Optimum thresholding using mean and conditional mean square error

Transcription

1 Optmum tresoldng usng mean and condtonal mean square error José E. Fgueroa-López Cecla Mancn! " #!$%&'! * * + *,+!! - +*,. + / /! *'%&%&

2 Optmum tresoldng usng mean and condtonal mean square error José E. Fgueroa-López and Cecla Mancn Marc 4, 7 Abstract We consder a unvarate semmartngale model for te logartm of an asset prce, contanng jumps avng possbly nfnte actvty IA. Te nonparametrc tresold estmator ˆ IV n of te ntegrated varance IV := T σ sds proposed n 6 s constructed usng observatons on a dscrete tme grd, and precsely t sums up te squared ncrements of te process wen tey are under a tresold, a determnstc functon of te observaton step and possbly of te coeffcents of X. All te tresold functons satsfyng gven condtons allow asymptotcally consstent estmates of IV, owever te fnte sample propertes of ˆ IV n can depend on te specfc coce of te tresold. We am ere at optmally selectng te tresold by mnmzng eter te estmaton mean square error MSE or te condtonal mean square error cmse. Te last crteron allows to reac a tresold wc s optmal not n mean but for te specfc pat at and. A parsmonous caracterzaton of te optmum s establsed, wc turns out to be asymptotcally proportonal to te Lévy s modulus of contnuty of te underlyng Brownan moton. Moreover, mnmzng te cmse enables us to propose a novel mplementaton sceme for te optmal tresold sequence. Monte Carlo smulatons llustrate te superor performance of te proposed metod. Keywords: Tresold estmator, ntegrated varance, Lévy jumps, mean square error, condtonal mean square error, modulus of contnuty of te Brownan moton pats, numercal sceme JEL classfcaton codes: C6, C3 Introducton We consder te model dx t = σ t dw t +dj t, were W s a standard Brownan moton, σ s a cadlag process, and J s a pure jump semmartngale SM process. Assume we ave at our dsposal a record {x,x t,..,x tn } of dscrete observatons of X spanned on te fxed tme nterval,t, defne Z or n Z te ncrement Z t Z t for any process Z, and defne tresold functon rσ, any determnstc non-negatve functon of te observaton step, and possbly of te coeffcents of X, suc tat for any value σ R rσ, lm rσ, =, lm log = +. We know tat ten te Tresold Realzed Varance or Truncated Realzed Varance ˆ IV n := X I { X rσ t, }, Department of Matematcs, Wasngton Unversty n St. Lous, MO, 633, USA fgueroa@mat.wustl.edu Department of Management and Economcs, Unversty of Florence, va delle Pandette 9, 57 cecla.mancn@unf.t

3 were := t t, gves a consstent estmator of te Integrated Varance IV := T as sup, as soon as σ s a.s. bounded away from zero on,t. In te case were te jump process J as fnte actvty FA and te observatons are evenly spaced, te estmator s also asymptotcally Gaussan. However te fnte sample propertes of IVn ˆ can depend on te specfc coce of te tresold TH. Te estmaton error s large wen eter te tresold s too small or wen t s too large. In te frst case too many ncrements are dscarded, ncluded te ncrements bearng only small and neglgble jumps, and TRV underestmates IV. In te second case too many ncrements are kept wtn TRV, ncluded many ncrements contanng jumps, leadng to an overestmaton of IV. In ts paper we look for an optmal tresold, by consderng te followng two optmalty crtera: mnmzaton of MSE, te expected quadratc error n te estmaton of IV; and mnmzaton of cmse, te expected quadratc error condtonal to te realzed pats of te jump process J and of te volatlty process σ s s. Assumng evenly spaced observatons, te two quanttes MSE and cmse are explct functons of te TH and under eac crteron t turns out tat for any semmartngale X, for wc te volatlty and te jump processes are ndependent on te underlyng Brownan moton, an optmal TH exsts, and s a soluton of an explctly gven equaton, te equaton beng dfferent under te two crtera. Furter, under eac crteron te optmal TH s unque, at least for gven classes of processes X. Te caracterzng equaton depends on te observaton step and so does ts soluton. Te optmal TH as to tend to as tends to zero and, under eac crteron, an asymptotc expanson wt respect to s possble for some terms wtn te equaton, wc n turn mples an asymptotc expanson of te optmal TH. Under te MSE crteron, wen X s Lévy and J as eter fnte actvty jumps or te actvty s nfnte but J s symmetrc strctly stable, te leadng term of te expanson s explct n, and n bot cases s proportonal to te modulus of contnuty of te Brownan moton pats and to te spot volatlty of X, te proportonalty constant beng Y, were Y s te jump actvty ndex of X Tus te ger te jump actvty s, te lower te optmal tresold as to be to dscard te ger nose represented by te jumps, n order to catc nformaton about IV. Te leadng term of te optmal TH does not satsfy te classcal assumptons under wc te truncaton metod as been sown n 6 to consstently estmate IV, owever at least n te fnte actvty jumps case t turns out tat te tresold estmator of IV constructed wt te optmal TH s stll consstent. Te assumptons needed for te cmse crteron are a lttle bt less restrctve, and we fnd tat, for constant σ and FA jumps, te leadng term of te optmal TH stll as to be proportonal to te modulus of contnuty of te Brownan moton pats and to σ. One of te man motvatons for consderng te cmse arses from a novel applcaton of ts to tuneup te tresold parameter. Te dea conssts n teratvely updatng te optmal TH and estmates of te ncrements of te contnuous and jump components Xt c = t σ sdw s and {J t } t, respectvely. We llustrate ts metod on smulated data. Mnmzaton of te condtonal mean square estmaton error n te presence of nfnte actvty jumps n X s object of furter researc. σ sds, An outlne of te paper s as follows. Secton deals wt te MSE: te exstence of an optmal tresold ε s establsed for a qute general SM X; for a Lévy process X, unqueness s also establsed Subsecton. and te asymptotc expanson for te optmal TH s found n Secton.3, n bot te cases of a fnte jump actvty Lévy X and of an nfnte actvty symmetrc strctly stable X. In Secton 3, for any fnte jump actvty SM X, consstency of IVn ˆ s verfed even wen te tresold functon conssts of te leadng term of te optmal tresold, wc does not satsfy te classcal ypotess. Secton 4 deals wt te cmse n te case were X s a SM wt constant volatlty and FA jumps: exstence of an optmal TH ε s establsed, ts asymptotc expanson s found, ten unqueness s obtaned. In Secton 5 te results of Secton 4 are used to construct a new metod for teratvely determne te optmal tresold value n fnte samples, and a relablty ceck s executed on smulated

4 data. Secton 6 concludes and Secton 7 contans te proofs not gven n te man text. Acknowledgements. José Fgueroa-López s researc was supported n part by te Natonal Scence Foundaton grants: DMS-4969 and DMS-636. Cecla Mancn s work as benefted from support by GNAMPA Italan Group for researc n Analyss, Probablty and ter Applcatons. It s a subunt of te INdAM group, te Itaan Group for researc n Hg Matematcs, wt ste n Rome and EIF Insttut Europlace de Fnance, subunt of te Insttut Lous Baceler n Pars. MEAN SQUARE ERROR: general results We compute and optmze te mean square error MSE of IVn ˆ passng troug te condtonal expectaton wt respect to te pats of σ and J: MSE := E IV ˆ n IV = E E IV ˆ n IV σ,j. Condtonng on σ, as well as assumng no drft n X, s standard n papers were MSE-optmalty s looked for, n te absence of jumps see e.g.. We assume evenly spaced observaton over a fxed tme orzon,t, so tat t = t,n = n, for any =...n, wt = n = T/n. Denote by ε te square root of a gven tresold functon: ε := rσ,. ˆ IVn and MSE are n fact functons of ε oter tan of, and we ndcate tem by ˆ IV n ε, MSEε. Note tat for ε = we ave ˆ IV n =, so MSEε = EIV ; as ε ncreases some squared ncrements X are ncluded wtn ˆ IV n, so ˆ IV n becomes closer to IV and MSEε decreases. However, f J, for ε + te quantty MSEε ncreases agan, snce ˆ IV n ncludes all te squared ncrements X and tus ˆ IVn estmates te global quadratc varaton IV + s T X s of X at tme T, and MSEε becomes close to E s T X s. We look for a tresold ε gvng MSEε = mn ε, MSEε. In ts secton we analyze te frst dervatve MSE ε and we fnd tat an optmal tresold exsts, n te general framework were X s a semmartngale satsfyng A below, and we furns an equaton to wc ε s a soluton, wle n Secton., we fnd tat ε s even unque. Te equaton as no explct soluton, but ε s a functon of and we can explctly caracterze te frst order term of ts asymptotc expanson n. Clearly we can always fnd an approxmaton of te optmal tresold wt arbtrary precson makng use of numercal metods. Let us denote X := XI { X ε }, σ := t t σ sds, m := J. To guarantee tat W remans a Brownan moton condtonally to σ and J, we need to assume te followng A. A.s. σ s > for all s, and σ, J are ndependent on W. Teorem. Under A and te fnteness of te expectaton of te terms below, for fxed and ε >, we ave MSE ε = ε Gε, were Gε := E a ε ε + ε m b j ε IV, a ε := e σ +e ε+m σ, σ j π b ε := E X σ,j = e ε m σ ε+m +e ε+m σ σ ε m + m +σ π m+ε σ π m ε σ e x dx. It clearly follows tat MSE ε > f and only f Gε > and, tus, to our am of fndng an optmal tresold, t suffces to study te sgn of Gε as ε vares. 3

5 Notaton. For brevty we sometmes omt to precse te dependence on ε of a ε and b ε. For a functon fε we sometmes use f+ for lm ε + fε. For two functons fx,gx of a non-negatve varable x wc tends to respectvely to +, by f g, or g f we mean tat f = og as x respectvely x +, by f g we mean tat bot f = Og and g = Of as x respectvely x +, wle by f g we mean tat f and g are asymptotcally equvalent.e. f/g. We denote φx = e x + π, Φx = φsds. x.o.t means ger order terms Proof of Teorem. Under A we ave tat condtonally to σ,j te ncrement X = t t σ s dw s + J s a Gaussan r.v. wt law Nm,σ, wc allows to compute te condtonal expectaton E IV ˆ n σ,j. We ave and E ˆ IV n σ,j = b ε = E ˆ IV n ε σ,j = = ε m + m +σ σ π e ε m E X 4 σ,j+ σ ε+m +e ε+m σ σ ε m π ε+m σ e t dt+ e t dt, E X j X σ,j j> e ε m σ σ ε 3 +m ε +m ε+m 3 +5m σ +3σε ε m σ + e t dt+ e ε+m σ σ ε 3 m ε +m ε m 3 5m σ +3σε ε+m σ e t dt m 4 +6m σ +3σ 4 + π avng used tat condtonally to σ and J, X and j X are ndependent. It follows tat MSEε = E e ε m σ σ ε 3 +m ε +m ε+m 3 +5m σ +3σε + e ε+m σ σ ε 3 m ε +m ε m 3 5m σ +3σε ε m σ + e t dt+ b εb j ε IV j> ε+m σ ε m + m +σ σ π e t dt m 4 +6m σ +3σ 4 π b b j, 3 j> e ε m σ ε+m +e ε+m σ σ ε m π ε+m σ e t dt+ e t dt +IV. MSEε s a dfferentable functons of ε, terefore to fnd te mnmum on,+ of MSEε we can study te sgn of ts frst dervatve MSE ε. Snce MSE ε = d dε E IV ˆ n ε IV d dε E IV ˆ n ε, we begn to compute d dε E IV ˆ n ε σ,j. Note tat d dε b ε = e ε m σ +e ε+m σ ε+m ε m σ π 4

6 e ε m σ +e ε+m σ σ + m +σ π σ π e ε m σ +e ε+m σ = ε e ε m σ +e ε+m σ σ π = ε a ε, so tat d dε E IV ˆ n n ε σ,j = ε a ε 4 s strctly greater tan zero for all values of ε >. As for d dε E IV ˆ n ε σ,j, note tat te term j> b b j n 3 can be wrtten as j b b j, so ts dervatve concdes wt j ε a b j +b ε a j, owever b a j = b a j b a j j = a b j a b = a b j so tat and j ε a b j +b ε a j = n Remark. If also J, we ave j j ε a b j, d dε E IV ˆ n ε σ,j = ε 4 n a ε+ b εb j ε 5 j> = ε 4 a +ε a b j d dε MSEε = ε j j E ε a +a b j IVa. j = ε E a ε + j b j IV = ε Gε. MSE = EIV > and, for small, lm MSEε >. ε + Corollary. Under te same assumptons of Teorem, even n te absence of jumps, an optmal tresold exsts and s soluton of Gε =. Proof. Note tat a ε and b ε are contnuously dfferentable functons of ε, and, wt fxed = T n, 6 a ε = σ 3 π m σ a = e, b =, σ π a + =, b + = E X σ,j = m +σ, σ ε m +e ε+m σ ε+m, b ε = ε a ε, e ε m so we fnd tat G = 4 σ π E e m σ IV <, and lm ε + Gε = +, so tere exsts ε + > : MSE ε > on ε +,+. On te compact set,ε + te contnuous functon MSE as necessarly absolute mnmum value MSE, and snce on ε +,+ MSE s ncreasng we ave tat on,+ te absolute mnmum s MSE. MSE ε s contnuous and assumes bot negatve and postve values, tus equaton Gε = as a soluton. Any mnmum pont of MSE on,+ as to be a statonary pont, so t as to solve te equaton. 5

7 Remark. In prncple MSEε could even ave many ponts ε were te absolute mnmum value MSE of MSE on,+ s reaced; MSE could even ave an nfnte number of local not absolute mnma. To determne te number of solutons to Gε =, we need to study te sgn of G ε correspondng to te convexty propertes of M SEε, but ts s not easy. Defne g ε := ε + j b j IV, so tat Gε = Ea εg ε. We can easly study te functons g, snce we know tat g = IV <, lm ε + g ε = + and g ε = ε+ε j a > for all ε >. However wtn te jont functon Gε te presence of te terms a ε makes t dffcult even to know weter a g s postve.. Wen X s Lévy Let us assume A. X s Lévy. We now ave tat σ > s constant and X are..d., so te equaton caracterzng MSE ε = s muc smpler: from 6, snce wtn a j b j te term m of a s ndependent on te terms m j of b j, we ave MSE ε = ε Gε = ε nea ε ε +n Eb ε IV. Teorem. If X s Lévy, equaton ε +n Eb ε IV = 7 as a unque soluton ε and, tus, tere exsts a unque optmal tresold, wc s ε. Proof. For ε > we ave MSE ε > f and only f Gε >, wc n turn s true f and only f were, settng m := m = J, we recall tat we ave Te sgn of gε s studed as follows: gε := ε +n Eb IV > σ Eb = E e ε m σ ε+m+e ε+m σ ε m π + m +σ ε m σ e t dt+ π g = σ T <, lm gε = +, ε + g ε = ε+n εea ε+m σ e t dt. so tat g ε > for all ε >, n >. Tat mples tat gε starts at ε = from a negatve value and strctly ncreases towards +, as ε ncreases, so tat tere exsts a unque ε suc tat gε < for ε,ε, gε = and gε > for ε ε,+. Tat mples n turn tat MSEε as a unque mnmum pont n ε, wc s ten te optmal tresold we were lookng for: ε s te unque soluton of equaton 7, correspondng to gε = Gε =. Te equaton n 7 as no explct soluton, owever we can gve some mportant ndcatons to approxmate ε. 6

8 . Asymptotc beavor of Eb ε For te rest of Secton we assume tat ε := ε = ε, even wen for brevty we omt to ndcate te dependence on. We stll are under A, so recall tat Eb ε = E σ n W + n J { σ n W+ n J ε}, s constant n. Note tat Eb ε s fnte for any Lévy process J, regardless of weter J as bounded frst moment or not. We consder two cases: te case were J s a fnte jump actvty process and te one were ts s a symmetrc strctly stable process. Te asymptotc caracterzaton of Eb ε wll be used n Subsecton.3 to deduce te asymptotc beavor of te optmal tresold ε. We antcpate tat n Subsecton.3 we wll also see tat an optmal tresold ε as to tend to as and n suc a way tat ε +... Fnte Jump Actvty Lévy process Teorem 3. Let X be a fnte jump actvty Lévy process wt jump sze densty f and wt jump ntensty λ. Suppose also tat te restrctons of f on, and, admt C extensons on, and,, respectvely. Ten, for any ε = ε suc tat ε and ε, as, we ave Eb ε = σ σε e ε σ +λ ε3 π 3 Cf+O +o ε e ε σ +o ε 3, were above Cf := f + +f. Proof. By defnton, Eb ε = E n X { n X <ε, n N=} +E n X { n X <ε, n N } =: G +L. 8 By Lemma S. and Lemma S.5 wt k = n 3, provded tat ε, we ave wc sows te result. L := E n X ε 3 { n X <ε, n N } λ Cf,, 9 3 G := σ σε e ε σ +O +o ε e ε σ, π.. Strctly stable symmetrc Lévy process Let us start by notng tat Eb ε = E σw +J { σw +J ε} = σ E W { σw +J ε} +σe W J { σw +J ε} +E J { σw +J ε} =: C ε+d ε+e ε. Te frst term above can be wrtten as were C ε = σ σ E W { σw +J >ε} = σ σ C + ε+c ε, C + ε = E W {W+σ / J >σ / ε}, C ε = E W {W+σ / J < σ ε}. / 7

9 By condtonng on J and usng te fact tat EW {W>x} = xφx+ Φx, for all x R, we ave φ + Φ. C ± ε = E ε σ J σ ε σ J σ ε σ J σ In wat follows, we determne te beavor of te above quanttes under te assumpton tat ε. Te proofs of te followng Lemma and Lemma are n an Appendx. Lemma. Suppose tat {J t } t s a symmetrc Y-stable process wt Y,. Ten, tere exst constants K and K suc tat: ε E φ σ J σ = e ε σ K ε Y 3 +.o.t. π E J φ ε J = K ε Y +.o.t.. Lemma. Suppose tat {J t } t s a symmetrc strctly stable process wt Lévy measure C x Y dx. Ten, te followng asymptotcs old: ε E Φ σ J σ = C Y ε Y +O ε Y ε +O E φ σ J σ, E J C { σw +J ε} = Y ε Y +O ε Y +O 4 Y +O Y. 3 We are ready to sow our man result n ts part: Teorem 4. Let X t = σw t +J t, were W s a Wener process and J s a symmetrc strctly stable Lévy process wt Lévy measure C x Y. Ten, for any ε = ε suc tat ε and ε, as, we ave Proof. From Lemmas and, Eb ε = σ σ π εe ε σ + C Y ε Y +.o.t.. ε C + ε = E σ J σ = ε σ ε = σ ε π e ε φ σ J σ e ε σ K ε Y 3 π σ K + Φ ε σ J σ σ σ / ε Y +.o.t., K ε Y + C Y ε Y +.o.t. were above we used tat ε Y / ε Y. Terefore, usng tat D = and Lemma, wt K 3 = C Eb ε = E σw +J { σw +J ε} = C ε+d ε+e ε = σ σ ε σ ε π e σ K σ / ε Y +K 3 ε Y +.o.t. were above we used tat ε Y 3/ ε Y. = σ σ π εe ε σ +K 3 ε Y +.o.t., Y,.3 Asymptotc beavor of ε We now assume A3. J and te support of any J t s R. We frstly see tat an optmal tresold ε = ε as to tend to as and n suc a way tat ε +. Ten we wll sow te asymptotc beavor of ε n more detal. 8

10 Remark 3. Note tat under A3, f ε mnmzes MSE, ten necessarly ε as. Indeed, f lmnfε = c >, ten on a sequence ε convergng to c we would ave IV ˆ n IV s T J si Js c n probablty, rater tan IV ˆ n IV ; snce P{ s T J si Js c > } >, te MSE could not be mnmzed. Lemma 3. Suppose X t = σw t +J t, were W s a Brownan moton and J s a pure-jump Lévy process of bounded varaton or, more generally, suc tat, for some Y,, /Y n J n P J, for a real-valued random varable J. Ten, ε n/ n, as n. Remark. If J as FA jumps and drft η, J t = ηt + N t k= γ k, ten we ave J P η and, tus, te above assumpton s satsfed wt Y =. ε Proof. We sow te result by contradcton. Suppose tat lmnf n n n <. For smplcty and wtout loss of ε generalty, we furter assume tat lm n n n =: L < as all te statements below are vald on a subsequence ε {n k } k. Let M, be suc tat sup n n n M. Also, for smplcty, let us wrte ε n for ε n and assume tat T = so tat n = /n. Consder te decomposton Eb ε = E σw +J { σw +J ε} = σ E W { σw +J ε} +σe W J { σw +J ε} +E J { σw +J ε} =: c ε+d ε+e ε. Note tat domnated convergence mples tat c n ε n = σ E W n { σw+ / n n J n / n ε n} σ E W { W L/σ} < σ, snce / n J n = Y n n /Y J n, n probablty. For d note tat σ W / n J n { σw+n / J n n / ε σ n} W +σ W n / ε n σ W +σ W M, terefore, agan by domnated convergence n d n ε n = σe Smlarly, snce / n J n { σw+ / W / n J n { σw+ / n J n / n ε n} n J n / n n. ε n} σ W + n ε n W +M, n e n ε n = E n / J n { σw+n / J n / n ε n} n. Fnally, let us wrte te equaton ε n +n Eb ε n n n σ = as ε n + n dn ε n + e n ε n = σ n c n ε n. 4 n n n n n Te rgt-and sde of te equaton converges to σ E W { W L/σ} >, wle te left and sde converges to and ts leads to a contradcton and terefore lm n n n ε =. We are now ready to sow more precsely te asymptotc beavor of ε. Te followng result covers te FA case. Proposton. Let J ave FA jumps and let ε = ε be te optmal tresold. Ten, ε σ ln, as. 9

11 Proof. For smplcty, n wat follows we take T = so tat = /n. Agan, recall tat ε s te soluton of ε +n Eb ε nσ =. Trougout, we sall use tat ε, as proved n te above lemma. For smplcty, we wrte ε nstead of ε. By te asymptotc beavor of Eb ε descrbed above, ε +n σ σε e ε σ +λ ε3 π 3 Cf+O +o ε e ε σ +o ε 3 nσ =, and, tus, usng tat = /n, ε σ 4 σ ε e ε σ +λ ε3 ε π 3 Cf+O+o e ε σ +o ε 3 =. 5 Now, snce = oε as assumed at te begnnng, we can wrte te prevous equaton as ε 4 σ ε e ε ε σ +o e ε σ +o ε =. π Dvdng by ε and rearrangng te terms, ε+o = 4 π σ e ε σ +o. 6 Ten, takng logartms of bot sdes and snce ln+o = o, lnε+o = ε σ 4σ ln+ln +o. 7 π wc can be wrtten as Defnng = ε /σ, we can wrte ε ln +o σ = ε 8 σ ln+ln +o π ln + ln ln π 8 o o = +. Terefore, makng and usng tat snce ε, ln Recallng tat = ε /σ, we conclude te result.. Te followng result specfes te asymptotc beavor of ε for symmetrc strctly stable processes. Proposton. Under te condtons of Teorem 4, te optmal tresold ε = ε s suc tat ε Yσ ln, as. Proof. For smplcty, we agan take T = so tat = /n and wrte ε nstead of ε. By te asymptotc beavor of Eb ε descrbed n Teorem 4, we can wrte ε +n Eb ε nσ = as ε +n σ σ ε e ε σ + C π Y ε Y +.o.t. nσ =, and, tus, usng tat = oε and ε = o ε Y, we ave 4C Y ε Y 4 σ ε e ε ε σ +o e ε σ +o ε Y =. 8 π

12 Dvdng by ε and rearrangng te terms, ε Y +o = Y C π σ e ε σ +o. Ten, takng logartms of bot sdes and snce ln+o = o, Ylnε+o = ε σ Yσ ln+ln C +o, π wc can be wrtten as Y ε ln σ + Y ln σ + Y Equvalently, wrtng = ε /σ and dvdng by, ln+o = ε σ ln+ln Yσ C π +o. Y ln + Yln K = + o. and usng tat snce ε, we get Yln. Recallng tat = ε /σ, we conclude te result. 3 Tresold crteron wen ε = Mlog Under te framework descrbed n 6, n te case of equally spaced observatons, te tresold crteron allows convergence of IV ˆ n := X I { X rσ t,} to IV T = T σ sds wen, for all =,...,n, we ave rσ t, = r and r s a determnstc functon of s.t. r,, as. Here we sow tat, under fnte actvty jumps, te same estmator s also r log consstent n te case rσ, = M log, were M are proper random numbers. Concretely, assume te followng A4. Let dx t = a t dt+σ t dw t +dj t, 9 were J t = N t γ for a non-explosve countng process N and real-valued random varables γ j, a,σ are càdlàg and a.s. σ := nf s,t σ s >. Recall tat a.s., te pats of a and of σ are bounded on,t. Defne σ := sup s,t σ s, ten, te followng Proposton and Corollary old true. Proposton 3. UnderA4,fwecooser = M log,wtanym ωsuctatm ω nf s t,t σ sω, σ, we ave: a.s. η >, for suffcently small : =,...,n, I { X +ηr } = I { N=}. Corollary. For all η >, we ave n X P I { X +ηr } IV, as. Proof of Proposton 3. In order to prove te proposton, we follow and modfy te proof of Teorem n 6, n tat we sow tat a.s., for all η >, for suffcently small, we ave =,...,n,i { N=} I { X +ηr }

13 =,...,n,i { N=} I { X +ηr }. Ten te tess follows. Call X = t t a s ds+ t t σ s dw s, ā = sup s,t a s, σ = sup s,t σ s and γω = mn l: Nl γ l ω, and note tat under our assumptons Pγ =. To sow a we use te followng key fact: sup sup {,...,n} B IVt B IVt IV log IV X M log sup sup IV log IV M log M ā + M log sup {,...,n} log log were B s a standard Brownan moton and we used te fact tat σ W s a tme canged Brownan moton 7, teorems.9 and., meanng tat we can represent σ W = B IVt B IVt. By te Paul Lévy law on te modulus of contnuty of te BM pats 5, teorem 9.5 and te monotoncty of te functon xln/x on,/e, t follows tat for suffcently small te frst two factors of te last lne of last dsplay are bounded above by, so tat sup X M log M := sup ā log M +sup M log log ā + log σ log + σ log wc tends to, as. Now, n order to sow, we defne {J} = { {,,...,n} : N }, and t s suffcent to prove tat for small enoug sup X {J} +η. Indeed, sup X r {J} = sup X r {J} sup X r {,..,n} M, r tus for all η > for suffcently small, t s ensured tat sup X {J} < +η, tat s: for all, f r N = ten necessarly we ave X < +η r, and follows. In order to sow we prove tat, for suffcently small, nf X {J} > +η. In fact frstly note tat for r suffcently small all te ncrements of N are eter or. It follows tat f N, ten N =, and J concdes wt te sze, say γ l, of a sngle jump J = γ l. Ten X γ l X and r r r + M, nf {J} X r γ σ log sup {J} X M log γ +η σ log and ts tends to + wen, tus nf X {J} > + η, meanng tat f r N ten necessarly X > r +η, as we needed. Proof of Corollary. Te proof of te Corollary s stragtforward, n tat a.s. we fx any η >, and for suffcently small we ave X I { X +ηr } = X I { N=} = X snce te last term tends to n probablty, as E n X I { N } N T O. X P I { N } IV T,

14 4 CONDITIONAL MEAN SQUARE ERROR: FA jumps case We now put ourselves under A. Te quantty of our nterest ere, cmseε =. E IV ˆ IV σ,j, s suc tat ω, cmse = IV and as soon as J ten cmse+ >, because IV ˆ ε + QV. Furter, from te proof of Teorem, we ave cmse ε = ε Fε, wt Fε =. g, g = ε a + b j IV. j We analyze te sgn of Fε: for n, fxed, σ and m also are fxed, and we ave F = IV n a <, snce b j =. Furter we ave F+ = + : to see t, frst note tat, from te expresson of b ε, b + = m +σ, ten g ε ε + j m j σ ε, as ε +. Moreover, eac a π / σ exp ε σ ε σ, tus, for for some suffcently large ε, F = n a g s a fnte sum of n postve terms a g Kπ / σ ε exp constant K and fxed σ, so Fε +, as ε +. Snce F s contnuous, t follows tat, even n te absence of jumps, an optmal tresold exsts and solves Fε =. We now assume also A3. Remark 4. As n Remark 3, f ε = ε mnmzes cmse, ten t as to be true tat ε, as. In wat follows we agan also fnd tat necessarly ε +. A4. We assume A4 wt a, constant σ > and n =. Wen consderng, we assume to ave a suffcently small so tat a.s. te number of jumps occurrng durng t,t s at most ; note tat for any t we ave m I t t,t J t, so wen consderng a jump tme t we assume tat s suffcently small so tat te sgn of any m I t t,t s te same as te one of J t, n partcular f J t ten te ncrements m approacng t are non-zero. 4. Asymptotc beavor of b ε and F Proposton 4. Under A, A3, A4, f ε = ε solves Fε = and ε = ε, ten ε +. Proof. ε s suc tat n a g =,.e. n a ε + j b j IV =. For smplcty let us rename ε by ε. If ε lmnf = L,+ we can fnd a subsequence tat we recall ε suc tat lm ε = L. Note tat.e. = Now we sow tat σ ε a + j b j σ n = ε ε n a j b j n a a + a b j σ n a, j n = n σ a n a j b j. = σ n n a j bj n tends to a strctly postve constant, wc n turn means tat equalty a s mpossble, snce on any sequence ε suc tat ε L te left term tends to L, wle te rgt one tends to +. Let us ten ceck tat σ n a j bj n a tends to a strctly postve constant. Snce J as FA, a.s. we only ave fntely many J t, and, for small, N T concdes wt n I m. Recallng te explct expresson of b j also reported below, we ave b j = j j,m j= b j + j,m j b j n N T σ ε εe ε σ +n N T σ σ π π ε σ e x dx 3

15 j,m j σ ε e ε m j π σ +e ε+ m j σ Now, te factors εe ε σ and ε postve, so j e ε m j + m j e ε m j σ + e ε+ m j σ ε b j n N T σ σ π ε σ σ e ε+ m j σ e x dx+ + m j e ε m j j,m j + j,m j σ e ε+ m j σ m j +σ mj+ε σ π were f ε L as ten te frst term of te rs above tends to d := σ m j ε σ L σ π L σ m j +σ mj+ε σ π e x dx, m j ε σ e x dx. of σ π are strctly e x dx < σ, wle eac term of te latter fnte sum tends to, snce mj, so te fnte sum tends to. It follows tat, for all, j b n n j d+o, were d < σ, so a j bj n d+o, and σ a j bj n σ d+o σ d >, a a as we wanted. We now ceck te asymptotc beavor of b and a wen ε = ε tends to as n suc a way tat ε +. To ts end, for fxed σ, we defne bε,m, := σ π e ε m σ ε+m+e ε+m σ ε m + m +σ m+ε σ π m ε σ e x / dx ε m aε,m, := e σ +e ε+m σ σ, π so tat b j ε = bε,m j, and a j ε = aε,m j,, and note tat, as, we ave see te Appendx for te smple proof, σ σ π ε e ε σ +.o.t., f m =, bε,m, = 3 σ m ε e m ε σ +.o.t., f m. π σ π e ε σ, f m =, aε,m, = 4 It follows tat g ε = ε + j = ε + π ε σ π b j ε IV = ε + j :m j e m ε σ +.o.t., f m, j :m j m σ j ε m j εe σ b j ε+ j :m j= j :m j= σe ε σ bj ε σ j :m j j :m j σ σ σ σ +.o.t.. 5 Gven any sequence ε = ε = ε, wc tends to as n suc a way tat ε +, we now sow tat Fε = F ε +Rε, were F ε s consttuted by te leadng terms of F, wle Rε gves te remander ger order terms. A soluton ε of F = non necessarly s suc tat F ε =, owever f wt te ε above we ave F ε ten te wole Fε, so t as to be true tat ε s close n a way tat wll become explct later to one of te solutons ε of F =. Proposton 5. Under A4, f ε as n suc a way tat ε + ten Fε = F ε +.o.t., were F ε := ε e ε σ ε ε e σ 4σ π σ π. 4

16 Proof. For smplcty, n wat follows, we omt te dependence on n te functons aε,m, and bε,m, defned n -. Let us recall tat, under Assumpton A4, N t s te number of jumps by tme t, {γ l } l are te consecutve jumps of J and {J} = {J} n := { : n N }. It follows tat, for s small enoug, Fε = aε,m bε,m j IV = / {J} + {J} ε + j aε,m ε + = n N T aε, aε,m ε + j :j {J} j :j {J} bε,m j + bε,m j + ε σ N T ++ NT j :j/ {J} j :j/ {J} bε,m j IV bε,m j IV bε,γ k +n N T bε, σ + N T + aε,γ l ε σ N T + k +n N T bε, σ k lbε,γ l= = n N T σ ε π e N T + l= σ k= ε σ N T + 4n N T σε π e ε σ N T + k= σ ε γ k e γ k ε σ π σ γ k ε π e σ ε σ N T 4n N T σε e ε σ π In wat follows we use te followng notaton: + k l σ ε e γ k ε σ γ k π v = ε, u l = v e γ l σ, s = e v π π Now, snce u l = s p l and p l, as, + +.o.t.. σ, p l = e γ l σ γl v. Fε = n N T N T s v σ N T ++σv s ε σ γ k p k n N T + k= + N T s p l v σ N T +σv s ε σ γ k p k n N T +.o.t. l= k l = n N T s v 4σv s n + N T s p l v 4σv s n +.o.t. σ σ l= = n N T + N T p l s v 4σv s n +.o.t. σ l= = n s v v 4σs n +.o.t. σ = ε e ε σ ε ε e σ 4σ π σ π. 6 5

17 Note tat v n, but s, so wc s te leadng term between v and ns depends on te coce of v. Remark 5. Te asymptotc beavor 6 also olds for any drft process {a t } t tat as almost surely locally bounded pats recall tat any cadlag a satsfes suc a requrement and tat s ndependent on W. Indeed, for nonzero drft, by condtonng also on a, we ave tat Fε = N T aε,ā ε σ N T ++ bε,γ k +ā k + bε,ā j σ + / {J} k= N T + aε,γ l +ā l ε σ N T + bε,γ k +ā k + k l l= j :j/ {J} j l :j/ {J} bε,ā j σ, were ā = t t a s ds/ and te ndces < < < NT are defned suc tat k J, wle J = for any oter / {,,..., NT }. Next, we can follow te same arguments as above usng te facts tat, f a as locally bounded pats, for any and k aε,ā = ε σ / φ +.o.t., aε,γ k +ā k = γk ε σ / φ e γ kā k σ +.o.t.. σ bε,ā = σ σε ε φ σ 4. Asymptotc beavor of ε Corollary 3. Under A, A3, A4 we ave tat +.o.t., bε,γ k +ā k = σ γ k ε φ ε Proof. In fact, from Proposton 4 and 6, we ave tat σ ln, as. F ε = σ n s v v n s 4σ +.o.t. =, σ γk ε σ e γ kā k σ +.o.t.. ε σ were v := ε / and s = e π. Tus, v n s 4σ +.o.t. =, 7 or, equvalently, ε σ ε e 4σ +.o.t. =, π wc s exactly te condton n 6, entalng tat ε σ ln, as. Now we am at approxmatng any optmal ε := ε, wc s suc tat F ε =, usng a sequence ε = v. To ts end, we am at makng Fε as quckly as possble, te only possble way beng renderng v and ns 6 of te same order. So we want to coose v suc tat wc s exactly te condton n 7. v = ns 4σ π +.o.t., 8 6

18 Remark 6. Tere exsts a determnstc functon w of suc tat w :,,+ and w + w 3 e w w π as. In fact, for example a functon of type w = ln lnln lny, wt any contnuous functon y tendng to π as, satsfes te 3 condtons. Ten v = σw satsfes 8. However te quckest convergence speed of F to would be reaced by coosng a functon w wc satsfes te followng tree more restrctve condtons, as, w + w 3 3 e w w π, were condton 3 means tat F ε. In fact suc a w exsts, snce te followng olds true. Teorem 5. Tere exsts a unque determnstc functon w of suc tat w :,,+ and te tree condtons, and 3 are satsfed. Suc a w turns out to be dfferentable and to satsfy also te ODE w = w, wc entals tat w +w w + log. 9 We fnally reac te unqueness of te optmal tresold ε as a consequence of te followng Proposton, wose proof s n Appendx. Proposton 6. Te frst dervatve d dε Fε of F s suc tat, wen evaluated at a functon ε of suc tat ε ε, +, and ε = 4σ s +.o.t., as, ten F ε = F ε +.o.t., as, were F ε = 4 σ π e Remark 7. Unquenessof ε. SnceF ε > foranyε,wereactatforsuffcentlysmallweave d dε Fε > on any sequence ε as n te above Proposton. Tat entals tat for any suffcently small te cmse optmal ε s unque. In fact f tere exsted two optmal ε < ε we would necessarly ave tat ε ε, + and ε = 4σ s π +.o.t., but ten, for small, on suc sequences F >, and ten on suc sequences F s strctly ncreasng, and tus F ε < F ε, wc s a contradcton, because n order to be optmal bot sequences ave to satsfy F ε =. ε σ ε 3. Remark 8. Te asymptotc beavor of te optmal tresold ε = ε for te cmse crteron s te same as te one of te optmal tresold ε for te MSE crteron under FA jumps. Ts s due to te fact tat ε solves F =, ε solves G =, F = F +.o.t., G = G +.o.t., and te leadng terms n F are te ones wt m =, wc do not depend on ω, tus tey are te same as for G. It follows tat, n te case of Lévy FA jumps, we ave F = F +.o.t. = EF +.o.t. = G +.o.t.. Also, an alternatve n eurstc justfcaton s tat we expect tat Fε = ag n n nea g, tus te asymptotc beavor of te ε satsfyng G = nea g = s te same as any ε satsfyng Fε =. Remark 9. Comparson wt te results n. In a FA jumps process X s consdered, eter of Lévy type, wt jumps szes avng dstrbuton densty satsfyng gven condtons, or of Itô SM type, wt determnstc absolutely We tank Andrey Sarycev for avng provded suc nce examples. We tank Salvatore Federco for avng provded a suc nce result. Te proof s avalable upon request. 7

19 contnuous local caracterstcs addtve process. Te estmators Ĵ n = XI { X >ε }, ˆNn = I { X >ε } are consdered, and, as, frstly t s sown tat te condton ε + s necessary and suffcent for te convergence to of bot MSE IV ˆ n IV stronger condton mplyng consstency of IVn ˆ and MSEĴn J T. Secondly, te autors sow tat ε σ MSE ˆN n N T e, ε meanng tat n order to ave L Ω,P convergence to of te estmaton error ˆN n N T a stronger condton on ε s needed, mplyng ε. Trdly, exstence and unqueness of an optmal tresold ˇε mnmzng E IV ˆ n IV + ˆN n N T for fxed s obtaned, and te asymptotc expanson n of ˇε as leadng term 3σ log. Te factor 3 s ger tan te factor of te leadng terms of ε and ε : tat s due to te fact tat te mnmzaton crteron for ˇε ncludes also te error on N T, wc requres tat ˇε s ger tan ε, and tus ˇε > ε s necessary. 5 A NEW METHOD FOR FINITE JUMP ACTIVITY PROCESSES In ts secton, we propose a new metod to tuneup te tresold parameter ε := rσ, of te Tresold Realzed Varance TRV ntroduced n. Ts s based on te condtonal mean square error cm SEε = E ˆ IV IV σ,j studed n Secton 4. We llustrate te metod for a drftless FA process wt constant volatlty σ. As proved teren, te optmal tresold ε s suc tat F ε = εg ε =, g ε = ε a + b j ε nσ, j were a ε and b ε are rewrtten ere for easy reference: ε m a ε := aε,m,σ := e σ +e ε+m σ σ, π b ε := bε,m,σ := σ π σ ε+m +e ε+m σ ε m e ε m + m +σ m+ε σ π For future reference we set m = m,...,m n and Fε;σ,m := aε,m,σ ε + bε,m j,σ nσ j m ε σ e x / dx. Te man ssue wt te optmal tresold ε les on te fact tat ts depends on σ and te ncrements m = m,...,m n of te jump process, wc we don t know. Note also tat, for small enoug, eac m wll be eter or one of te jumps of te process and a good proxy of m s actually n X { n X > ε}. Te dea s ten to teratvely estmatng ε, σ, and m as follows: 3. Start wt some ntal guesses of σ and m, wc we call ˆσ and ˆm. In te sequel, we obtan ˆσ by assumng tat tere s no jump; tat s, we set ˆm =,..., and ˆσ = T n n X. 3 To be consstent wt secton 4, I corrected ε ere wt ε and put ˇε for te optmal tresold of. Ceck weter you approve. 8

20 . Usng ˆσ and ˆm, we ten fnd an ntal estmate for te optmum ε tat we denote ε. Tus, under te no-jump ntal guess of te prevous tem, ε s suc tat F ε ;ˆσ, ˆm = or, more specfcally, ε solves te equaton: ε +n ˆσ e ε ε ˆσ ε+ ˆσ ˆσ e x / dx nˆσ =. 3 π π It s easy to see tat ε s of te form v nˆσ, were vn s te unque soluton of te equaton: vn +4n v n e v n vn + e x / dx n =. 3 π π ε ˆσ Fgure sows tat v n ranges from about 3 to 4 wen n ranges from to. 3. Once we ave an ntal estmate of ε, we can update our estmates of σ and m usng te estmators: ˆσ := ˆ IV n ε := X { X ε }, ˆm := n X { n X > ε },..., n nx { n n X > ε } We contnue ts procedure teratvely by settng ε k suc tat F ε k ;ˆσ k, ˆm k =, wc s ten used to get ˆσ k+ := X { X ε k }, ˆm k+ := n X { n X > ε k },..., n nx { n X > ε k }. 34 We stop wen te sequence of estmates ˆσ k+ stablzes e.g., wen ˆσ k+ ˆσ k tol, for some desred small tolerance tol. vn n, number of observatons Fgure : Te soluton v n of equaton 3 as a functon of n. Te prevous procedure resembles te one ntroduced n, wc s based on coosng te tresold ε so to mnmze E IV ˆ n IV + ˆN n N T, or equvalently te expected number of jumps mss-classfcatons: n Lossε := E { n X >ε, n N=} + { n X ε, n N>}. 35 It was proved teren tat, for a FA Lévy processes, te optmal tresold, denoted ˇε, s asymptotcally equvalent to 3σ ln/, as. Usng ts nformaton, an teratve metod was proposed, n wc, gven an ntal estmate ˇσ of σ, t was set ˇε k := 3ˇσ k ln, ˇσ k+ := X { X ˇε k }, k. 36 9

21 In te lgt of te procedure used n, we adopt ere also te followng smpler one, oter tan te procedure 3-34 descrbed above. Snce, as proved n Secton 4, te optmal tresold ε as te asymptotc beavor σ ln/, as, t s natural to consder te followng teratve metod to estmate ε: ε k := σ k ln, σ k+ := X { X ε k }, k, 37 startng agan from an ntal guess σ of σ. It can be proved tat f we take bot ˇσ and σ equal to te realzed quadratc varaton T n n X n bot 36 and 37, ten te sequences of estmates { σ k } k, {ˇσ k } k s nonncreasng and, tus, eventually ˇσ k = ˇσ k+ and σ k = σ k+, for some k. So, we can and wll set te tolerance tol to. 5. Smulaton results We now proceed to assess te metods ntroduced above. We take a Lévy Merton s log-normal model of te form: N t X t = at+σw t + γ j, were N s a Posson process wt ntensty λ and {γ } s an ndependent sequence of ndependent normally dstrbuted varables wt mean and standard devaton µ Jmp and σ Jmp, respectvely. We consder te followng estmators:. σ := T n n X ;. Te estmator ˆσ as defned n 33 wt ntal guesses ˆσ = T n n X and ˆm =,...,; 3. ˆσ k found wt te new metod descrbed by te teratve formulas 34. We stop wen ˆσ k ˆσ k tol = 5 ; 4. Te estmator ˇσ as n 36 wt k =, usng te tresold 3ˇσ log/ wt ˇσ = T n n X ; 5. Te estmator ˇσ k defned by 36 wt k suc tat ˇσ k = ˇσ k, k ; 6. Teestmator σ asn37wtk =,usngtetresold ε = σ log/wt σ = T n n X ; 7. Te estmator σ k defned by te teratve formulas 37 and wt k suc tat σ k = σ k, k ; 8. Tresold Realzed Varance usng te tresold ε = ω wt ω = Tresold Realzed Varance usng te tresold ε = ω wt ω =.495. Realzed Bpower Varaton BPV. Tresold Realzed Varance usng a tresold of te form 4 ω BPV/T wt ω =.49 ts s used n te recent work 4;. Te estmator ˆσ := n X { X ε } gven n 33 were ε s suc tat F ε ;ˆσ, ˆm =, but ts tme takng ˆσ = T n n X { n X ε } and ˆm := n X { n X > ε },..., n nx { n n X > ε } wt ε as defned n te tem 6 above. 3. Te estmator ˆσ k defned n 34, were ε k s suc tat F ε k ;ˆσ k, ˆm k =, were ε s gven as n te tem above, and k s suc tat ˆσ k ˆσ k tol = 5. j=

22 Te adopted unt of measure s year 5 days and we consder 5 mnute observatons over a mont tme orzon wt a 6.5 ours per day open market. For our frst smulaton, we use te followng parameters: σ =.4, σ Jmp = 3, µ Jmp =, λ =, = Te dependence of σ Jmp on σ Jmp on was done for easer comparson wt standard devaton of te ncrements of te contnuous component, wc s.4. So, te standard devaton of te jumps s about 7.5 tmes te standard devaton of te contnuous component ncrement. Te parameter values n 38 yeld an expected annualzed volatlty of.45, wc s reasonable. Table below sows te sample means and standard devatons based on smulatons below Loss equals te number of jump msclassfcatons as defned by 35, wle N s te number of teratons needed to fnd te estmator s value. As sown teren, te new proposed estmator tems 3, 3 performs te best, followed by te teratve metod 7 based on 37. It takes on average teratons to fns f we take as an ntal guess for te tresold te soluton of Eq. 3. However, f we take advantage of te asymptotc beavor of ε as n te metod above, one teraton suffces. Metod ˆσ stdˆσ Loss stdloss ε stdε N stdn Table : Estmaton of te volatlty σ =.4 for a log-normal Merton model based on smulatons of 5-mnute observatons over a mont tme orzon. Te jump parameters are λ =, σ Jmp = 3 and µ Jmp =. We now double te ntensty of jumps and consder te followng parameter settng: σ =.4, σ Jmp = 3, µ Jmp =, λ =, = 56.5, wc yelds an expected annualzed volatlty of.5. Te results are sown n Table. We agan notce tat te metods 3 and 3 outperforms all te oters, followed by metod 7 based on te asymptotc beavor ε σ ln/. Fnally, we consder a jump ntensty of jumps per year but we reduce σ and σ Jmp n order to obtan an expected annualzed volatlty of.39. Concretely, we set: σ =., σ Jmp =.5, µ Jmp =, λ =, = 56.5, TeresultsaresownnTable3. Inspteofbengatougsettng, tenewmetoddoesagoodjobandoutperforms all oters, except metod 7, wc s based on te asymptotcs ε σ ln/. Note tat n ts case t takes on average 5 teratons for te teratve metods to converge.

23 Est ˆσ stdˆσ Loss stdloss ε stdε N stdn Table : Estmaton of te volatlty σ =.4 for a log-normal Merton model based on smulatons of 5-mnute observatons over a mont tme orzon. Te jump parameters are λ =, σ Jmp = 3 and µ Jmp =. Est ˆσ stdˆσ Loss stdloss ε stdε N stdn Table 3: Estmaton of te volatlty σ =. for a log-normal Merton model based on smulatons of 5-mnute observatons over a mont tme orzon. Te jump parameters are λ =, σ Jmp =.5 and µ Jmp =. 6 Conclusons We consder te problem of estmatng te ntegrated varance IV of a semmartngale model X wt jumps for te log prce of a fnancal asset. In vew of adoptng te truncated realzed varance of X, we look for a teoretcal and practcal way to select an optmal tresold n fnte samples. We consder te followng two optmalty crtera: mnmzaton of MSE, te expected quadratc error n te estmaton of IV; and mnmzaton of cmse, te expected quadratc error condtonal to te realzed pats of te jump process J and of te volatlty process σ s s. Under gven assumptons, we fnd tat for eac crteron an optmal TH exsts, s unque and s a soluton of an explctly

24 gven equaton, te equaton beng dfferent under te two crtera. Also, under eac crteron, an asymptotc expanson wt respect to te step between te observatons s possble for te optmal TH. Te leadng terms of te two expansons turn out to be proportonal to te modulus of contnuty of te Brownan moton pats and to te spot volatlty of X, wt proportonalty constant Y, Y beng te jump actvty ndex of X. It turns out tat te tresold estmator of IV constructed wt te optmal TH s consstent, at least n te fnte actvty jumps case. Te results obtaned for te cmse crteron allow for a novel numercal way to tuneup te tresold parameter n fnte samples. We llustrate te superorty of te new metod on smulated data. Mnmzaton of te condtonal mean square estmaton error n te presence of nfnte actvty jumps n X s object of furter researc. 7 Appendx: addtonal proofs Proof of Lemma. Trougout, p t denotes te densty of J t and recall tat te caracterstc functon of J t s of te form E e ujt = e ct u Y. Let us also recall tat te Fourer transform and ts nverse are defned by Fgx = π R gze zx dz and F Gx = π R Gzezx dz. In wat follows, we set u := Let us start by notng tat ε E φ σ J σ = F φ σ ε σ u = π x φ σ ε σ e ux dx. x φ σ ε σ p xdx = Fxp xdx = ufp udu, were, snce J s a symmetrc stable process, Fp u = π / e c u Y. Terefore, we obtan te representaton ε E φ σ ± J σ = σ/ π e c u Y σ u +εu du. 39 In order to prove, let us make te cange of varables w = σ / u and, ten, expand n a Taylor s expanson exp cσ Y Y/ w Y as follows: π e cσ Y Y/ w Y w + ε σ /w dw = π e w + ε σ /w dw + I k,n, k= were I k,n := k! ck σ ky k Y/ π = k! ck σ ky k Y/ π w ky e w + ε σ /w dw w ky e w cos ε σ /w dw. Te frst term of s ten clear. For te subsequent terms, let us apply te formula for te cosne ntegral transformaton of w ky e w / as well as te asymptotcs for te generalzed ypergeometrc seres or Kummer s functon Ma,b,z: I k,n = { k! ck σ ky k Y/ π +ky Γ + ky = k! ck σ ky k Y/ π +ky Γ Γ ε Γ ky σ ky + + ky Γ Γ + ky M e ε σ + ky ; } ε ; σ ε σ ky +.o.t.. 3

25 In te asymptotc formula for te Kummer s functon above, te frst term respectvely, second term vanses f Γ ky/ respectvely, Γ/ + ky/ are nfnty. Ts appens wen ky/ or / +ky/ are nonpostve ntegers. It s now evdent tat tere exsts nonzero constants a k and b k suc tat Note tat a k I k,n = Γ ky ε ky k+ + b k Γ + ky e ε σ ε ky k Y +.o.t.. Terefore, ε Y + I k,n, for all k >. We now sow. Note tat ε E J φ σ J σ = were ε ky k+ ε k+y k++ ε /Y = ε /, ε Y + ε ky k+ e ε σ ε ky k Y. x φ σ ε σ xp xdx = Fxp xu = d du Fp u = d Y π du e c u Terefore, we ave te followng representaton: ε E J φ σ J σ = σ Yc 3/ π Furtermore, ε E J φ J = σ Yc π 3/ Y Yc = σ 3 Y π ufxp xudu, = π e c u Y Ysgnuc u Y. sgnu u Y e c u Y σ u +εu du. u Y e cuy σ u snεudu Next, we expand n a Taylor s expanson exp cσ Y Y/ w Y as follows: were π w Y e cσ Y Y/ w Y w sn σ ε / w dw. w Y e cσ Y Y/ w Y w sn σ ε / w dw = I k,n := k! ck σ Yk k Y/ π I k,n, k= w k+y e w sn ε / w dw. Ten, we agan apply te followng formula for te sne ntegral transformaton of w k+y e w / : I k,n = { k! ck σ Yk k Y/ π +k+y k +Y ε k +Y Γ + M + ; 3 } ; ε. Fnally, we use te relatonsp M k +Y + ; 3 ; ε = Γ 3 Γ k+y + Γ Γ 3 + k+y ε k+y k+y + e ε σ ε σ +.o.t., wc, n turn sows tat, We ten conclude te result of te Lemma. I k,n I,n ε Y. 4

26 Proof of Lemma. Let ε I n ± := E Φ σ J σ { ± ε σ J σ } For I n, + let us note tat for a constant K, Φz Kφz for all z and, tus, ε I n + KE φ σ J σ { } ε σ J = O E φ σ For te oter term, we decompose t as follows: In = φup ε R σ J σ,u ε σ J σ = φup ε du+ φup = P J Y ε + σ J σ φup du J Y ε σu Y ε σ J σ u ε σ J σ du.. Te frst term above s well-known to be P J /Y ε = Y C /Y ε Y + O ε Y. For te second term, let us frst recall tat tere exsts a constant K suc tat for all x >, Ex := PJ x C Y x Y Kx Y. 4 Terefore, For te frst term above, note tat ε Y φup J Y ε σu Y du = C Y + du φu Y Y ε σu Y du φue Y ε σu Y du. φu Y Y ε σu Y du = φu σuε / Y du, wc, by te domnated convergence teorem, converges to /, because ε /, as n. Smlarly, usng 4, we ave φue Y ε σu Y du K φu Y Y ε σu Y du = O ε Y. Terefore, we fnally conclude tat In = Y Cε Y +O ε Y, wc mples. We now sow 3. To ts end, let us frst consder E, ε := E J { σw +J ε,j,w } εσ = /Y = Y ε σ φx Y ε σ Y x ε φ w σ Y ε w u p ududx u p ududx. Let Eu := p u Cu Y and let us recall tat, for a constant K, Eu K u Y u Y Ku Y, for all u >. Next, E, ε = C Y + Y ε σ ε σ ε φ σ ε φ w σ Y ε w w u Y dudx Y ε w u Eududw 5

27 For te frst term above, note tat ε Y φ w Y Y Y ε w dw = Y Y σ We dvde te second term n two cases. If Y, ten ε Y ε w φ w u Eududx σ K Y ε Y Y Y Y Y Y K Y ε Y Y Y K Y ε Y ε φ σ σ ε w w Y dw. ε Y φ w σ Y ε w dw ε Y ε φ σ. ε σ w w Y dw Note tat te last lmt s vald provded tat w Y dw <, wc olds true wen Y. For Y >, let us frst observe tat z u u Y u Y du Y + z Y {z>} Y Y + Y. 4 Terefore, for a constant K, ε Y ε w φ w u Eududx σ K ε σ φ w dw K. σ ε We conclude tat Next, we consder E, ε = C Y ε Y +O ε Y +O Y. E, ε := E J { σw +J ε,j,w } = /Y = C /Y + /Y φx φx φx Y ε σ Y x σ Y x Y ε σ Y x σ Y x Y ε σ Y x σ Y x Te frst term on te rgt-and sde above can be wrtten as C Y Y Y /Y Y ε φx σ ε x σ ε x u p ududx u Y dudx u Eududx. Y dx C Y ε Y, were te last asymptotc relatonsp follows from domnated convergence teorem and te facts tat / /ε and x Y φxdx <. For te second term of E, ε, we ave two cases. For Y, we ave Y ε σ Y x Y ε σ Y x Y φx u Eududx σ Y x K Y φx u Y dudx σ Y x K Y Y Y = Y Y Y ε φx σ ε x σ ε x dx K ε Y, 6

28 were agan we used domnated convergence and use te fact tat φx x Y dx <. For Y >, we just use 4 to deduce tat Y φx Y ε σ Y x σ Y x u Eu dudx K Y φxdx, for a constant K. Fnally, we conclude tat Fnally, let us consder E, = C Y ε Y +O ε Y +O Y. E 3, ε := E J { σw +J ε,j,w } σ ε = /Y + /Y σ ε φx u p ududx σ Y x φx Y ε σ Y x σ Y x u p ududx. Usng te fact tat p u Ku Y for a constant K and all u >, te frst term above s suc tat σ ε /Y φx σ Y x Smlarly, te second term can be wrtten as /Y σ ε σ ε u p ududx K /Y = K Y σ Y K 4 Y Y σ Y x φx u p ududx K σ Y x Y ε Y Puttng togeter te prevous results, we obtan tat = o 4 Y φx σ Y x u Y dudx Y σ ε φxx Y dx φxx Y dx = o ε Y. Y σ Y = o ε Y. σ ε E ε = E J { σw +J ε} = E, ε+e, ε+e 3, ε = C Y ε Y +O ε Y +O 4 Y +O Y. φxx Y dx Proof of 3. Let and recall tat, for x >, Nx = x φzdz, Rx = Nx φx φx, Rx x x 3. x φzdz φx x 7

29 Ten, for fxed m > and small enoug suc tat ε < m, we ave bε,m, = σ m ε m φ σ m ε m σ m+ε m φ σ m+ε m σ m ε φ σ ε σ m+ε φ σ ε m ε m+ε +σ 3 3/ φ σ σ 3 3/ φ m ε σ ±m +σ m ε R m+ε σ = σ m ε φ m σ ε σ m+ε φ m σ ε σ m ε + φ mm ε σ ε 3 σ m+ε φ mm+ε σ ε 3 + σ3 m ε m ε 3/ φ σ σ3 m+ε m+ε 3/ φ σ ±m +σ m ε R σ It s now clear tat 3 olds true. We can smlarly deal wt te case m <. Te asymptotc beavor for aε,m, s drect. Proof of Proposton 6. Let us fx, and n =, ten d dε Fε = n a g +a g = σ 3 3 π e ε m σ ε m +e σ ε+ m g + ε+ m e ε m σ +e ε+ m σ σ π ε+ ε a j. j We now evaluate F ε at ε suc tat ε wt ε, as. Snce agan wen m we ave e ε m σ e ε+ m σ and ε m, ten F ε π = {J} σ e ε m σ m σ g +ε+ε j a j + Note tat wtn g n 5 we ave tat te fnte sum π = s ε σ j :j {J} m p j j sneglgblewrts snce ε j :j {J}. Terefore p j a.s. m j {J} ε σ ε e σ σ εe mj ε σ j :j {J} m j ε π j :j {J} e g σ ++ε a j +.o.t. j = j :j {J} σ m j εu j σ = n N T I { {J}} +n N T I { {J}} s, g = ε 4σ π εs n N T I { {J}} +n N T I { {J}} σ N T I { {J}} +N T +I { {J}} +.o.t.. Furter, N T n and ε, ten for all g = ε 4σ π εs +.o.t.. Moreover from 7 we reac tat j a j = j,j {J} s σ + u j j,j {J} σ +.o.t., and agan te second sum s neglgble wrt te frst one, tus, for all, ε j Now, usng 8, from {J} a g = NT σ s l= p l NT σ s l= p l a j = s ε σ n N TI {m } +n N T I {m=}+.o.t. = s ε σ +.o.t.. v 4σv s n v σ N T +σv s ε k l +.o.t. γ k p k n N T +.o.t. = 8

30 we reac tat {J} a g m σ = NT σ s l= p l v 4σv γ s n l σ +.o.t. and from {J} a g = n N T σ s v σ N T ++σv s s n N T σ v 4σv s n +.o.t. we reac tat {J} a gε σ = n N T σ s v 4σv ε s n σ +.o.t.. Tus ε N T k= γ k p k n N T = F π = v v N T γ l 4σs n u l σ σ n N T ε s σ σ l= +ε + s ε σ u σ + s σ +.o.t.. J J If now our sequence ε s suc tat v = 4σns +.o.t., and notng tat also j J p j γ j a.s. and tat nε = n v = v + ten F ε NT s π = v ons σ 3 p l γ l nε + εs σ + s ε σ n N T +.o.t. l= s = nεv ons σ 3 + 4nεs σ + s ε σ +.o.t. now v = 4σns +ov means also s = ε +oε, and tus s ε = ε +o ε +, terefore F ε π = ε ε ons ε s ε ns 8 s σ 3 + 4nεs σ σ σ +o+.o.t. = 8 σ s ε s ε +.o.t. = +.o.t.. References Band, F. & Russel, J.R. 7. Realzed covaraton, realzed beta, and mcrostructure nose. Workng paper of te Graduate Scool of Busness, Te Unversty of Ccago, January 5 Fgueroa-López J. & Nsen, J. 3: Optmally tresolded realzed power varatons for Lévy jump dffuson models. Stocastc Processes and ter Applcatons 37, Fgueroa-López J. & Nsen, J. 6: Second-order propertes of tresolded realzed power varatons of FJA addtve processes. Preprnt, avalable at ttps://pages.wustl.edu/fgueroa/publcatons. 4 Jacod, J.& Todorov, V.4. Effcent Estmaton of Integrated Volatlty n Presence of Infnte Varaton Jumps. Te Annals of Statstcs 43: Karatzas, I. & Sreve, S.E Brownan moton and stocastc calculus, Sprnger. 6 Mancn, C.: Non-parametrc tresold estmaton for models wt stocastc dffuson coeffcent and jumps. Scandnavan Journal of Statstcs, 36, Revuz, D. & Yor, M.. Contnuous martngales and Brownan moton, Sprnger. 9