ARemarkonthe1=H-Variation of the Fractional Brownian Motion

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1 ARemarothe1=H-Variatio of the Fractioal Browia Motio Maurizio Pratelli Abstract We give a elemetary proof of the followig property of H-fractioal Browia motio: almost all sample paths have ifiite 1/H-variatio o every iterval. Keywords Fractioal Browia motio p-variatio Ergodictheorem 1 Itroductio ad Statemet of the Result Let.B t / t0 be the Fractioal Browia Motio with Hurst (or self-similarity) parameter H; 0 < H < 1 (we refer for istace to [2] or[5] or[8] p. 273 for the defiitios): fix t>0ad let t D t for iteger ad D 0;:::;: It is well ow (see e.g. [5]or[9]) that lim!1 D0 X 1 p Bt C1 B t L 1. / D However, if we defie the radom variable 8 ˆ< C1 p<1=h te B11=H p D 1=H : ˆ: 0 p > 1=H V.!/ D V 1=H Œ0;t.!/ D ;0t 1 <t 2 <:::<t t X 1 BtiC1.!/ B.!/1=H ti id1 the V.!/ D C 1 a.s. (ote that V is measurable sice the paths of the fractioal Browia motio are cotiuous ad therefore the ca be tae over ratioals t i ). M. Pratelli ( ) Dipartimeto di Matematica, Largo B. Potecorvo 5, Pisa, Italy pratelli@dm.uipi.it C. Doati-Marti et al. (eds.), Sémiaire de Probabilités XLIII, Lecture Notes i Mathematics 2006, DOI / , c Spriger-Verlag Berli Heidelberg

B t / t0 be the Fractioal Browia Motio with Hurst (or self-similarity) parameter H; 0 < H < 1 (we refer for istace to [2] or[5] or[8] p.

2 216 M. Pratelli This property is well ow i the case of stadard Browia Motio (i.e. H D 1=2): it was stated (but without a rigorous proof) by P. Lévy i [6] p. 190, ad also the excellet boo Revuz-Yor quotes the result without a proof (see [8] p. 28). A setch of the proof (always i the case H D 1=2) was give by D. Freedma i [4] p. 48. For the stadard Browia Motio, there is a more precise (ad more techical) result due to Taylor (see [11]): give a icreasig fuctio W Œ0; C1/! Œ0; C1/, we ca defie the variatio of the fuctio f o the iterval Œa; b as ;adt 1 <t 2 <:::<t Db jf.tic1 / f.t i /j (whe.t/ D t p it is called the p variatio). Taylor showed that the correct fuctio for the variatio of the paths of the BM is the fuctio 1.s/ D s 2 =2 log log s (where log s D max.1 ; j log sj/) ithe sese that V ;Œa;b.!/ D ;adt 1 <t 2 <:::<t Db jbtic1.!/ B ti.!/j is a fiite r.v. but is ifiite if 1 is replaced by ay fuctio such that.s/= 1.s/!C1as s! 0C. The impact of the p-variatio of the paths for (stochastic) itegratio is well highlighted by L. Couti (see [2]) for the case of FBM ad by Dudley ad Norvaisa (see [3]) for more geeral stochastic processes. I the geeral case of the Fractioal Browia Motio, it is well ow that p D 1=H is a limit case, ad that sample paths are -hölder cotiuous for ay <H. The result of Theorem 1 is ow sice it is a cosequece of Theorem IV.5.1 of [1]: their proof, however, is based o a complex techology (the theory of Besov spaces). The aim of this short ote is to give a elemetary complete proof suggested by the argumet preseted i [4]; I wat to tha Sara Biagii ad Giorgio Letta for a discussio o the subject. Let us fix H with 0<H <1,let.B t / t0 be a FBM with parameter H ad cotiuous sample paths: defie V Œa;b.!/ D The statemet of the result is as follows BtiC1.!/ B.!/1=H ti ;adt 1 <t 2 <:::<t Db Theorem 1. There exists a ull-set N such that, if! N, the for every a<b, V Œa;b.!/ DC1.

For the stadard Browia Motio, there is a more precise (ad more techical) result due to Taylor (see [11]): give a icreasig fuctio W Œ0; C1/!

3 A Remar o the 1=H -Variatio of the Fractioal Browia Motio The Proof I the sequel, is the Lebesgue measure o IR C ad p is a shorthad for 1=H. Let U be a fiite uio of disjoit ope itervals s i ;t i Œ of IR C with s i ;t i 2 Q ad let U be the collectio of such subsets of IR C. For U D [ id1 s i ;t i Œ ( 0 s 1 < t 1 s 2 < ::: < t ), let q U.!/ D P id1 jb t i.!/ B si.!/j p ; ote that V Œa;b.!/ D q U.!/ U 2U ;UŒa;b Lemma 1. Fix m > 0 ad let p m D P.jB 1 j p m/. Let Z D I fjb B 1 j p mg :them D Z 1CCZ coverges a.s. to EŒZ 1 D p m. Proof. The sequece of oe step icremets of B, X D B B 1, is statioary, cetered Gaussia ad with covariace fuctio R./ D EŒX 1 X C1 which teds to 0 whe goes to ifiity (see e.g. [7] p. 274): therefore.x/ 1 is ergodic (see [10] p. 413). Now Z ca be writte i the form Z D g.x / with a borel fuctio g ad therefore also.z / 1 is ergodic. As a cosequece of the ergodic theorem (Theorem 3.3 p. 413 of [10]), the sequece of the empirical meas M D.M / 1 coverges a.s. (ad i L 1 )toeœz 1. The ey of the proof is the followig result: Lemma 2. Let I D s; tœ be a ope iterval with s; t 2 Q C ad fix m>0:let p m D PfjB 1 j p mg ad r<p m. The there exists a measurable A m with P.A m / D 1 such that for all! 2 A m there exists U! 2 U with the properties: 1. U! I 2..U! />r.i/ 3. q U!.!/ m.u! / Proof. For >1ad i D 0;:::;let t i S D X id1 D s C i.t s/ ad J i D ti 1 ;t i Œ.Set I jb t B t ic1 i jp m.t s/ S.!/ couts the umber of subitervals Ji o which q J i.!/ > m.ji /. Thas to P the self-similarity property of fbm (see e.g. [7] p. 275), S is distributed as Z D id1 I fjb ic1 B i j p mg. By the Lemma 1, Z Modulo a subsequece,! p m almost surely, whece S lim S D p m a.s. teds to p m i probability. Call A m the set o which the above sequece coverges: if! 2 A m the there exists! such that S.!/ > r for!.

!/ D P id1 jb t i.!/ B si.!/j p ; ote that V Œa;b.!/ D q U.!/ U 2U ;UŒa;b Lemma 1. Fix m > 0 ad let p m D P.jB 1 j p m/. Let Z D I fjb B 1 j p mg :them D Z 1CCZ coverges a.s. to EŒZ 1 D p m. Proof.

4 218 M. Pratelli Select! ad amog the subitervals Ji ad let U! be their uio. The m.t s/ exactly those such that q J i.!/.u! / D S.!/ t s >r.i/ ad.t s/ q U!.!/ S.!/ m D m.u! /: The same result holds evidetly for a elemet U 2 U.SiceU i coutable, we have immediately the followig result Corollary 1. Fix m>0ad set r D p m =2 : there exists a measurable set C m with P.C m / D 1 such that, if! 2 C m ad V 2 U, there exists U! 2 U ad U! V such that.u! />r.v/ad q U!.!/ m.u! /. Lemma 3. Fix 0 a<bi a; b 2 Q :thev Œa;b.!/ DC1a.s. Proof. Choose m>0adapplylemma2 to I D a; bœ : the for every! 2 C m there exists U! 1 I such that q U! 1.!/ m.u1! / ad.u! 1 / r.i/. Now iterate the procedure, that is apply Corollary 1 to I U 1 1! U! is the closure of U! 1 : thus there exists U 2! I U 1! with qu 2.!/ m.u2!! / ad.u! I 2 />r U 1!. At the. C1/-th step, we have a subset U! C1 of I.U 1! [ :::[ U! / such that.u! C1 />r I U 1! [ :::[ U! ad qu C1.!/ m.u C1!! /. Call V! D U! 1 [ :::[ U!,theV! 2 U ad q V!.!/ m.v! / Moreover, by iductio.i V! /.1 r/.b a/ ad therefore q V!.!/ m lim.v! / D m.b a/ Now, the itersectio C D T m2in C m has probability oe ad ay! 2 C satisfies V Œa;b.!/ m.b a/ 8m 2 IN whece the thesis. If N Œa;b is the ull-set! 2 VŒa;b.!/ < C1, the the coutable uio of all N Œa;b with a<bi a; b 2 Q, satisfies the hypothesis of Theorem 1.

2 U ad U! V such that.u! />r.v/ad q U!.!/ m.u! /. Lemma 3. Fix 0 a<bi a; b 2 Q :thev Œa;b.!/ DC1a.s. Proof. Choose m>0adapplylemma2 to I D a; bœ : the for every! 2 C m there exists U!

5 A Remar o the 1=H -Variatio of the Fractioal Browia Motio 219 Refereces 1. Ciesielsi, Z., Keryacharia, G., Royette, B.: Quelques espaces foctioelles associés a des processus Gaussies. Studia Math. 107(2), (1993) 2. Couti, L.: A itroductio to (stochastic) calculus with respect to fractioal Browia motio. I: Sémiaire de ProbabilitésXL, Lecture NotesiMathematics, vol. 1899, pp Spriger (2007) 3. Dudley, N.R.: A itroductio to P -variatio ad Youg itegrals. Tech. rep. 1., Maphysto, Ceter for Mathematical Physics ad Stochastics, Uiversity of Aarhus (1998) 4. Freedma, D.: Browia Motio ad Diffusio. Holde-Day (1971) 5. Guerra, J., Nualart, D.: The 1=H -variatio of the divergece itegral with respect to the fractioal Browia motio for H>1=2ad fractioal Bessel processes. Stoch. Process. Appl. 115(1), (2005) 6. Lévy, P.: Processus stochastiques et Mouvemet Browie. Gauthier-Villars (1965) 7. Nualart, D.: The Malliavi Calculus ad Related Topics. Spriger, New Yor (2006) 8. Revuz, D., Yor, M.: Cotiuous Martigales ad Browia Motio, 3rd ed. Spriger, Berli (1999) 9. Rogers, L.C.G.: Arbitrage with fractioal Browia motio. Math. Fiace 7, (1997) 10. Shiryaev, A.N.: Probability, 2d ed. Spriger, Berli (1996) 11. Taylor, S.J.: Exact asymptotic estimates of Browia path variatio. Due Math. J. 39, (1972)

$: Browia Motio ad Diffusio. Holde-Day (1971) 5. Guerra, J., Nualart, D.: The 1=H -variatio of the divergece itegral with respect to the fractioal Browia motio for H>1=2ad fractioal Bessel processes.$

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval