338 A^VÇÚO 1n ò Lke n Mancn (211), we make te followng assumpton to control te beavour of small jumps. Assumpton 1.1 L s symmetrc α-stable, were α (,

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1 A^VÇÚO 1n ò 1oÏ 215c8 Cnese Journal of Appled Probablty and Statstcs Vol.31 No.4 Aug. 215 Te Speed of Convergence of te Tresold Verson of Bpower Varaton for Semmartngales Xao Xaoyong Yn Hongwe (Department of Matematcs, Nancang Unversty, Nancang, 3331) Abstract In ts paper, we consder te speed of convergence of te tresold verson of bpower varaton for a semmartngale, wc s drven by a standard Brownan moton and a pure jump Lévy process wt possbly nfnte actvty of te small jumps. Keywords: Speed of convergence, bpower varaton, semmartngales, ntegrated volatlty. AMS Subject Classfcaton: 6F5. 1. Introducton To represent te prce of a fnancal asset, we start wt a semmartngale (X t ) t [,T, drven by a standard Brownan moton W and a pure jump Lévy process L defned on a fltered probablty space (Ω, (F t ) t [,T, F, P) X t = x + b s ds + σ s dw s + L t, t [, T, (1.1) were b t, σ t are F t -adapted càdlàg processes, and L t = J t + M t wt J t := xµ(dx, ds) = Nt t γ j, M t := x[µ(dx, ds) ν(dx)dt. (1.2) x >1 j=1 Here µ s a Posson random measure on R [, T wt ntensty measure ν(dx)dt, N x 1 s a Posson process wt ntensty ν({x : x > 1}) <, and {γ j } are..d. random varables and ndependent of N suc tat J s a compound Posson process. We denote by µ(dx, dt) := µ(dx, dt) ν(dx)dt te compensated Posson random measure. In ts paper we allow for te nfnte actvty case, were small jumps of L occur nfntely often,.e. ν(r) =. In order to measure te degree of actvty of te small jumps, we use te Blumental-Getoor (BG) ndex of L, defned as { } α := nf δ : δ, x δ ν(dx) < 2. x 1 Te project was supported by te Natural Scence Foundaton of Jangx Provnce (2151BAB2121). Receved October 12, 212. Revsed February 6, 215. do: /j.ssn

2 338 A^VÇÚO 1n ò Lke n Mancn (211), we make te followng assumpton to control te beavour of small jumps. Assumpton 1.1 L s symmetrc α-stable, were α (, 2) s te BG ndex. Ts assumpton mples tat (Cont and Tankov, 24) ν as a densty of te form A/( x 1+α ) for some constant A R, and L as fnte varaton (FV) f and only f α (, 1). In te sequel, let IV = T σ2 sds be te ntegrated volatlty of X and set X c t = x + b s ds + σ s dw s, X J t = X c t + J t. For a semmartngale Z, we set Z = Z t Z t 1 and Z t = Z t Z t. Moreover, f(ω, ) p g(ω, ) means tat bot f(ω, ) = O p (g(ω, )) and g(ω, ) = O p (f(ω, )) as. We assume to be n te g-frequency stuaton wt observatons made at regular tmes t =, were = 1/n and = 1, 2,..., nt. Wen te underlyng process s contnuous (L ), for every t [, T, t s sowed n Proposton 1 of Mykland and Zang (26) tat nt n =1 ( ( X) 2 ) σsds 2 L S 2 σ 2 sdw s, (1.3) were W s anoter Brownan moton defned on an extenson of (Ω, (F t ) t [,T, F, P) and beng ndependent of F, and L S means F -stable convergence n law. Wen X nt exbts jumps, Mancn (29) as ntroduced ÎV := ( X) 2 I { X u }. Under Assumpton 1.1, by te Paul Lévy law of te Brownan moton pats ( P lm sup 1 nt =1 W 2 log( 1 ) 1 ) = 1, (1.4) t s proved n Corollary 2 and Teorem 4 of Mancn (29) tat ÎV P IV as, and te speed of convergence s dscussed n Mancn (211). Alternatvely, anoter way to estmate IV s to use bpower varaton BV nt 1 t := X +1 X. A lmt teorem for BV t s gven by Vetter (21) wen te semmartngale exbts jumps n te FV case. In addton, a tresold verson of bpower varaton =1 BIV t := nt 1 =1 X I {( X) 2 u } +1 X I {( +1 X) 2 u } (1.5) s constructed and ts asymptotc beavour s studed n Vetter (21).

3 1oÏ Ùö : 'uœv C Âñ Ý 339 In ts paper, nspred by Mancn (211), we want to study te speed of convergence of te tresold verson of bpower varaton, wc as smlar asymptotc beavour as te tresold estmator of IV ntroduced n Mancn (29). It s sown tat ts tresold estmator s also able to separate te ntegrated varance IV from te sum of te squared jumps. ten 2. Man Teorem and Its Proof Teorem 2.1 Under Assumpton 1.1, let u = c β for β (, 1) and c R +, BIV T m 2 1IV p Z + K I {1 α<2}, as, (2.1) L S were Z Z, K Cu 1/2 α/4 varable and m 1 = E Z = 2/π. for a constant C, Z s a standard normal random Lemma 2.1 (Lemma 6.1 n Cont and Mancn (211)) Let u = c β for β (, 1), ten we ave P{ X > u } = P{ M > u /4} + O( 1 αβ/2 ), lm sup Proof of Teorem 2.1 BIV T m 2 1IV nt 1 αβ/2 nt =1 P{ X > u } C. We ave te decomposton = X I {( X) 2 u } +1 X I {( +1 X) 2 u } m 2 1IV =1 [ = X J I {( X J ) 2 4u } +1 X J I {( +1 X J ) 2 4u } m 2 1IV [ + X J +1 X J (I {( X) 2 u,( +1 X) 2 u } I {( X J ) 2 4u,( +1 X J ) 2 4u }) [ + ( X +1 X X J +1 X J )I {( X) 2 u }I {( +1 X) 2 u } =: 3 I j (). j=1 Snce te jumps of X J ave fnte varaton, t follows by Teorem 3.3 n Vetter (21) tat I 1 ()/ converges stably n law to a mxed normal random varable, wc mples stable convergence of I 1 ()/ p(2)iq to a standard normal random varable wt p(2) = 1 + 2m 2 1 3m4 1.

4 34 A^VÇÚO 1n ò Next we sow I 2 () = o p ( )I {α 1} + o p (u 1 α/2 )I {1<α<2}. Note tat I {( X) 2 u,( +1 X) 2 u } I {( X J ) 2 4u,( +1 X J ) 2 4u } I {( X) 2 u,( X J ) 2 >4u }I {( +1 X) 2 u } + I {( X) 2 u }I {( +1 X) 2 u,( +1 X J ) 2 >4u } + I {( X) 2 >u,( X J ) 2 4u }I {( +1 X J ) 2 4u } + I {( X J ) 2 4u }I {( +1 X) 2 >u,( +1 X J ) 2 4u } =: Snce on {( X) 2 u, ( X J ) 2 > 4u }, we ave wc mples tat M > u. 4 I 2j (). 2 u M < X J M X u, (2.2) j=1 On te oter and, f X J > 2 u, ten from (1.4) and te assumpton wtout loss of generalty tat b, σ are bounded on Ω [, T (or troug localzaton smlarly to Lemma 4.6 n Jacod (28), we ave N a.s. (we omt a.s. below). Snce E[( M) 2 = t 1 x 1 x2 ν(dx)dt = A x 1 x 1 α dx C, t follows tat { 1 P } X J +1 X J I 21 () nt P{ M > u, N } = nt P{ N }P{ M > u } no() E( M) 2 ( ) = O. u u Moreover, we ave 1/2 X J +1 X J I 22 () P by te smlar analyss. Next we andle X J +1 X J I 23 (). On {( X J ) 2 4u } for all, we ave N = for suffcently small. In fact, snce J X c X J 2 u, so f N = 1 (te event of N > 1 s neglgble as ), ten J 1 by defnton of J, tus we would ave 1 J 2 u + X c and u + X c 3 u, but t s mpossble for small. Terefore, t olds tat X J +1 X J I 23 () X c +1 X c I {( X) 2 >u,( X J ) 2 4u } = X c +1 X c I {( X) 2 >u } X c +1 X c I {( X) 2 >u,( X J ) 2 >4u } =: I 23 (1) I 23 (2). (2.3)

5 1oÏ Ùö : 'uœv C Âñ Ý 341 Frst we sow I 23 (2)/ s neglgble. Snce { X J > 2 u } { N = 1} for suffcently small, t follows tat [ I23 (2) E 1 [ E X c +1 X c I { N=1} CE[N T, 1 [ NT E X c +1 X c =1 were we ave used te fact tat W and N are ndependent. Next we deal wt I 23 (1). For all, f s suffcently small, ten X c u /2. So f J + M + X c X > u, ten J + M > u X c > u /2, wc mples tat eter J (tus N = 1) or M > u /2. However, 1 [ E X c +1 X c I { N=1} CE[N T. In addton, from Lemma 2.1, we ave [ E X c +1 X c I { M > u /2} C P{ M > u /2} 1 αβ/2 = o p ( )I {α 1} + o p (u 1 α/2 )I {1<α<2}. On te oter and, a smlar result olds for X J +1 X J I 24 (). In a word, we ave sown tat I 2 () = o p ( )I {α 1} + o p (u 1 α/2 )I {1<α<2}. Next we sow I 3 () Cu 1/2 α/4 for < α < 2. Frst we rewrte I 3 () as [ X J I {( X) 2 u }( +1 X +1 X J )I {( +1 X) 2 u } [ + ( X X J )I {( X) 2 u } +1 X J I {( +1 X) 2 u } [ + ( X X J )I {( X) 2 u }( +1 X +1 X J )I {( +1 X) 2 u } =: K 1 () + K 2 () + K 3 (), (2.4) and defne k +1 := +1 X +1 X J +1 M. Snce [ X J = [ X J X c + X c σ s dw s + t 1 t follows tat σ s dw s, t 1 K 1 () + + J I {( X) 2 u } +1 M I {( +1 X) 2 u } b s ds +1 M I {( +1 X) 2 u } t 1 σ s dw s I {( X) 2 u }k +1 I {( +1 X) 2 u } =: t 1 3 K 1j (). j=1

6 342 A^VÇÚO 1n ò For te frst summand, snce J I {( X) 2 u } contrbutes only wen N, wc mples M > u, t follows tat P{ 1/2 K 11 () } nt P{ M > u, N }. For K 12 (), t as been proved tat durng te proof of Teorem 2.1 n Mancn (211). nequalty, we ave ( K 12 () ( M) 2 I {( X) 2 u } p u 1 α/2 (2.5) b s ds t 1 ) 2 Tus, usng te Caucy-Scwarz ( +1 M) 2 I {( +1 X) 2 u } p u 1/2 α/4. Next we deal wt K 13 (). From (2.5), we ave K 13 () σ s dw s +1 M I {( +1 X) 2 u } t 1 ( ) 2 σ s dw s ( +1 M) 2 I {( +1 X) 2 u } t 1 p IVu 1/2 α/4 u 1/2 α/4. Summng up, we ave K 1 () Cu 1/2 α/4. Te same concluson can be drven for K 2 (). Now we consder K 3 (). Usng (2.5) and te Caucy-Scwarz nequalty, we ave K 3 () M I {( X) 2 u } +1 M I {( +1 X) 2 u } [ 1/2 [ 1/2 ( M) 2 I {( X) 2 u } ( +1 M) 2 I {( +1 X) 2 u } p u 1 α/2. Terefore, we obtan tat I 3 () C 1 u 1/2 α/4 + C 2 u 1 α/2 Cu 1/2 α/4. Specally, f α (, 1), ten { x 1} x ν(dx) <. We are gong to sow I 3() = o p ( ). For any δ >, we rewrte X as X t = X 1 t + X δ t, X δ t = J δ t + M δ t, were X 1 t = x + J δ t = δ< x 1 ( ) b s xν(dx) ds + { x 1} xµ(dx, ds), M δ t = σ s dw s + J t, x δ xµ(dx, ds).

7 1oÏ Ùö : 'uœv C Âñ Ý 343 Notce tat t does not matter f we replace X J wt X 1 n I 3 (), so we study Ĩ 3 () := ( X +1 X X 1 +1 X 1 )I {( X) 2 u }I {( +1 X) 2 u } nstead. Snce X +1 X X 1 +1 X 1 X +1 X δ + X δ +1 X 1, (2.6) t follows tat te two summands n (2.6) can be andled n te same way. consder te frst one. Snce I {( X) 2 u } I { N=} + I { N=1, X δ >1 2 u } + I { N 2}, t follows tat ( P 1/2 ) X +1 X δ I { N=1, X δ >1 2 u }I {( +1 X) 2 u } ( P ( N = 1, X δ > 1 2 ) u ) Smlarly, nt P( N = 1)P( X δ > 1 2 u ) C E( X δ ) 2 ( P 1/2 (1 2 = O(). u ) 2 ) X +1 X δ I { N 2}I {( +1 X) 2 u } nt P( N 2) = O(). Next we As for Ĩ31() := X +1 X δ I { N=}I {( +1 X) 2 u }, te rest work we need to do s smlar to te proof of Teorem 3.3 of Step 3 n Vetter (21). In fact, t olds tat X +1 X δ X +1 J δ + X +1 M δ. Obvously, on {( +1 X) 2 u }, we ave +1 X δ +1 X + +1 X 1 u + u = 2 u for suffcently small, t follows tat I { +1 X u } I { +1 J δ <4 u } + I { +1 J δ 4 u, +1 M δ 2 u }. However, for some δ small enoug, we ave δ > 4 u (tus +1 J δ = ) for all < δ. Ten for Ĩ 31 (1) := 1/2 X +1 J δ I { N=}I {( +1 X) 2 u },

8 344 A^VÇÚO 1n ò we only ave to consder t on { +1 J δ 4 u, +1 M δ 2 u }. Now we ntroduce te fltraton (F t ) wc s te smallest fltraton contanng (F t ), and t makes J δ F - measurable suc tat W remans a Brownan moton wt respect to (F t ) and X 1 as te same representaton as before bot wt respect to (F t ) and (F t ). Terefore, we ave E [ +1 J δ I { +1 M δ 2 1/2 u } F t < Cu E [ +1 J δ +1 M δ F t = Cu 1/2 E [ +1 J δ E [ +1 M δ F t F t Cu 1/2 2, were we ave used te fact tat and E [ +1 M δ F t < C, E [ +1 J δ F t < C, E [ [ X I { N=} Ft 1 E X c [ Ft 1 + E M Ft 1 < C. Next by takng successve expectatons, t s easy to sow tat Ĩ31(1) P. Now we turn to Ĩ31(2) := 1/2 X +1 M δ I { N=}I {( +1 X) 2 u }. Notce tat E [ +1 M δ [ +1 Ft = E t [ +1 E t x δ x δ xµ(dx, dt) F t x ν(dx)dtf t Cδ 1 α, terefore, by takng successve expectatons and lettng δ, we ave Ĩ31(2) P. Fnally, t olds tat Ĩ31() Ĩ31(1) + Ĩ31(2) P. For te second term n (2.6), t olds tat X δ +1 X 1 J δ +1 X 1 + M δ +1 X 1. Proceedng n te same way as we dd n our study of te frst term above, fnally we ave Ĩ 3 () = o p ( ) for < α < 1. Now te wole proof of Teorem 2.1 s complete. 3. Dscusson about te Better upper Bounds of K n (2.1) From te proof of Teorem 2.1, we know tat BIV T m 2 1 IV p K Cu 1/2 α/4 1 α < 2, were K p K 13 () + K 3 () and K 13 () σ s dw s +1 M I {( +1 X) 2 u } =: II. t 1 for

9 1oÏ Ùö : 'uœv C Âñ Ý 345 Fx any q > 1 and defne Ñ t = s t I { Xs > u /q}, ξ n, := Ten we can wrte t 1 { x u /q} x µ(dx, dt) { xν(dx). u /q< x 1} I {( +1 X) 2 u } = I { +1 Ñ=} I { +1 Ñ=,( +1 X) 2 >u } + I { +1 Ñ 1,( +1 X) 2 u }. Notce tat on { +1 Ñ = }, we ave +1 M = ξ n,+1, tus II = σ s dw s +1 M I { +1 Ñ 1,( +1 X) 2 u } t 1 + σ s dw s ξ n,+1 (1 I { +1 Ñ 1} I { +1 Ñ=,( +1 X) 2 >u } ) t 1 σ s dw s +1 M I { +1 Ñ 1,( +1 X) 2 u } t 1 + σ s dw s ξ n,+1 =: II 1 + II 2. t 1 For II 1, snce we can proceed smlarly as Mancn (211; p ) wo as sown ( M) 2 I { Ñ 1,( X) 2 u } = o p(1), u (1 α/2) t follows tat u (1 α/2) II 1 = o p (1) by replacng ( M) 2 I { Ñ 1,( X) 2 u } wt σ s dw s +1 M I { +1 Ñ 1,( +1 X) 2 u }. t 1 As for II 2, we ave were Snce we ave E(II 2 ) C 1/2 E ξ n,+1 C + C 1/2 E ξ, Eξ 2 = ξ := { x u /q} { x u /q} x µ(dx, dt). (3.1) x 2 ν(dx)dt = Cu 1 α/2, t follows tat E(II 2 ) Cu 1/2 α/4. Te problem s tat we do not know wat te precse value of E ξ s. Terefore, for 1 α < 2, we ave te better upper bounds BIV T m 2 1IV p K C 1 1/2 E ξ + C 2 u 1 α/2. (3.2)

10 346 A^VÇÚO 1n ò References [1 Mancn, C., Te speed of convergence of te tresold estmator of ntegrated varance, Stocastc Processes and Ter Applcatons, 121(4)(211), [2 Cont, R. and Tankov, P., Fnancal Modellng wt Jump Processes (Second Edton), London: CRC Press, 24. [3 Mykland, P.A. and Zang, L., ANOVA for dffusons and Itô processes, Te Annals of Statstcs, 34(4)(26), [4 Mancn, C., Non-parametrc tresold estmaton for models wt stocastc dffuson coeffcent and jumps, Scandnavan Journal of Statstcs, 36(2)(29), [5 Vetter, M., Lmt teorems for bpower varaton of semmartngales, Stocastc Processes and Ter Applcatons, 12(1)(21), [6 Cont, R. and Mancn, C., Nonparametrc tests for patwse propertes of semmartngales, Bernoull, 17(2)(211), [7 Jacod, J., Asymptotc propertes of realzed power varatons and related functonals of semmartngales, Stocastc Processes and Ter Applcatons, 118(4)(28), 'uœv C Âñ Ý Ùö (HŒÆêÆX, H, 3331), éudioùk$äúxalévyl ÄŒ, 3#Nakà ¹5œ¹e, ïä ÙV C Âñ Ý. ' c: Âñ Ý, V C, Œ, È ÅÄÇ. Æ a Ò: O211.4.