Maximal Inequalities for the Ornstein-Uhlenbeck Process

Transcription

1 Poc. Ame. Math. Soc. Vol. 28, No., 2, (335-34) Reseach Reot No. 393, 998, Det. Theoet. Statist. Aahus Maimal Ineualities fo the Onstein-Uhlenbeck Pocess S.. GRAVRSN 3 and G. PSKIR 3 Let V = (V t ) t be the Onstein-Uhlenbeck velocity ocess solving dv t = V t dt + db t with V =, whee > and B = (B t ) t is a standad Bownian motion. Then thee eist univesal constants C > and C 2 > such that C log (+ ) jv t j C 2 log (+ ) fo all stoing times of V. In aticula, this yields the eistence of univesal constants D > and D 2 > such that D log +log (+ ) jb t j +t D 2 log +log (+ ) fo all stoing times of B. This ineuality may be viewed as a stoed law of iteated logaithm. The method of oof elies uon a vaiant of Lenglat s domination incile [2] and makes use of Itô calculus.. Intoduction Conside the andom movement of a Bownian aticle susended in a liuid. The instein- Smoluchowski theoy suggests the standad Bownian motion B t N (; t) as a model fo the osition of the aticle. The Onstein-Uhlenbeck theoy [6] elies uon Newtonian mechanics and suggests that the osition of the Bownian aticle should be modelled as X t = R t V d whee R V t = e t db e N (; 2 (e2t )) is the Bownian velocity solving the Langevin euation: (.) dv t = V t dt + db t ( > ) (see [3] fo moe details). The instein-smoluchowski theoy may be seen as an idealised Onstein- Uhlenbeck theoy, and edictions of eithe cannot be distinguished by eeiment. Howeve, if the Bownian aticle is unde influence of an etenal foce, the instein-smoluchowski theoy beaks down, while the Onstein-Uhlenbeck theoy emains successful (see [3].53-78). Pehas one of the main easons that the instein-smoluchowski model is so oula in stochastic calculus today is due to the fact that the standad Bownian motion is a matingale. Conside the standad Bownian motion B = (B t ) t. Then the celebated Bukholde-Gundy ineuality [] states that thee eist univesal constants A > and A 2 > such that (.2) A jb t j A 2 3 Cente fo Mathematical Physics and Stochastics, suoted by the Danish National Reseach Foundation. AMS 98 subject classifications. Pimay 6J65, 6G4, 65. Seconday 6J6, 6G5. Key wods and hases: The Onstein-Uhlenbeck velocity ocess, imum ocess, stoing time, imal ineuality, Lenglat s domination incile, Bownian motion, diffusion ocess, Gaussian ocess, the Langevin stochastic diffeential euation. goan@imf.au.dk

2 fo all stoing times of B. In othe wods, and less fomally, this ineuality states that the imal osition of the Bownian aticle, taken u to a andom instant of time which does not anticiate the futue, in aveage behaves as. In this note we addess the same uestion fo the velocity ocess V = (V t ) t. Ou main esult (Theoem 2.5) shows that the imal velocity of the Bownian aticle, taken u to a andom instant of time which does not anticiate the futue, in aveage behaves as log (+ ). In view of the evese dift tem in (.), which is due to a fictional foce towads the oigin (euilibium state of velocity zeo), the uantitative diffeence in the esult is in ageement with ou intuition. The esult of Theoem 2.5 can also be estated in tems of the standad Bownian motion B, and this may be viewed as a stoed law of iteated logaithm (Coollay 2.7). 2. The esult and oof The following domination incile was initially oved in [2] in the case H() = fo < <. Its etension to moe geneal functions 7! H() follows along the same lines and can be found in [5] (.55-56). This etension aeas cucial in ou teatment below, and we esent the oof fo comleteness. Poosition 2. (Lenglat) Let (; F ; (F t ) t; P ) be a filteed obability sace, let X = (X t ) t be an (F t )-adated non-negative ight-continuous ocess, let A = (A t ) t be an (F t )-adated inceasing continuous ocess satisfying A =, and let H : R +! R + be an inceasing continuous function satisfying H() =. Assume that (2.) (X ) (A ) fo all bounded (F t )-stoing times. Then (2.2) su H(X t ) fo all (F t )-stoing times, whee (2.3) e H() = Z fo all. Poof. By Fubini s theoem we obtain (2.4) su H(X t ) Z s H P su eh(a ) dh(s) + 2H() X t = Z su X t > s ; A s 8 su X t > s9 dh(s) o + P n A > s dh(s) since s 7! H(s) is inceasing and continuous. Conside the following stoing times: 2

3 (2.5) = inf f t > j X t > s g 2 = inf f t > j A t > s g. Then by Makov s ineuality and (2.) we find: (2.6) P su X t > s ; A s Z n o n o P ; 2 P X ^ 2^ s A s ^ 2^ wheneve is bounded. Fom (2.4) and (2.6) we can conclude: (2.7) su H(X t ) A s 8A s9 + 2P Z A s dh(s) + 2 A n A > s H(A ) o = eh(a dh(s) ) fo all bounded. Finally, obseve that 7! H() e is inceasing, and ass to the limit when k! to each any though bounded ones ^ k. This comletes the oof. Remak 2.2: If H() = with <<, then H() e = ((2)=()) ; if H() =, then H() e +, and the bound on the ight-hand side in (2.2) is non-inteesting; geneally, the ight-hand side in (2.2) gives a non-tivial bound if H() tends to infinity as slow as fo some < < ; the bound is bette (asymtotically otimal) if the eo in (2.) is smalle (negligible).. The initial esult which we state now is motivated by the consideations in [4]. This is addessed in moe detail in Remak 2.4 following the oof below. Theoem 2.3 Let V = (V t ) t be the Onstein-Uhlenbeck velocity ocess solving (.) with V =, whee is a standad Bownian motion. Intoduce the following functional: B = (B t ) t (2.8) I t = Z t e V 2 d. Then thee eist univesal constants A > and A 2 > such that (2.9) log + I A fo all stoing times of V. Poof. If 7! F () (2.) F jvt j = F () + jv t j A 2 is even and C 2, then by Itˆo fomula we find: Z t IL V F (V ) d + whee IL V denotes the infinitesimal geneato of V : Z t F (V ) db log + I 3

4 (2.) IL V 2 2 Motivated by ou consideations in [4], we shall set (2.2) F (v) = e v2 Then it is easily veified that IL V (F ) = c whee c(v) = e v2. By alying the otional samling theoem in (2.), it follows easily that (2.3) F jv j = I fo all bounded stoing times of V. This shows that the condition (2.) is satisfied with X t = F (jv t j) and A t = I t. Denote H() = F () and obseve that (2.4) H() = H(; ) = log (+) whee by H(; ) we indicate the deendence on. By (2.3) we then have (2.5) Z H(; e ) = dh(s; ) + 2H(; ). s Conside the following function: (2.6) G(; ) = Obseve that fo all we have: H(; ) (2.7) G(; ) = G(; ).. Z s dh(s; ). Thus, if we want to comute the limit of G(; ) when! o!, it is no estiction to assume that =. Note that (2.8) Z ds G(; ) = 2 log( + ) log( + s) ( + s) s lementay calculations show that (2.9) lim G(; ) =! (2.2) lim! G(; ) = (2.2) G(; ) (8 > ). Fom (2.5) we then find: (2.22) e H (; ) 3 H (; ) 4

5 fo all, and hence the ight-hand ineuality in (2.9) follows fom (2.2) and (2.4). To ove the left-hand ineuality in (2.9), we shall note by (2.3) that (2.23) I F jv t j fo all bounded stoing times of V. Thus, the left-hand ineuality in (2.9) follows fom (2.2) and (2.4) uon the identification X t = I t and A t = t F (jv j). The oof is comlete. Remak 2.4: It was oved in [4] that (2.24) jv tj C log e V 2 fo all stoing times of V fo which the ocess (e V 2 Itˆo fomula this ineuality is euivalently witten as follows: (2.25) jv tj C log +(I ) ^t ) t is unifomly integable; by whee C > is some constant. Ou esult (2.9) shows that the second eectation sign in (2.25) can be ulled out in font of the suae-oot and logaithm sign; in view of Jensen s ineuality this bound is bette, although not easily comuted; as the ineuality (2.9) above is two-sided, this also detects the eal size of the eo in the teminal-value bound (2.24); obseve also that ou oof above establishes (2.9) with A = =3 and A 2 = 3 ; thus C in (2.24) can be taken as small as A main disadvantage of the ineuality (2.9) is the comlicated fom of the functional I. In ou attemt to undestand bette its size, we now ove that I in (2.9) can be elaced by. In view of the obvious ineuality I, and that I is actually much lage than, this fact may seem suising at fist. Howeve, noting that we also have the logaithm function in (2.9), and ecalling that the vaiance of V t N ( ; 2 (e2t )) emains bounded ove all t, we see that eveything agees well with ou intuition. Theoem 2.5 Let V = (V t ) t be the Onstein-Uhlenbeck velocity ocess solving (.) with V =, whee B = (B t ) t is a standad Bownian motion. Then thee eist univesal constants C > and C 2 > such that (2.26) C fo all stoing times of V. log + jv t j C 2 log + Poof. If 7! F () is even and C 2, then by Itô fomula we know that (2.) holds. Motivated by this eession, conside the euation: (2.27) IL V (F ) = 5

6 with IL V as in (2.). The geneal solution of (2.27) is given by (2.28) F () = Z Z u e 2 u2 e v2 dv + K du + K 2 whee K and K 2 ae constants. Motivated by the fact that A fom Poosition 2. should satisfy A =, we shall imose the condition F () =, which imlies that K 2 =. Imosing futhe that F () =, which imlies that K =, we obtain the following solution of (2.27): (2.29) F () = 2 Z Z u e u2 e v2 dvdu. Obseve that 7! F () is even and C 2, and thus (2.) holds. Alying the otional samling theoem in (2.), and using (2.27), we see that (2.3) F jv j = ( ) fo all bounded stoing times of V. Thus the condition (2.) is satisfied with X t = F (jv t j) and A t = t. Fom (2.3) we also see that (2.3) ( ) F jv t j fo all bounded stoing times of V. Thus the condition (2.) is also satisfied with X t = t and A t = t F (jv j). Denoting H() = F (), it is ossible to ove that (2.32) log (+) H() D log (+) fo all, whee D > is some constant. The left-hand side in (2.32) is veified staightfowadly, while the ight-hand side euies some moe effot. Ou calculations show that one may take D = : The esult now follows fom (2.3)-(2.32) and (2.2) above uon veifying that eh()=h() 3 fo all > ; obseve that F () 2 and F () 2 so that H() and H () =(2 ) fo all >. The oof is comlete. Remak 2.6: Obseve fom the oof above that in (2.26) one may take C = =3 and C 2 = 3D = 3: Note also that (.2) is obtained fom (2.26) by letting #. Coollay 2.7 Let B = (B t ) t be standad Bownian motion. Then thee eist univesal constants D > and D 2 > such that (2.33) D log +log (+ ) fo all stoing times of B. jb t j D 2 log +log (+ ) +t 6

7 Poof. In the setting of Theoem 2.5 above, we shall use the well-known fact that (2.34) V t = 2 e B(e 2t ) which is deived by a standad time-change agument. Set t = e 2t ; then is a stoing time of V if and only if is a stoing time of B. Fom (2.34) we see that (2.35) 2 jv t j = jb t j +t. Set H(; ) = (= ) log (+) ; then (2.26) above can euivalently be ewitten as follows: (2.36) Substituting t = u in (2.36), we see that (2.37) Obseve that (2.38) u( ) 2 H(; ) = 2 jb t j 2 H ;. +t jb u j 2 H ( ) ;. +u log (+) 2 H e ; = 2 and (u) = (=2) log(+u). Thus log + 2 log +e whee e = ( ) is a stoing time of B. Howeve, since clealy (2.39) log log log +log + when tends to o, we see fom (2.37) and (2.38) that (2.33) holds. The oof is comlete. Coollay 2.8 Let M = (M t ) t be a continuous local matingale with the uadatic vaiation ocess M. Then thee eist univesal constants t t D > and D 2 > such that (2.4)! D log +log (+ M ) jm t j D 2 log +log (+ M ) fo all stoing times of M. + M Poof. It follows fom Coollay 2.7 by a standad time-change agument (see [5]). t RFRNCS [] BURKHOLDR, D. L. and GUNDY, R. F. (97). taolation and inteolation of uasilinea oeatos on matingales. Acta Math. 24 (249-34). 7

8 [2] LNGLART,. (977). Relation de domination ente deu ocessus. Ann. Inst. H. Poincaé Pobab. Statist. 3 (7-79). [3] NLSON,. (967). Dynamical Theoies of Bownian Motion. Pinceton Univ. Pess. [4] PSKIR, G. (998). Contolling the velocity of Bownian motion by its teminal value. Reseach Reot No. 39, 998, Det. Theoet. Statist. Aahus ( ). Analytic and Geometic Ineualities and thei Alications, Math. Al. Vol. 478, Kluwe Academic Publishes, 999 ( ). [5] RVUZ, D. and YOR, M. (994). Continuous Matingales and Bownian Motion. Singe- Velag. [6] UHLNBCK, G.. and ORNSTIN, L. S. (93). On the theoy of Bownian motion. Physical Review 36 (823-84). Svend ik Gavesen Deatment of Mathematical Sciences Univesity of Aahus, Denmak Ny Munkegade, DK-8 Aahus matseg@imf.au.dk Goan Peski Deatment of Mathematical Sciences Univesity of Aahus, Denmak Ny Munkegade, DK-8 Aahus home.imf.au.dk/goan goan@imf.au.dk 8