Nonparametric tests for pathwise properties of semimartingales

Transcription

1 Bernoull 17(2, 211, DOI: 1.315/1-BEJ293 Nonparametrc tests for patwse propertes of semmartngales RAMA CONT 1 and CECILIA MANCINI 2 1 IEOR Department, Columba Unversty, New York and Laboratore de Probabltés et Modèles Aléatores, CNRS-Unversté Pars VI, France. E-mal: Rama.Cont@columba.edu 2 Dpartmento d Matematca per le Decson, Unverstà d Frenze, Italy. E-mal: cecla.mancn@dmd.unf.t We propose two nonparametrc tests for nvestgatng te patwse propertes of a sgnal modeled as te sum of a Lévy process and a Brownan semmartngale. Usng a nonparametrc tresold estmator for te contnuous component of te quadratc varaton, we desgn a test for te presence of a contnuous martngale component n te process and a test for establsng weter te jumps ave fnte or nfnte varaton, based on observatons on a dscrete-tme grd. We evaluate te performance of our tests usng smulatons of varous stocastc models and use te tests to nvestgate te fne structure of te DM/USD excange rate fluctuatons and SPX futures prces. In bot cases, our tests reveal te presence of a non-zero Brownan component and a fnte varaton jump component. Keywords: g frequency data; jump processes; nonparametrc tests; quadratc varaton; realzed volatlty; semmartngale 1. Introducton Contnuous-tme stocastc models based on dscontnuous semmartngales ave been ncreasngly used n many applcatons, suc as fnancal econometrcs, opton prcng and stocastc control. Some of tese models are constructed by addng..d. jumps to a contnuous process drven by Brownan moton [16,22], wle oters are based on purely dscontnuous processes wc move only troug jumps [8,18]. Even wtn te class of purely dscontnuous models, one fnds a varety of models wt dfferent pat propertes fnte/nfnte jump ntensty, fnte/nfnte varaton wc turn out to ave an mportance n applcatons, suc as optmal stoppng [5] and te asymptotc beavor of opton prces [9,1]. It s terefore of nterest to nvestgate wc class of models dffuson, jump-dffuson or pure-jump s te most approprate for a gven data set. Nonparametrc procedures ave been recently proposed for nvestgatng te presence of jumps [2,6,17] and studyng some fne propertes of te jumps [3,4,25,26]nasgnal. Here, we address related, but dfferent, ssues: for a semmartngale wose jump component s a Lévy process, we propose a test for te presence of a contnuous martngale component n te prce process, wc allows us to dscrmnate between pure-jump and jump-dffuson models, and a test for determnng weter te jump component as fnte or nfnte varaton. Our tests are based on a nonparametrc tresold estmator [2] for te ntegrated varance (defned as te contnuous component of te quadratc varaton based on observatons on a dscrete-tme ISI/BS

2 782 R. Cont and C. Mancn grd. Wtout mposng restrctve assumptons on te contnuous martngale component, we obtan a central lmt teorem for ts tresold estmator (Secton 3 and use t to desgn our tests (Secton 4. Usng smulatons of stocastc models commonly used n fnance, we ceck te performance of our tests for realstc sample szes (Secton 5. Appled to tme seres of te DM/USD excange rate and SPX futures prces (Secton 6, our tests reveal, n bot cases, te presence of a non-zero Brownan component, combned wt a fnte varaton jump component. Tese results suggest tat tese asset prces may be modeled as te sum of a Brownan martngale and a jump component of fnte varaton. 2. Defntons and notaton We consder a semmartngale (X t t [,T ], defned on a (fltered probablty space (, (F t t [,T ], F, Pwt pats n D([,T], R, drven by a (standard Brownan moton W and a pure-jump Lévy process L: t t X t = x + a s ds + σ s dw s + L t, t ],T], (1 were a, σ are adapted processes wt rgt-contnuous pats wt left lmts (cadlag processes, suc tat (1 admts a unque strong soluton X on [,T] wc s adapted and cadlag [11]. L as Lévy measure ν and may be decomposed as L t = J t + M t, were J t := t x >1 N t xμ(dx,ds = γ l, M t := l=1 t x 1 x[μ(dx,ds ν(dxdt]. (2 J s a compound Posson process representng te large jumps of X, μ s a Posson random measure on [,T] R wt ntensty measure ν(dxdt, N s a Posson process wt ntensty ν({x, x > 1}<, γ l are..d. and ndependent of N and te martngale M s te compensated sum of small jumps of L. We wll defne μ(dx,dt ν(dxdt =: μ(dx,dt, te compensated Posson random measure assocated to μ. We allow for te nfnte actvty (IA case ν(r =, were small jumps of L occur nfntely often. For a semmartngale Z, we denote by Z = Z t Z t 1 ts ncrements and by Z t = Z t Z t ts jump at tme t. TeBlumental Getoor (BG ndex of L, defned as { } α := nf δ, x δ ν(dx < + 2, x 1 measures te degree of actvty of small jumps. A compound Posson process as α =, wle an α-stable process as BG ndex equal to α ], 2[. Te gamma process and te varance gamma (VG process are examples of nfnte actvty Lévy processes wt α =. A pure-jump Lévy process wt BG ndex α<1 as pats wt fnte varaton, wle for α>1, te sample pats ave nfnte varaton a.s. Wen α = 1, te pats may ave eter fnte or nfnte varaton [7]. Te normal nverse Gaussan process (NIG and te generalzed yperbolc Lévy moton (GHL

3 Nonparametrc tests for patwse propertes of semmartngales 783 ave nfnte varaton and α = 1. Tempered stable processes [8,1] allow for α [, 2[. We call IV = T σ u 2 du te ntegrated varance of X and IQ = T σ u 4 du te ntegrated quartcty of X, and we wrte t t X t = a s ds + σ s dw s, X 1t = X t + J t. We wll use te followng assumpton. Assumpton A1. α [, 2] x ε x 2 ν(dx ε 2 α as ε, (3 were f( g( means tat f(= O(g( and g( = O(f ( as. Ts assumpton mples tat α s te BG ndex of L. A1 s satsfed f, for nstance, ν as a K densty wc beaves as ± wen x ±, were K x 1+α ± >. In partcular, A1 olds for all Lévy processes commonly used n fnance [1]: NIG, varance gamma, tempered stable processes or generalzed yperbolc processes. Typcally, we observe X t n te form of a dscrete record {x,x t1,..., X tn 1,X tn } on a tme grd t = wt = T/n. Our goal s to provde, gven suc a dscrete observatons, nonparametrc tests for: detectng te presence of a contnuous martngale component n te prce process; analyzng te qualtatve nature of te jump component, tat s, weter t as fnte or nfnte varaton. 3. Central lmt teorem for a tresold estmator of ntegrated varance Te realzed varance n ( X 2 of te semmartngale X converges n probablty [24] to T [X] T := σt 2 dt + T R {} x 2 μ(dx,ds. A tresold estmator [19,2] of te ntegrated varance IV = T σ t 2 dt s based on te dea of summng only some of te squared ncrements of X, tose wose absolute value s smaller tan some tresold r : ˆ IV := n ( X 2 I {( X 2 r }. (4

4 784 R. Cont and C. Mancn Te term T R {} x2 μ(dx,ds, due to jumps, vanses as for an approprate coce of te tresold. P. Lévy s law for te modulus of contnuty of te Brownan pats mples tat ( W P lm sup 1 = 1 2 ln 1/ {1,...,n} and allows suc a tresold to be cosen. It s sown n [2], Corollary 2, Teorem 4, tat, under te above assumptons, f we coose a determnstc tresold r suc tat lm r ln = and lm =, (5 r P ten IVˆ IV as. If te jumps ave fnte ntensty, ten te tresoldng procedure allows as, a jump to be detected n ]t 1,t ]. In fact, snce a and σ are cadlag (or caglad, ter pats are a.s. bounded on [,T], so It follows from [2] tat lm sup lm sup sup t t 1 a s (ω ds A(ω < sup t t 1 σs 2 (ω ds (ω < and a.s. (6 a.s. sup t t 1 a s ds + t t 1 σ s dw s A :=. (7 2 log 1/ Snce realstc values of σ for asset prces belong to [.1,.8] (n annual unts, we ave tat for small, te r.v. as order of magntude of 1, tus, n te fnte jump ntensty case, a.s. for suffcently small, ( X 2 >r > 2 log 1 ndcates te presence of jumps n ]t 1,t ]. Wen L as nfnte actvty, n ( X 2 I {( X 2 r } beaves lke n ( X 2 I { N=, M 2 r } for small (Lemma A.2. Moreover, for any δ>, te jumps contrbutng to te ncrements X suc tat ( X 2 r for small ave sze smaller tan c r + δ ([2], Lemma 1, so ter contrbuton vanses wen. Note tat r = c β satsfes condton (5 for any β ], 1[ and any constant c. Snce 2σ 1 n most applcatons, we use c = 1. Defne η 2 (ε := x 2 ν(dx, d(ε := xν(dx. (8 x ε ε< x 1 Let us remark tat f lm r =, ten, by A1, we ave, as, η 2( 2 r = x 2 x 2 ν(dx r 1 α/2, r x 2 x k ν(dx r (k α/2, r k = 3, 4, (9

5 Nonparametrc tests for patwse propertes of semmartngales r < x 1 2 r < x 1 xν(dx [ [ c + r (1 α/2 ] I{α 1} + ln ν(dx r α/2, 1 2 r ] I {α=1}, were α s te BG ndex of L. Te followng lemma, proved n te Appendx, states tat under (5, eac ncrement M suc tat M 2 r only contans jumps of magntude less tan 2 r f α 1, or smaller tan 2 1/(2α log 1/(2α 1 f α>1. Lemma 3.1. Defne, for >, v := 1/(2α log 1/(2α 1. Under (5. tere exsts a sequence k = T/n k tendng to zero as k suc tat, for k suffcently large and { k,k k }: ( f α 1, ten for all = 1,...,n, MI {( M 2 4r } ( t = x 2 x μ(dx,dt r t 1 t t 1 2 r < x 1 xν(dxdt I {( M 2 4r } a.s.; ( f α>1, ten for all = 1,...,n, we ave MI {( M 2 4r } ( t = x μ(dx,dt x 2v t 1 t t 1 2v < x 1 xν(dxdt I {( M 2 4r } a.s. Remark 3.2. Note tat v r 1/4 so tat n te case ( above (α >1, for all = 1,...,n,te jumps of M on {( M 2 4r } are bounded by r 1/4. Defnton. Defne L ( t := t t M ( := t 1 x μ(dx,dt x 2 4 r x μ(dx,dt. x 2 4 r t 2 4 r < x 1 xν(dxdt, (1 By Lemma 3.1, on a subsequence, a.s. for suffcently small, = 1,...,n,on{( M 2 4r }, we ave M = L ( = M ( d ( 2 4 r. (11 M ( s te compensated sum of jumps smaller n absolute value tan 2 4 r, wle d(2 4 r s te compensator of te (mssng jumps larger tan 2 4 r.

6 786 R. Cont and C. Mancn In [2], a central lmt teorem for IVˆ was sown n te case of fnte ntensty jumps and cadlag adapted σ. Teorem 3.5 extends ts to te case of nfnte actvty wtout extra assumptons on σ. In partcular, wen α<1, te error IVˆ IV as te same rate of convergence and asymptotc varance as n te case of fnte ntensty jumps. Te followng proposton gves te asymptotc varance of ( IVˆ IV/ 2 wen α<1. Proposton 3.3. If r = β wt 1 >β> 2 α/2 1 [1/2, 1[, ten, as, IQˆ := ( X 4 I T {( X 2 r } P IQ = σt 4 dt. 3 Te followng result wll be used to prove Teorem 3.5. Teorem 3.4. Under Assumpton A1, as, n ( t t 1 x ε x μ(dx,dt t t 1 x ]ε,1] xν(dxdt2 Tl 2, ε 2 α Tl 2 1, ε2 2α I {α 1} T l4, ε 2 α/2 (12 d N(, 1, were ε = u,<u 1/2, l j, = x ε xj ν(dx/ε j α for j = 2, 4 and l 1, = ε< x 1 xν(dx/ [(c + ε 1 α I {α 1} + ln 1 2ε I {α=1}] tend to non-zero constants dependng on ν. We are now ready to state our central lmt teorem for te estmator ˆ IV. A sequence (X n s sad to converge stably n law to a random varable X (defned on an extenson (, F,P of te orgnal probablty space f lm E[Uf (X n ]=E [Uf (X] for every bounded contnuous functon f : R R and all bounded random varables U. Ts s obvously stronger tan convergence n law [15]. Teorem 3.5. Assume A1 and σ ; coose r = β wt β> 2 α/2 1 [1/2, 1[. Ten: (a f α<1, we ave, wt st denotng stable convergence n law, ˆ IV IV 2 ˆ IQ st N(, 1; (13 (b f α 1, ten ˆ IV IV 2 ˆ IQ a.s. +. Remark. For α<1, Jacod [13], Teorem 2.1(, as sown a related central lmt result for te tresold estmator of IV, were L s a semmartngale, but under te addtonal assumpton

7 Nonparametrc tests for patwse propertes of semmartngales 787 tat σ s an Itô semmartngale. Te proof of Teorem 3.5 n te case α<1 does not rely on [13], Teorem 2.1(. An alternatve proof under te Itô semmartngale assumpton for σ could combne te results [2] wt [13], Teorem 2.1(, n tat IVˆ IV IV(X = ˆ 1 IV were + ˆ IV(X 1. = IV(M + ˆ + n ( X 1 2 (I {( X 2 r } I {( X 1 2 r } n ( M 2 (I {( X 2 r } I {( M 2 r } n X 1 MI {( X r }, n ( X 1 2 I {( X 1 2 r }, ˆ IV(M. = n ( M 2 I {( M 2 r }. Te frst term converges stably n law by [2], te second one converges stably to zero by [13], Teorem 2.1(. Tat te remanng terms are neglgble requres some furter work (see te proof of Teorem Statstcal tests 4.1. Test for te presence of a contnuous martngale component We now use te above results to desgn a test to detect te presence of a contnuous martngale component t σ t dw t, gven dscretely recorded observatons. Our test s feasble n te case were L as BG ndex α<1, tat s, te jumps are of fnte varaton (see Secton 4.2. Te test proceeds as follows. Frst, we coose a coeffcent β [1/2, 1[ closeto1.ifweavean estmate ˆα of te BG ndex [3,25,26], ten we may coose β> 1 2 ˆα (recall tat 1 2 α [1/2, 1[. We coose a tresold r = β and use te estmator IQˆ of te ntegrated quartcty defned n Proposton 3.3. We ave sown n Teorem 3.5 tat, wen σ n te case α<1, te estmator IVˆ s asymptotcally Gaussan as. However, f σ, ten bot te numerator and te denomnator of (13 tend to zero. To andle ts case, we add an..d. nose term: X v := X + v Z, Z..d. N(, 1. As, n ( X v 2 [X P v ] T = T σs 2 ds + v2 T + T R {} x 2 μ(dx,ds and I {( X v 2 r } removestejumpsofxv so tat under te assumptons of Teorem 3.5, as, n T IVˆ v := ( X v 2 P I {( X v 2 r } σs 2 ds + v2 T.

8 788 R. Cont and C. Mancn Under te null ypotess σ, we ave (3 P v 4 T and ˆ IV v IQ v P v 2 T, ˆ IQ v := ( X v 4 I {( X v 2 r } / IV U := ˆ v2 T st N. (14 2 ˆ Note tat f, on te contrary, σ, ten we ave tat te lmt n probablty of ˆ larger tan v 2 T and, by Lemma A.2, passng to a subsequence, a.s. lm IQˆ v = 1 3 lm IV v ( X v 4 I {( X v 2 r } = 1 3 lm ( X v 4 I { N=,( M 2 2r } 1 3 lm ( X + M + v 4I{( Z M 2 2r } c 3 lm ( X 4 + c 3 lm s strctly ( M 4 I {( M 2 2r } + c 3 lm ( 4. v Z Usng te facts tat lm ( M 4 I {( M 2 2r } lm 2r ( M 2 I {( M 2 2r } =, by (44, ( X 4 / P c T σ s 4 ds and (v Z 4 / a.s. cv 4, we ave, as, IQˆ v P. Terefore, under te alternatve (H 1 σ, U + and P { U > 1.96} 1, so te test s consstent. Local power of te test. To nvestgate te local power of te test U, we consder a sequence of alternatves (H1 σ = σ, were σ. We denote by IQˆ v σ,u σ te statstcs analogous to IQˆ v,u, but constructed from Xt = x + t a s ds + t σ s dw s + L t,t ],T]. In te case of constant σ and σ, and fnte jump ntensty, usng standard results on convergence of sums of a trangular array [14], Lemmas 4.1 and 4.3, we ave IQˆ v ucp σ v 4 T, U σ d (σ 2 lm T + 2v 2 Z T, were ucp denotes unform convergence n probablty on compacts subsets of [,T] [24] and Z s a standard Brownan moton. So, eter U σ tends n dstrbuton to c + 2v 2 Z T,fσ = O( 1/4, or U σ,f 1/4 = o(σ.tus,fc s a (possbly zero constant, we ave: f σ c, 1/4 ten f σ +, 1/4 ten { P {U σ > 1.64 H 1 } P Z 1 > 1.64 } c2 T ; 2Tv 2 P {U σ > 1.64 H 1 } 1. For values of v n Secton 5,weave1.64/ 2Tv 2 = O(1 8 and tus te local power of te test s small f σ = O( 1/4.

9 Nonparametrc tests for patwse propertes of semmartngales Testng weter te jump component as fnte varaton To construct a test for dscrmnatng α<1 from α 1, Teorem 3.5 suggests te use of ( IVˆ IV/ 2IQˆ, but ts requres knowng te process σ to compute IV. We propose a feasble alternatve. Consder, nstead, te estmator ˆ H := n XI {( X 2 >r } = X T n XI {( X 2 r }. Proposton 4.1. Wen α<1, ˆ H s a consstent estmator of J T + mt, m := 1 1 xν(dx. Consder Z = W v, were W v s a Wener process ndependent of W,L, and defne ˆ H v := XI {( X 2 >r } + v Z and H v T := J T + mt + vw v T. Under te null ypotess α<1, IVˆ H v := ( ˆ H v 2 I {( ˆ H v 2 r } s an estmator of te ntegrated varance v 2 T of H v, so, under te null ypotess (H α<1, we can fnd β> 2 α 1 ]1 2, 1[ suc tat U (α := IVˆ H v v 2 T 2IQˆ H v d N(, 1, (15 were IQˆ H v := 3 1 ( Hˆ v 4 I {( Hˆ v 2 r } and r = β. In partcular, P { U (α > 1.96} 5%. If, on te contrary, α 1, ten reasonng as n Teorem 3.5, for any β ], 1[, weave U (α P +, so te test s consstent. If U (α > 1.96, ten we reject (H α<1at te 95% confdence level. Remark. To apply ts test, we frst need to decde weter α<1, usng te prevously descrbed test. 5. Numercal experments 5.1. Testng te fnte varaton of te jump component We smulate n ncrements X of a process X = σw + L, were L s a symmetrc α-stable Lévy process, σ =.2. We generate 1 ndependent samples contanng n ncrements eac

10 79 R. Cont and C. Mancn Table 1. Testng for fnte varaton of jumps: α-stable process plus Brownan moton. pct s te percentage of outcomes were U (α (j > 1.96 n v α pct α pct 1 5 mn mn mn mn mn our day mn mn and compute U (α as n (15 for a range of values of v, (1 mnute, 5 mnutes, 1 our, 1 day and number of observatons n. Table 1 reports te percentage (pct of outcomes were U (α (j > 1.96, j = 1,...,1, for tresold exponent β =.999. Note tat wt n = 1 and equal to fve mnutes ( = 1/(252 84, we ave T<1year; for α =.6, te lower bound for β s 1 2 α =.71; wen n = 1,= 1/( and te BG ndex of L s.6 (resp., 1.6, te rato of v = 1 4 to te standard devaton of te ncrements X s.74 (resp.,.22. Te test results are observed to be relable f we use n = 1 observatons, a tme resoluton of fve mnutes and v = 1 4. In fact, wen te data-generatng process as BG ndex.6, te test leads us to accept te ypotess (H α<1n about 94 cases out of 1. On te contrary, wen te process as BG ndex 1.6, te test tells us to reject (H n 92 cases out of Test for te presence of a Brownan component We smulate 1 ndependent pats of a process X t = t σ u dw u + L, for dfferent Lévy processes L and constant or stocastc σ,onatmegrdwtn steps. We take tresold r =.999. For eac tral j = 1,...,1, we compute U (j gvenn(14 and report te percentage (pct of cases were U (j > Example 5.1 (Brownan moton plus compound Posson process, BG ndex α =. We consder ere constant σ and L = N t B, a compound Posson process wt..d. N(,.6 2 szes of jump and jump ntensty λ = 5(asn[1]. Table 2 llustrates te performance of our test for varous tme steps, numbers of observatons n and nose levels v: Note tat wen σ = (resp.,.2, n = 1 and = 1/( te rato of v = 1 4 to te standard devaton of te returns X equals.7 (resp.,.52. We fnd tat te test s relable for values n = 1, = 5 mnutes and v = 1 4 snce t correctly accepts (H n 95 cases out of 1 and rejects (H n all cases wen t s false.

11 Nonparametrc tests for patwse propertes of semmartngales 791 Table 2. Testng for te presence of a Brownan component: case of Brownan moton plus compound Posson jumps (Example 5.1 n v σ pct σ pct 1 5 mn mn mn mn mn our day mn mn Example 5.2 (Brownan moton plus α-stable jumps: α ], 2[. Here, L s a symmetrc α- stable Lévy process and σ s constant. Te results n Table 3 confrm te satsfactory performance of te test wen α =.3 < 1forn = 1, = 5 mnutes and v = 1 4. Table 4, for te case α = 1.2 > 1, confrms tat we cannot rely on te test results n ts case: even wen σ, te statstc U dverges f α 1. Te man pont ere s tat we may use a model-free coce of tresold. Example 5.3 (Stocastc volatlty plus varance gamma jumps: α =. Let us now consder a model X wt stocastc volatlty σ t, correlated wt te Brownan moton drvng X and wt jumps gven by an ndependent varance gamma process: dx t = (μ σ 2 t /2 dt + σ t dw (1 t + dl t, Table 3. Testng for te presence of a Brownan component: case of Brownan moton plus α-stable Lévy process wt α =.3 (Example5.2 n v σ pct σ pct 1 5 mn mn mn mn mn our day mn mn

12 792 R. Cont and C. Mancn Table 4. Testng for te presence of a Brownan component: case of Brownan moton plus α-stable Lévy process wt α = 1.2 (Example5.2 n v σ pct σ pct 1 5 mn mn mn mn mn our day mn mn were σ t = e K t, dk t = k(k t Kdt + ς dw t (2, d W (1,W (2 = ρ dt, (16 t W (l are standard Brownan motons, l = 1, 2, 3, and L t = cg t + ηw (3 G t s an ndependent varance gamma process, a pure-jump Lévy process wt BG ndex α = [18]; G s a gamma subordnator ndependent of W (3 wt G Ɣ(/b,b. Forσ, we coose K = ln(.3, k =.9, K = ln(.25, ς =.5 to ensure tat σ fluctuates n te range.2.4. As for te jump part of X, we use Var(G 1 = b =.23, η =.2, c =.2, estmated from te S&P 5 ndex n [18]. Te remanng parameters are ρ =.7 and μ =. Te followng results n Table 5 confrm te relablty of te test for te presence of a Brownan component wt n = 1, = 5 mnutes and v = 1 4. Remark. In [21], a varable tresold functon s used to estmate te volatlty, n order to account for eteroscedastcty and volatlty clusterng, wt results very smlar to te ones ob- Table 5. Testng for te presence of a Brownan component: stocastc volatlty process wt varance gamma jumps (Example 5.3 n v σ pct σ pct 1 5 mn.1.32 Stoc mn.1.17 Stoc mn.1.27 Stoc mn.1.54 Stoc mn.1.34 Stoc our Stoc day.1 1. Stoc mn.1.49 Stoc mn Stoc. 1

13 Nonparametrc tests for patwse propertes of semmartngales 793 taned wt a constant tresold. Ts s justfed by te fact tat n most applcatons, values of σ are wtn te range [.1,.8], tus te order of magntude of n (7 so1. 6. Applcatons to fnancal tme seres We apply our tests to explore te fne structure of prce fluctuatons n two fnancal tme seres. We consder te DM/USD excange rate from October 1st, 1991 to November 29t, 1994 and te SPX futures prces from January 3rd, 1994 to December 18t, From g-frequency tme seres, we buld fve-mnute log-returns (excludng, n te case of SPX futures, overngt log-returns. Ts samplng frequency avods many mcrostructure effects seen at sorter tme scales (e.g., seconds, wle leavng us wt a relatvely large sample Deutsce Mark/USD excange rate Te DM/USD excange rate tme seres was compled by Olsen & Assocates. We consder te seres of equally spaced fve-mnute log-returns, wt = , dsplayed n Fgure 1. Barndorff-Nelsen and Separd [6] provde evdence for te presence of jumps n ts seres usng nonparametrc metods. Usng as tresold r =.999, we apply te test of Secton 5.1 to te degree of actvty of te jump component. As n te smulaton study, we dvde te data nto 64 non-overlappng batces of n = 1 observatons eac and compute, for eac batc, te statstc U (α (j, j = 1,...,64, wt v = 1 4.Only4.7% of te values observed are outsde te nterval [ 1.96, 1.96], ence we cannot reject te assumpton (H α<1. Gven ts result, we can now use te test n Secton 5.2 for te presence of a Brownan component n te prce process. Computaton of te statstc U sows values muc larger tan 1.96 for all batces: we reject (H σ. Tese results ndcate, for nstance, tat a varance gamma model, wt no Brownan component, would be nadequate for te DM/USD tme seres. Fgure 1. Left: DM/USD fve-mnute log-returns, October 1991 to November Center: plot of XI {( X 2 r }, = 1,...,n. Rgt: ncrements wt jumps XI {( X 2 >r }, = 1,...,n.

14 794 R. Cont and C. Mancn Fgure 2. Left: SPX fve-mnute log-returns, January 1994 to December Center: plot of XI {( X 2 r }, = 1,...,n. Rgt: ncrements wt jumps XI {( X 2 >r }, = 1,...,n S&P 5 ndex We consder a seres of non-overlappng fve-mnute log-returns, as dsplayed n Fgure 2. Usng as tresold r =.999, we decompose te seres nto perods dsplayng jumps and oter perods, as dsplayed n Fgure 2 (central and rgt panels. We dvde te data nto 78 non-overlappng batces of n = 1 observatons eac and compute, for eac batc, te statstc U (α (j, j = 1,...,64, wt v = % of te values observed are outsde te nterval [ 1.96, 1.96]: for ts perod, we cannot reject te assumpton (H α<1. Gven ts result, we can use te test for te presence of a Brownan component n te prce process. Computaton of te statstc U sows values muc larger tan 1.96 for all batces: we reject (H σ. Te test tus ndcates te presence of a Brownan martngale component. We note tat our fndngs contradct te concluson of Carr et al. [8] wo model te (log- SPX ndex from 1994 to 1998 as a tempered stable Lévy process plus a Brownan moton and propose a pure-jump model usng a parametrc estmaton metod. Under less restrctve assumptons on te structure of te process and usng our nonparametrc test, we fnd evdence for a non-zero Brownan component n te ndex. Appendx: Tecncal results and proofs Proof of Lemma 3.1. By [23], Teorem 25.1, tere exsts a sequence (n k suc tat sup ( j M 2 ( M s 2 a.s., (17 t j (n k s ]t j 1,t j ]

15 Nonparametrc tests for patwse propertes of semmartngales 795 were (nk s te partton of [,T] on wc te ncrements ( M 2 are constructed. Let us rename n k as n. Usng Itô s formula, we ave ( M 2 s ]t 1,t ] ( M s 2 = 2 t t 1 (M s M t 1 dm s. ( For α<1, our statement s proved n [21], Lemma A.2, wc uses te fact tat te speed of convergence to of n t t 1 (M s M t 1 dm s s sown n [12]tobeu n = n.forα = 1, te same reasonng can be repeated snce u n = n/(log n 2 does not cange te concluson. ( If α>1, we ave u n = (n/ log n 1/α and can only conclude tat a.s. for small, t sup (M s M t 1 dm s cu 1 n t 1 wt c>, so tat a.s. for small, we ave ( sup ( M s I 2 {( M 2 4r } sup ( M 2 s ]t 1,t ] ( cu 1 n + 4r = O Lemma A.1. Under (5: s ]t 1,t ] ( M s 2 + sup ( M 2 δ 1/α log 1/α 1 ( tere exsts a strctly postve varable suc tat for all = 1,...,n, ( ( n te case r = β, β ], 1[, we ave I { } I {( X 2 >r } = a.s.; (18 c>,np{ N,( M 2 >cr } ; (19 lm sup αβ/2. n P {( X 2 >r } c. (2 Proof. Equalty (18 s a consequence of (7, wle (19 s a consequence of te ndependence of N and M, and of te Cebysev nequalty: as, np { N,( M 2 >cr } no( E[( M 2 ( ] = O. cr r Te proof of (2 can be aceved as n [3], Lemma 6, but we gve a smpler proof under our assumptons. It s suffcent to sow tat P {( X 2 >r } c 1 αβ/2. (21

16 796 R. Cont and C. Mancn Frst, we sow tat P { X > r } = P { M > r /4 } + O( 1 αβ/2 (22 so tat for (21, t s suffcent to prove tat P { M > r /4 } c 1 αβ/2. (23 To sow (22, note tat f X > r, ten eter J or M > r /4 snce, for small, r < X X + J + M r /2 + J + M a.s. (24 Tus, P { X > r } P { J }+P { M > r /4 } and snce P { J }=O( = o( 1 αβ/2, (22 s verfed. In order to verfy (23, defne Ñ t := s t I { M s > r /4} and wrte P { M > r /4 } = P { Ñ =, M > r /4 } + P { Ñ 1, M > r /4 } (25 P { Ñ 1}+P { Ñ =, M > r /4 }. Note tat Ñ t = t x > r /4 μ(dx,dt s a compound Posson process wt ntensty ν{ x > r /4} =O(r α/2, sop { Ñ 1} =O(ν{ x > r /4} = O( 1 αβ/2 and tus te frst term above s domnated by 1 αβ/2, as requred. Fnally, on { Ñ = }, M does not ave jumps bgger tan r /4 on te nterval ]t 1,t ],so terefore t M = t 1 x x μ(dx,dt xν(dx, r /4 r /4< x 1 P { Ñ =, M > r /4 } P { M > r /4, M s r /4 for all s ]t 1,t ] } 4 E[( M 2 I { Ms r /4foralls ]t 1,t ]}] r r ( η 2 (r /4 = O = O( 1 αβ/2 and (23 s verfed.

17 Nonparametrc tests for patwse propertes of semmartngales 797 Proof of Proposton 3.3. ( X 4 I {( X 2 r } 3 = ( X 1 4 I {( X 1 2 4r } + 1 ( X 1 4( I 3 3 {( X 2 r } I {( X 1 2 4r } := + 4 k=1 3 I j (. j=1 ( 4 ( X 1 4 k ( M k I {( X 2 r } k 3 By Proposton 1 n [2], I 1 ( tends to T σ t 4 dt n probablty. We sow ere tat te oter terms tend to zero n probablty. Let us consder I 2 ( := 1 3 ( X 1 4 (I {( X 2 r } I {( X 1 2 4r } :on{( X 2 r,( X 1 2 > 4r }, we ave r X > X 1 M > 2 r M, (26 so M > r. Moreover, f X 1 > 2 r, ten we necessarly ave N snce X + J X 1 > 2 r (27 and, by (18, a.s. for suffcently small, for all = 1,...,n, X r, tus J > 2 r X r. It follows tat { 1 P } ( X 1 4 I {( X 2 r,( X 1 2 >4r } np { M > r, N }, by Lemma A.1. On te oter and, for all = 1,...,n on {( X 1 2 4r }, we ave, for suffcently small, N = because J X X 1 2 r, (28 so f N, ten a.s. for small, we n fact ave N = 1 and J s 1, by te defnton of J. Terefore, f N, we would ave 1 J 2 r + r = 3 r, wc s mpossble for small. It follows tat {( X 2 >r,( X 1 2 4r } {( X + M 2 >r } { ( X 2 > r } 4 { ( M 2 > r 4 }.

18 798 R. Cont and C. Mancn Ts mples, by (18 and (23, tat a.s. as, 1 ( X 1 4 I {( X 2 >r,( X 1 2 4r } ( X 4 I {( M 2 >r /4} 4 ln 2 1 I {( M 2 >r /4} We can conclude tat I 2 ( P as. Now, consder I 3 ( := 4 k=1 ( 4 k P. I 3,k (, were s decomposable as I 3,k ( := ( X 1 4 k ( M k I {( X 2 r }, k = 1,...,4, ( X 1 4 k ( M k I {( X 2 r,( M 2 4r } ( X 1 4 k ( M k I {( X 2 r,( M 2 >4r }. (29 We ave, a.s. for small, tat for all on {( X 2 r,( M 2 > 4r }, N snce 2 r X 1 < M X 1 X r and ten X 1 > r and, smlarly as n (27, J > 3 r /4. So, te probablty tat te second term of (29 dffers from zero s bounded by (19 and tends to zero. As for te frst term, a.s. for suffcently small, for all on {( X 2 r,( M 2 4r }, we ave N = because X 1 M X r, tus X 1 < 3 r and we proceed as n (28. So, te frst term n (29 s a.s. domnated by X 1 4 k M k I { N=,( M 2 4r } X 4 k M k I {( M 2 4r }. 3 3 Now, for k = 4, we apply to M property (C.19 n [4], Lemma 5, wt β tere beng α ere, u n = r = β/2, p = 4, v = φ for a proper exponent φ we specfy below and β =. Result (C.19 of [4] ten mples tat 1 E [ n M ( 4 I { M 2 r } ] M v 4 I { Mv 2 r } c (β/2(4 α 1 η 4,n, v T were η 4,n = ( β/2 v α + 2 αβ/2 ( β/2 v 3α + αβ/2 ( β/2 2α + (2 β/2 α + 1/4 ((4 α/4β/2 + v (4 α/4. As soon as β>1/(2 α/2 and we coose φ ], 1 β 3 [,so

19 Nonparametrc tests for patwse propertes of semmartngales 799 tat for all α ], 2[ we ave φ<(2/α β/3, t s guaranteed bot tat (β/2(4 α 1 and tat (β/2(4 α 1 η 4,n. Tus, lm M 4 t I {( M 2 4r } t 1 x 2 r x 4 μ(dx,dt = lm 3 3 and [ E t t 1 ] x 2 x 4 μ(dx,dt/3 r ( = O x 2 x 4 ν(dx/ r = O ( (β/2(4 α 1, gven tat β>1/(2 α/2. To sow, furter, tat te terms X 4 k M k I {( M 2 4r } 3 tend to zero n probablty for k = 1, 2, 3, we use te fact tat, by (11, eac term s domnated by (recall te notaton n (1 Now, a.s. c X 4 k M ( k + c X 4 k d(2 4 r k. 3 3 X 4 k d(2 4 r k ( ln 1 3 (4 k/2 [ c n k 1 (1 α/4 + r k I {α 1} + ln k 1 ( c k/2 ln 1 (4 k/2 ( + c k/2 ln 1 = o(1 + c k[1/2+β(1 α/4] log (4 k/2 1 r 1/4 I {α=1} ] (4 k/2 r k(1 α/4 + /2 ln 2 k/2 1 r 1/4 for all k = 1, 2, 3asr = β, β ], 1[.Asfor X 4 k M ( k, (3 3 we need to deal separately wt eac of k = 1, 2, 3. Note tat snce a and σ are locally bounded on [,T], we can assume tat tey are bounded wtout loss of generalty, so

20 8 R. Cont and C. Mancn E[( t t 1 σ s dw s 2k ]=O( k for eac k = 1, 2, 3, usng, for nstance, te Burkolder nequalty [24], page 226, and a.s. ( t t 1 a s ds 2k = o( k. Terefore, E[( X 2k ]=O( k for eac of k = 1, 2, 3. For k = 1, te expected value of (3 s bounded by (n/3 E[( X 6 ] E( M ( 2 = O(r (1/4(1 α/2 and tus tends to zero as. As for k = 2, wose expected value s gven by ( X 2 ( M ( 2 ln 1 ( M ( 2, (31 ln 1 η2 (2r 1/4 as snce r = β, wt β>. Concernng k = 3, we ave X M ( 3 c ( X 2( M ( 2 c ( + M ( 4, so tat ts step s reduced to te steps wt k = 2, 4 wc we dealt wt prevously. Proof of Teorem 3.4. Let us defne K n := ( t t 1 x ε x μ(dx,dt ε< x 1 xν(dx2.we apply te Lndeberg Feller teorem to te double array sequence H n gven by te normalzed versons of te varables K n, = 1,...,n, and n = T/. Usng relatons (9, we ave ( 2 E[K n ]=l 2, ε 2 α + xν(dx ε< x 1 = l 2, ε 2 α + l 2 1, 2 [(c + ε 1 α 2 I {α 1} + ( ln 2 1 ] I {α=1}. ε Takng ε = u and any u ], 1/2], we obtan tat [( t 4 ] vn 2 := var[k n]=e x μ(dx,dt xν(dx t 1 x ε ε< x 1 En 2 x 4 ν(dx = l 4, ε 4 α as. Ten, consder x ε H n := K n E[K n ] nvn K n l 2, ε 2 α l 2 1, 2 [(c + ε 1 α 2 I {α 1} + (ln 2 1/εI {α=1} ] T l4, ε 2 α/2. (32 We now sow tat for any δ>, tere exsts a q>1suc tat n E [ Hn 2 I { H n >δ}] cε α/(2q (33

21 Nonparametrc tests for patwse propertes of semmartngales 81 as, so te Lndeberg condton s satsfed and mples tat n d H n N(, 1. (34 Notng tat /ε 2 α/2 and (ε 1 α /(ε 2 α/2 I {α 1} + ( ln 2 (1/ε/(ε 2 α/2 I {α=1} tend to zero as, (34 leads to (12. To sow nequalty (33, consder ne [ Hn1 2 I ] { H n1 >δ} ne 1/p [H 2p n1 ]P 1/q { H n1 >δ}, (35 as for te last factor above, we note tat H n1 >δf and only f eter ( K n1 <l 2, ε 2 α + l 2 1, [(c 2 + ε 1 α 2 I {α 1} + ln 2 1 ] I {α=1} δ Tl 4, ε 2 α/2 ε = ε 2 α/2( o(1 cδ, were c denotes a generc constant, or K n1 >l 2, ε 2 α + l 2 1, 2 [(c + ε 1 α 2 I {α 1} + ( ln 2 1 ] I {α=1} + cδε 2 α/2 = O(ε 2 α/2. ε However, K n1, wle for suffcently small, te rgt-and term of te frst nequalty above s strctly negatve, terefore H n1 >δf and only f K n1 >cε 2 α/2, tat s, eter cε 1 α/4 (c + ε 1 α I {α 1} + I {α=1} ln 1 t1 ε cε1 α/4 > x μ(dx,dt or, for suffcently small, t 1 x ε x μ(dx,dt>cε1 α/4, and so H n1 >δf and only f t1 x μ(dx,dt >cε1 α/4. x ε Ts entals tat for suffcently small, { t1 P { H n1 >δ}=p c E[ t 1 x ε } x μ(dx,dt >cε1 α/4 x ε x ε x μ(dx,dt 2 ] ε 2 α/2 = 1 (αu/2. Te frst two factors of te rgt-and sde of (35 are domnated by cn E1/p [(K n1 l 2, ε 2 α 2 l 2 1, [(c + ε1 α 2 I {α 1} + (ln 2 1/εI {α=1} ] 2p ] ε 4 α cn E1/p [K 2p n1 ]+(ε2 α (1 ε 1 α ln 4 1/ε ε 4 α.

22 82 R. Cont and C. Mancn Te last tree terms gve no contrbuton to (35 snce n (ε2 α (1 ε 1 α ln 4 1/ε ε 4 α (1 αu/2(1/q. On te oter and, by coosng, for example, p = 5/4, we ave E[K 2p n1 ]=O(ε5 α, so we are left to deal wt n (ε5 α 1/p (1 αu/2(1/q = ε α/(2q and te nequalty n (33 s ε proved. 4 α Lemma A.2. As, f r, n = T/and sup,...,n a =O(r, ten a I {( X 2 r } a I {( M 2 4r, N=} P. Proof. On {( X 2 r }, we ave L X X r and, tus, by (7, for small, L 2 r, so tat a.s. However, as and tus a.s. lm lm a I {( X 2 r } lm a I {( L 2 4r }. a I {( X 2 r } lm a I {( L 2 4r, N } sup a.s. a N T (36 We now sow tat, on te oter and, te postve quantty In fact, lm a I {( L 2 4r, N=} = lm a I {( M 2 4r, N=}. a ( I {( L 2 4r, N=} I {( X 2 r } = a.s. {( L 2 4r, N = } {( X 2 r } ={( L 2 4r, N =,( X 2 >r } { L 2 r, N =, X + M > r } { X > r /2 } { M 2 r, M > r /2 }.

23 Nonparametrc tests for patwse propertes of semmartngales 83 Snce, by (18, a.s. for suffcently small a I { X > r /2} =, we a.s. ave lm a ( I {( L 2 4r, N=} I {( X 2 r } lm owever, by Remark 3.2, as, a I { M 2 r, M > r /2}; [ ] E a I { M 2 r, M > r /2} O(r np { M 2 r, M > r /2 } O(r np { M I { M 2 r } > r /2 } O(r n E[( M 2 I { M 2 r }] r = O(r n η2 (2r c1/4 r. Lemma A.3. Under te assumptons of Teorem 3.5, for all α [, 2[, n n ( M 2 I {( M 2 r /16} o P ( 1 α/2 ( M 2 I {( X 2 r,( M 2 4r } n ( M 2 I {( M 2 9r /4} + o P ( 1 α/2 (37 a.s. Proof. Let us frst deal wt n ( M 2 I {( X 2 >r,( M 2 4r }. As n (24, on {( X 2 >r }, we ave eter J > r /4or M > r /4, so However, n ( M 2 I {( X 2 >r,( M 2 4r } n ( M 2 I {( X 2 >r, J,( M 2 4r } + n ( M 2 I {( X 2 >r,( M 2 >r /16,( M 2 4r }. [ n ( M 2 ] ( I {( M E 2 4r, N } η 2 (r 1/4 = O N T 1 α/2 1 α/2,

24 84 R. Cont and C. Mancn so n ( M 2 I {( X 2 >r,( M 2 4r } o P ( 1 α/2 + = o P ( 1 α/2 + = o P ( 1 α/2 + n ( M 2 I {( M 2 4r,( M 2 >r /16} (38 n n ( M 2 I {( M 2 4r } ( M 2 I {( M 2 4r,( M 2 r /16} n n ( M 2 I {( M 2 4r } ( M 2 I {( M 2 r /16}. Now, consder n ( M 2 I {( M 2 4r,( M 2 >9r /4} :on{2 r M > 3 2 r }, eter N, n wc case n ( M 2 I {( M 2 4r, N } 1 α/2 P, as before, or N =, n wc case X > M X > 3 2 r 1 2 r = r,so n ( M 2 I {( X 2 >r,( M 2 4r } + o P ( 1 α/2 Terefore, n ( M 2 I {( M 2 4r,( M 2 >9r /4}. n ( M 2 I {( X 2 >r,( M 2 4r } n n o P ( 1 α/2 + ( M 2 I {( M 2 4r } ( M 2 I {( M 2 9r /4}. (39 Now combnng (38 and (39, we obtan (37 snce n ( M 2 I {( X 2 r,( M 2 4r } n n = ( M 2 I {( M 2 4r } M ( 2 I {( X 2 >r,( M 2 4r }.

25 Nonparametrc tests for patwse propertes of semmartngales 85, te assumptons of Proposton 3.3 are sat- Proof of Teorem 3.5. Note tat under β> 1 sfed. Snce X = X 1 + M, we decompose 2 α/2 IVˆ IV = 2IQˆ = n ( X 2 I {( X 2 r } IV 2 ( X 4 I {( X 2 r } /3 n ( X 1 2 I {( X 1 2 4r } IV (2/3 ( X 4 I {( X 2 r } (4 := 2IQ + (2/3 ( X 4 I {( X 2 r } [ n ( X 1 2 (I {( X 2 r } I {( X 1 2 4r } 2IQ n (41 X 1 MI n {( X r } ( M 2 ] I {( X + 2 r } 2IQ 2IQ 4 I j (. j=1 Te proof of [2], Teorem 2, sows tat I 1 ( converges stably n law to a standard Gaussan random varable. To sow tat te remanng terms eter tend to zero or to nfnty, we can assume wtout loss of generalty tat bot a and σ are bounded a.s. If ( X 2 r and ( X 1 2 > 4r, ten M > r and N, exactly as for I 2 ( n Proposton 3.3. It follows tat { n ( X 1 2 } I {( X P 2 r,( X 1 2 >4r } 2IQ np { N, M > r }, by (19. Te man factor of te remanng part of I 2 ( s n ( X 1 2 I {( X 2 >r,( X 1 2 4r } 2IQ. We recall tat on { X 1 2 r }, we ave N =, tus ( X 1 2 = ( X 2. Moreover, n ( t t 1 a u du 2 I {( X 2 >r,( X 1 2 4r } 2IQ = O P (

26 86 R. Cont and C. Mancn and, by (2, 1 2IQ n t t 1 a u du t t 1 σ u dw u I {( X 2 >r,( X 1 2 4r } c = O ln 1 ( 1 αβ/2 n ln 1 I {( X 2 >r }. Terefore, n probablty, n lm I ( t t 2( = lm 1 σ u dw u 2 I {( X 2 >r,( X 1 2 4r }. 2IQ We now sow tat term I 3 (/2 n(41 tends to zero n probablty. Frst, recall tat X 1 = X + J and, wtn te sum n J MI {( X 2 r } /,teterm contrbutes only wen N, n wc case we also ave ( X 1 2 > 4r and tus M > r, as n (26. Tat mples { n } J MI {( X P 2 r } np { N, M > } r. 2IQ As for n X MI {( X 2 r }, as n te proof of Lemma A.2,weave n X MI n {( X 2 r } X MI {( X = 2 r,( L 2 4r }. (42 However, snce bot P { 1 n 1 X MI {( X 2 r,( L 2 4r, N } } and P { n X MI {( X 2 r,( M 2 4r, N } } are domnated by np { N, ( M 2 >cr }, we ave lm 1 n X MI {( X 2 r,( L 2 4r } = lm 1 = lm 1 = lm 1 n X MI {( X 2 r,( L 2 4r, N=} n X MI {( X 2 r,( M 2 4r, N=} n X MI {( X 2 r,( M 2 4r }.

27 Nonparametrc tests for patwse propertes of semmartngales 87 Moreover, by te Caucy Scwarz nequalty, we ave n t t 1 a u du MI {( X 2 r,( M 2 4r } n ( t t 1 a u du 2 n ( M 2 I {( M 2 4r } (43 c n ( M 2 I {( M 2 4r }, wc tends to zero n probablty snce, by Remark 3.2,as, [ n ] E ( M 2 I {( M 2 4r } = On te oter and, 1 n = 1 ( t T x 2r 1/4 σ u dw u MI {( X 2 r,( M 2 4r } t 1 n 1 ( t n x 2 ν(dx = Tη 2 (r 1/4. (44 σ u dw u t 1 M ( I {( X 2 r,( M 2 4r } (45 ( t σ u dw u d ( 2 4 r I{( X 2 r,( M 2 4r }, t 1 were, usng te fact tat t t 1 σ u dw u and M ( are martngale ncrements wt zero quadratc covaraton, te L 1 ( -norm of te frst rgt-and term s bounded by [ n ( t t E 1 σ u dw u 2 ( M ( 2 ], wc s dealt wt smlarly as n (31 and tends to zero. Moreover, [ 1 n ( t E σ u dw u d ( ] 2 4 r I{( X 2 r,( M 2 4r } t 1 = c [ ( (1 α/4 I α 1 c + r + Iα=1 ln 1 ] r 1/4

28 88 R. Cont and C. Mancn [ n ( t ] E σ u dw u I {( X 2 r,( M 2 4r } t 1 c [ ( (1 α/4 I α 1 c + r + Iα=1 ln 1 r 1/4 ] E [ n ( t t 1 σ u dw u 2 ]. Usng te fact tat 2IQ 2/3 ( X 4 I {( X 2 r } tends to 1 n probablty, treatng I 4 ( as n (42 and puttng togeter te smplfed verson of I 2 (, we obtan tat ( IVˆ IV/ 2IQˆ s te sum of a term wc converges n dstrbuton to an N(, 1 r.v. plus a neglgble term and a remander n ( t t 1 σ u dw u 2 I n {( X 2 >r,( X 1 2 4r } ( M 2 I {( X + 2 r,( M 2 4r }. (46 2IQ 2IQ (a If α<1, te frst term of (46 s neglgble wt respect to r1 α/2 2IQ, n fact, were n ( t t 1 σ u dw u 2 I {( X 2 >r,( X 1 2 4r } r 1 α/2 Terefore, (46 can be wrtten as [ n ln(1/i {( X E 2 >r } r 1 α/2 2IQ [o P (1 + r 1 α/2 n ln(1/i {( X 2 >r }, r 1 α/2 ] 1 β ln 1. n ( M 2 I {( X 2 r,( M 2 4r } r 1 α/2 Usng (37, Lemma 3.1( and Teorem 3.4, we arrve at n ( M 2 I {( X 2 r,( M 2 4r } r 1 α/2 n ( M 2 I {( M 2 9r /4} + o P ( 1 α/2 r 1 α/2 n ( M 2 I {( M 2 9r /4} r 1 α/2 ]. (47

29 Nonparametrc tests for patwse propertes of semmartngales 89 ( t t 1 x 3 r /2 x μ(dx,dt 3 r /2< x 1 xν(dx2 r 1 α/2 ( α/2 = R + Tc+ Tc 1 α/2 P Tc, r were te term R as varance cr α/2 and so converges to zero n probablty. Snce r 1 α/2, we arrve at ˆ IV IV 2 ˆ IQ st N(, 1. (b If α>1, defne R t := s t I { M s > }. Ten, by (37, te last term (tmes 2IQn(46 domnates n ( M 2 I {( M 2 r /16} o P ( 1 α/2 = 1 [ ( M 2 I { R=} + ( M 2[ I {( M 2 r /16} I { R=}] ] o P ( 1/2 α/2 o P ( 1/2 α/2 + ( t t 1 x x μ(dx,dt < x 1 xν(dx 2 ( M 2 I {( M 2 >r /16, R=}. (48 Frst, ( M 2 I {( M 2 >r /16, R=} = [ t ] [M]+2 (M s M t 1 dm s I {( M 2 >r /16, R=}. t 1 As n Lemma 3.1, te sum of te rgt-and terms wtn brackets s of order u n = (n/ log n 1/α so tat t t 1 (M s M t 1 dm s = u n t t 1 (M s M t 1 dm s P u n snce u n = ( n (1 α/2 log n 1/α +. Teorem 3.4 appled wt u = 1/2 yelds tat wt ε = 1/2, ( t t 1 x x μ(dx,dt < x 1 xν(dx 2 = ε 2 α/2 Y + Tcε 2 α + Tcε 2 2α,

30 81 R. Cont and C. Mancn were var(y 1. Terefore, n (48, we reman wt 1/2 α/2 [ o P (1 + α/4 Y + Tc+ Tc 1 α/2 [M]I {( M 2 >r /16, R=} 1 α/2 were te dvergence s due to te fact tat 1/2 α/2 + wle tends to zero n probablty snce ts expected value s domnated by avng used te fact tat ] a.s. +, [M]I {( M 2 >r /16, R=} 1 α/2 n 1 α/2 E1/2[( 2 ] [M]I { R=} P 1/2 {( M 2 >r /16, R = } n ( 1/2 1 α/2 x ν(dx 4 (2 α/2 β1/2 = (1 β/2, x P {( M 2 >r, R = }=P { ( M 2 I { R=} >r } E[( M 2 I { R=}] r (49 = x x2 ν(dx r = 2 α/2 β. On te oter and, te frst term n (46 s neglgble wt respect to 1/2 α/2 (te speed of dvergence of ( n ( M 2 I {( M 2 r /16} o P ( 1 α/2 / because n ( t t 1 σ u dw u 2 I {( X 2 >r,( X 1 2 4r } log(1/ αβ/2 1/2 α/2 1 α/2 = (α/2(1 β log 1. Terefore, (46 explodes to +. Fnally, f α = 1n(46, ten te frst term s neglgble, as n ( t t 1 σ u dw u 2 I ( {( X 2 >r,( X 1 2 4r } = O p (1 β/2 log 1. For te second term, we take a δ> suc tat 2/3 <β+ δ<1, we coose ε = (β+δ/2 and we use te same steps as were used to reac (48 forα>1, but we consder R t = s t I { M s >ε} n place of R t. Also usng Teorem 3.4, we obtan tat te second term n (46 domnates Y ε 3/2 2IQ + ε 2IQ [M]I {( M 2 >r /16, R=} 2IQ 2 n t t 1 (M s M t 1 dm s I {( M 2 >r /16, R=} 2IQ, were te varance of Y tends to 1 so tat Y ε 3/2 / tends to zero n probablty. Te second term tends to + at rate ε/. Te trd term s neglgble wt respect to ε/ : applyng (49

31 Nonparametrc tests for patwse propertes of semmartngales 811 wt R n place of R and te Caucy Scwarz nequalty, we get [ 1 t ] E x 2 μ(dx,dti ε {( M 2 >r /16, R=} = O( δ/2. t 1 x 1 Fnally, te last term s also neglgble snce te speed of convergence to zero of te numerator s u n = n/ log 2 n (as n te proof of Lemma 3.1 and u n +. So, even for α = 1, te normalzed bas ( IVˆ IV/ 2IQˆ dverges to +. Proof of Proposton 4.1. As n Lemma A.2 wt r n place of r as bound for max,...,n a n, usng te fact tat α<1 and applyng Lemma 3.1(, we deduce tat Hˆ as te same lmt n probablty as X T n ( X + MI { N=,( M 2 r } wen. Moreover, snce a.s. N T < and n X I {( M 2 >r } = O P ( (1 αβ/2 log(1/, takng R t = s t I { M s > r }, te above term as lmt n probablty equal to n ( X T lm X + MI {( M 2 r } [ n t = X T X T lm t 1 x x μ(dx,dt T r lm M ( I {( M 2 r } I { R=}. r < x 1 xν(dx ] Usng te fact tat P { R 1} =O( 1 αβ/2, as was used after (25, we deduce tat MI {( M 2 r, R 1} = O P ( (1 αβ/2. Usng te Hölder nequalty wt exponents p = q = 2, we ave MI {( M 2 >r, R=} = O P (r (1 αβ/2. Fnally, T x r x μ(dx,dt L2 and r < x 1 xν(dx m so tat P Hˆ,T JT + mt. Acknowledgements A prevous verson of ts workng paper appeared as Nonparametrc test for analyzng te fne structure of prce fluctuatons, Columba Fnancal Engneerng Report Ts researc was supported n part by te European Scence Foundaton program Advanced Matematcal Metods n Fnance, by Isttuto Nazonale d Alta Matematca and by MIUR Grants Nos and We tank Jean Jacod and Suzanne Lee for mportant comments.

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