 i holds ha r 2 j <. ii For ay < ɛ < 2 β fro A.3 here exiss a aural uber ɛ such ha r j Cr j, j for all ɛ. iii Se ρ = 1 ad ρ j = 2 1 j 1 β 2 j β + j + 1 β for j 1. The i holds ha r j ρ j for ay j. iv For < β < 3 2 ad ay l 2 we have ha 1 r l j ρ l j. Proof of Lea 1. Par i of Lea 1 is rivial. By assuios A1 ad A2 we deduce he ideiies
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1855 r 1 = 1 + 2 β 1 L 2 L 1, r j = 1 j 2 + θ β 2 j L2 j+θ j L 1, j 2, where θ j are soe real ubers wih θ j < 1. Recall assuio A3 ad se a = 1 b, 1. Whe is large eough we have r 1 < 1 because L is a slowly varyig fucio ad β, 2 ad for 2 j [ a ] we obai r j C j 1 β 2 by assuio A3. For [ a ] j we obai by 2.7 he followig aroxiaio r j + 1 1 2 j β 2 L 2 j+θ j L 1 j β 2+ɛ aɛ L 2 j+θ j L 1 C j β 2+ɛ. Thus, asserio ii follows. Nex, by assuio A1 ad A.2 we obai he forula r j = j 1β L j 1 2 j β L j + j + 1β L j+1 2L 1, j 1. We ca readily deduce ar iii, because he fucio L is slowly varyig. Nex, assue ha < β < 3 2. We use < ɛ < 3 2 β i defiiio A.3. Sice β+ɛ 2 < 1 2, we deduce ha r l j < for ay l 2, by ar i. By ars ii, iii ad he doiaed covergece heore we obai iv, ad he roof is colee. Proof of Proosiio 1. We firs show he oiwise covergece V G, he ideiy E[H k U 1 H l U 2 ] = δ k,l ρ l l!, U 1, U 2 N, where δ k,l deoes he Kroecker sybol. For ay > we have E[V G, ] = µ + O 1 ad by A.4, 5.1 ad 5.2 we obai he ideiy VarV G, = µ 2 µ 2 [] 2 + 2 [] 1 1 2 [] jcov G = l!al 2 b l + O 1, l=2 P µ. Recall 1 ρ, A.4 ρ 1 τ A.5 1+ j, G. τ
1856 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 where he coefficies a l are give by 5.2 ad he cosas b l are defied by b l = 2 [] 1 2 [] jr l j. By i ad ii of Lea 1 we deduce for ha A.6 b l 2 [] 1 r l j 2 [] 1 r 2 j, for ay l 2. This ilies he oiwise covergece V G, P µ. A.7 The uc covergece follows iediaely, because V G, is icreasig i ad he lii rocess µ is coiuous. Proof of Theore 2. The basic idea behid he roof of Theore 2 is he aroxiaio of he rocess φ by a sequece of se fucios ad he alicaio of Proosiio 1. I [15] a roof of 3.3 is give for he case of fracioal Browia oio, ad we will basically follow heir ideas. Cosider firs he case 1. For ay, we obai he decoosiio where ad V X, µ A = 1 τ B, = 1 τ C, = 1 τ [] [] [] D [] = µ 1 I j = For ay fixed, C, su φ s ds = A i X φ i 1 φ i 1 + B, i G, [] i G φ j 1 i I j φ j 1 + C, + D, φ j 1 i G, i I j [] i G µ 1 φ s ds { i j 1 i, j ]}, j 1. C, [T ], φ j 1, coverges i robabiliy o, uiforly i, as, i.e. φ j 1 1 τ i I j i G µ 1 P A.8 haks o he uifor covergece V G, uc µ. Nex, observe ha he uber of jus of φ ha are bigger ha ε is fiie o coac iervals, because φ is regulaed. This
ilies O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1857 su D µ 1 as. For he er B, su B, 1 [T ] τ su φ + [T ] we obai he iequaliy i I j φ j 1 + su φ su su φ j 1 s j 1, j ] φ i 1 1 i G φ s τ 1 [] i 1 []+1 1 [T ] τ su φ j 1 φ s s j 2, j ] + su φ 1 su i I j i G τ 1 [] i 1 []+1 By Proosiio 1, he laer exressio coverges i robabiliy o E = µ 1 su φ + [T ] su φ j 1 s j 2, j ] φ s P as. As above, we obai E as. For he er A, we deduce by Youg s iequaliy for 1 su A 1 [] τ su i X φ i 1 i G 1 [T ] τ i X φ i 1 i G C τ [T ] var q i G P i G. i 1 φ;, i ] i 1 var 1/ β 2 ε G;, i ] = C F T, where < ε < β/2. Nex, we fix δ > ad cosider he decoosiio F T 1 τ i 1 var q φ;, i ] var 1/ β 2 ε Observe ha [T ] var q + δ τ i:var q φ; i 1, i ]>δ [T ] i 1 G; var 1/ β 2 ε, i ]. i 1 φ;, i ] q var q φ; [, T ] q < ; i 1 G;, i ]
1858 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 ] cosequely, he uber of idices i for which var q φ; i 1, i > δ is bouded by var q φ; [, T ] q /δ q. Recallig 2.8 ad 2.9 we obai F T var qφ; [, T ] q+ τδ q ax i 1 1 i [T ] G; var 1/ β 2 ε, i ] + δ [T ] τ i 1 G; var 1/ β 2 ε, i ] varq φ; [, T ] q+ C τδ q β 2 ε + δ τ β 2 ε. Choose < ε < 1 2, ε < α < 1 ɛ ad se δ = α. By 2.7 we deduce ha F T P, which colees he roof of Theore 2 for 1. For > 1 we use Mikowski s iequaliy o obai he aroxiaio [] 1/ V X, 1/ µ φ s ds 1 1/ τ 1 [] 1/ + 1/ φ i 1 φ j 1 i G τ i I j + 1 [] 1/ φ j 1 1/ [] i G µ 1 τ i I j + µ 1/ [] 1/ 1/ 1 φ j 1 φ s ds. i X φ i 1 φ j 1 1/ i G 1/ By he sae ehods as reseed above we obai he asserio of Theore 2 for > 1. Proof of Theore 6. We se Z = 1 [] H i G. A.9 τ Se 1: Le us show he ighess of he sequece of rocesses G, Z. For ay > s we have E[Z Zs 4 ] = 1 [] 2 E H i G 4. i=[s]+1 By Proosiio 4.2 i [31] ad ar iv of Lea 1 we kow ha, for ay N 1, 1 N N 2 E H i G 4 C τ τ 2 r 2 i C i= 2 ρ i 2. i=
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1859 Sice he rocess G has saioary icrees, we obai E[Z Zs 4 ] C [] [s] 2. For ay 1 2, he Cauchy Schwarz iequaliy ilies ha E[Z 2 Z 2 Z Z 1 2 [2 ] [] [] [1 ] ] C C 2 1 2. The ighess of G, Z follows ow by Theore 15.6 i [13]. Se 2: Fially, we eed o rove he covergece of fiie-diesioal disribuios of G, Z. Defie he vecor Y = Y 1,..., Y dt by Y k = 1 [b k ] i=[c k ]+1 H i G, A.1 τ where c k, b k ], k = 1,..., d, are disjoi iervals coaied i [, T ]. Clearly, i suffices o rove ha G bk G ck, Y k D G bk G ck, τw bk W ck 1 k d 1 k d, where τ is give by 5.4. Nex, we wa o aly Theore 5. Le H 1 be he firs Wieer chaos associaed wih he riagular array j G/τ 1,1 j [], i.e he closed subsace of L 2 Ω, F, P geeraed by he rado variables j G/τ 1,1 j []. Noice ha H 1 ca be see as a searable Hilber sace wih a scalar roduc iduced by he covariace fucio of he rocess j G/τ 1,1 j []. This eas we ca aly he heory of Secio 4 wih he caoical Hilber sace H = H 1. Deoe by H he h Wieer chaos associaed wih he riagular array j G/τ 1,1 j [] ad by I equied wih he or! H 1 he corresodig liear isoery bewee he syeric esor roduc H 1 ad he h Wieer chaos. Fially, we will deoe by J he rojecio oeraor o he h Wieer chaos. Sice E[G bk G ck Y l ] = for ay 1 k, l d because H is a eve fucio, i is sufficie o check he followig codiios. i For ay 1 ad k = 1,..., d, he lii li E[ J Y k 2 ] = τ,k 2 exiss ad =1 su E[ J Y k 2 ] <, ii For ay 1 ad k h, li E[J Y k J Y h] =, iii For ay 1, k = 1,..., d ad 1 1, we have ha li I 1 J Y k I 1 J Y k =. Uder codiios i iii we he obai by Theore 5 he ceral lii heore D Nd, τ 2 diagb 1 c 1,..., b d c d, A.11 Y where τ 2 is give by 5.4. Sice he icrees of he rocess G are saioary we will rove ars i ad iii oly for k = 1, c 1 = ad b 1 = 1.
186 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 i We have J Y 1 = a H i G. τ Hece, we obai see A.4 E[ J Y 1 2 ] =!a 2 1 1 + 2 i r i. By ar iv of Lea 1 we deduce ha li E[ J Y 1 2 ] =!a 2 1 + 2 ρ i, ad =2 su E[ J Y 1 2 ] <. Furherore, we obai ha li E 1 H i G 2 = li ii For ay 1 k, h d wih b k c h we have E[J Y k J Y h ] =!a2 τ =2 [b k ] [b h ] r j=[c k ]+1 i=[c h ]+1 E[ J Y 1 2 ] = τ 2. i j. Assue w.l.o.g. ha c k =, b k = c h = 1 ad b h = 2 he case b k < c h is uch easier. By ar ii of Lea 1 wih < ɛ < 3 2 β i he defiiio of r see A.3 we obai he aroxiaio E[J Y k J Y h ]!a 2 1 jr 1 j + r + j. I follows ha r j j 1 1 δ for soe δ > ad for all, j 2. Hece, we obai E[J Y k J Y h ] as. iii Fix 1 1. We obai he ideiy I 1 J Y 1 I 1 J Y 1 = 1 j G i G τ 1 j,i τ = 1 r j i j G i G, 1 j,i τ τ
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1861 where deoes he syerizaio. Cosequely, we eed o rove ha he quaiy r j l r h k 2 1 j,l,h,k j G l G, h G k G τ τ coverges o zero as. I suffices o cosider a er of he for r j l r h k 2 1 j,l,h,k r α j h r α τ l h r α j k r α l k, where α. The laer er is saller ha 1 r j l r kr α α jr lr α j k r α l k. j,l,k 1 τ H 2 1 Wihou ay loss of geeraliy we ca assue ha = = 1 ad α = or α = 1. For α = ad ay < ε < 1 we ge 2 2 1 r j l r l 1 r j l r l j 1 l 1 j [ε] l 1 2 + 2 1 r j l r l [ε]< j 1 l [ε/2] 2 + 2 1 r j l r l 2ε [ε]< j 1 l< 1 r l 2 2 + 6 [ε/2]<l 1 l< 1 r l 2 [ε/2]<l< r l 2 which coverges o 2ε l< ρ2 l 2 as by Lea 1. The desired resul follows by leig ε ed o zero. This colees he roof of Theore 6. Proof of Theore 7. Theore 7 is deduced fro Theore 6 by he sae ehods as reseed i [15] see Theore 4 herei. For ay we obai he decoosiio V X, µ φ s ds = A + B, + C, where A, C, ad D are defied i A.8 ad B, is give by B, = 1 [] τ 1 τ φ i 1 [] [] i G µ 1 φ j 1 i I j φ i 1 [] i G + µ 1 φ j 1. + D,
1862 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 We firs rove he sable covergece for he er C,. Defie Y, j = 1 τ i G µ. i I j For ay fixed, we obai by Theore 6 ad he roeries of sable covergece, Y, j F G s, τ j W φ j 1 as. Hece, C, For he laer we have 1 j [] F G s [] τ [] τ φ j 1 j W uc τ φ j 1 φ j 1 j W. φ s dw s. 1 j [] Now, we show ha he oher ers are egligible. Recallig ha φ is Hölder coiuous of order a we obai he iequaliy su D µ [T ] su φ + φ j 1 φ j 1 µ µ su φ + 1 su [T ] φ 1 + su where j 1 j 1, j. Hece su D P, because a 1 > 1 2. For he er B, B, = φ + µ T 1 φ 1 a we obai he iequaliy [] i I j [] + [] φ j 1 φ i 1 i [] su φ j 1 1 φ i 1 τ i G µ 1 τ i I j φ j 1 1 φ 1 + 1/2 a 1, i G 1 τ i G µ µ
O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 1863 [] φ s 1 i I j τ i G µ [] φ j 1 1 τ i G µ i I j + su [T ] [] i [] φ i 1 su φ s φ j 1 s j 2, j ] + su [] φ i 1 i [] where s j 2, j ]. The, by Theore 6, we obai li su P su B, τ su φ su > ɛ P τ 1 τ i G µ Y j, + µ 1 τ i G µ [T ] W W []/ > ɛ su φ, su φ s φ j 1 s j 2, j ] j W for ay ɛ >. Sice φ is Hölder coiuous of order a wih a 1 > 1/2 i holds, for ay δ >, ha [T ] su φ s φ j 1 s j 2, j ] 1C φ 1 a su j W φ 1 + a 1+1/2+δ, which coverges o as if δ is sall eough. This ilies ha li li su P su B, > ɛ. Fially, le us show ha su A P. We have A 1 [] 12 2 + τ + 1 τ 12 2 + φ j 1 j G 1 + [] j X φ j 1 j X φ j 1 j G. j G 1
1864 O.E. Bardorff-Nielse e al. / Sochasic Processes ad heir Alicaios 119 29 1845 1865 By 2.9 ad Youg s iequaliy we deduce as i Theore 2 su A C β 2 ɛ 1 + su φ 1 + τ [T ] j 1 + [T ] var 1/a var 1/a j 1 φ;, j ] var 1/ β 2 ɛ G; ] 1, j j 1 φ;, j ] j 1 var 1/ β 2 ɛ G;, j ] C β 2 ɛ 1 + 1 β 2 ɛ+a+1 su φ 1 + + β 2 ɛ+a+1 τ which coverges o as, rovided ɛ < 1 a 1 2 1. This colees he roof. Refereces [1] D.J. Aldous, G.K. Eagleso, O ixig ad sabiliy of lii heores, A. Probab. 6 2 1978 325 331. [2] O.E. Bardorff-Nielse, S.E. Graverse, J. Jacod, M. Podolskij, N. Shehard, A ceral lii heore for realised ower ad biower variaios of coiuous seiarigales, i: Yu. Kabaov, R. Liser, J. Soyaov Eds., Fro Sochasic Calculus o Maheaical Fiace. Fesschrif i Hoour of A.N. Shiryaev, Sriger, Heidelberg, 26,. 33 68. [3] O.E. Bardorff-Nielse, S.E. Graverse, J. Jacod, N. Shehard, Lii heores for biower variaio i fiacial ecooerics, Ecooeric Theory 22 26 677 719. [4] O.E. Bardorff-Nielse, J. Schiegel, Tie chage, volailiy ad urbulece, Thiele Research Reor 12/27. [5] O.E. Bardorff-Nielse, N. Shehard, Realised ower variaio ad sochasic volailiy odels, Beroulli 9 23 243 265. [6] O.E. Bardorff-Nielse, N. Shehard, Power ad biower variaio wih sochasic volailiy ad jus wih discussio, J. Fiac. Ecoo. 2 24 1 48. [7] O.E. Bardorff-Nielse, N. Shehard, Ecooeric aalysis of realised covariaio: High frequecy covariace, regressio ad correlaio i fiacial ecooics, Ecooerica 72 24 885 925. [8] O.E. Bardorff-Nielse, N. Shehard, Iac of jus o reurs ad realised variaces: Ecooeric aalysis of ie-defored Lévy rocesses, J. Ecoo. 131 26 217 252. [9] O.E. Bardorff-Nielse, N. Shehard, Variaio, jus, arke fricios ad high frequecy daa i fiacial ecooerics, i: R. Bludell, T. Persso, W.K. Newey Eds., Advaces i Ecooics ad Ecooerics. Theory ad Alicaios, Nih World Cogress, i: Ecooeric Sociey Moograhs, Cabridge Uiversiy Press, 27,. 328 372. [1] O.E. Bardorff-Nielse, N. Shehard, M. Wikel, Lii heores for uliower variaio i he resece of jus, Sochasic Process. Al. 116 26 796 86. [11] A. Basse, Gaussia ovig averages ad seiarigales, De. Mah. Scieces, Uiversiy of Aarhus, 27. Uublished aer. [12] A. Basse, Secral rereseaios of Gaussia seiarigales, De. Mah. Scieces, Uiversiy of Aarhus, 27. Uublished aer. [13] P. Billigsley, Covergece of Probabiliy Measures, Wiley, New York, 1968. [14] N.H. Bigha, C.M. Goldie, J.L. Teugels, Regular Variaio, Cabridge Uiversiy Press, 1987. [15] J.M. Corcuera, D. Nualar, J.H.C. Woerer, Power variaio of soe iegral fracioal rocesses, Beroulli 12 4 26 713 735. [16] T. Geiig, M. Schlaher, Sochasic odels ha searae fracal diesio ad he Hurs effec, SIAM Rev. 46 24 269 282. [17] L. Guyo, J. Leo, Covergece e loi des H-variaios d u rocessus gaussie saioaire sur R, A. l I.H.P. B25 1989 265 282.
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