Lecture 15 First Properties of the Brownian Motion

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1 Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies of he Browia moio ad is rajecories. May oher properies which require various ools from sochasic aalysis will be scaered hroughou he remaider of he oes. Proposiio 15.1 (Symmeries of he Browia moio). Le {B } [0,) be a Browia moio. The he followig processes are also Browia moios (oly o [0, 1] i (5)): 1. { B } [0,), (reflecio), 2. { 1 α B α } [0,), for α > 0, (scalig), 3. {B 0 + B 0 } [0,), for all 0 0, (shifig), 4. {X } [0,), where X 0 = 0 ad X = B 1/, for > 0, (iversio), 5. {B 1 B 1 } [0,1], (ime reversal). Proof. I is easy o see ha all of he above are ceered Gaussia processes wih he Browia covariace srucure. Coiuiy of he pahs is clear i 2., 3. ad 5., ad everywhere excep a = 0 i 4. To deal wih ha case, we may use he Kolmogorov-Česov heorem (how?), or use our ex resul. Proposiio 15.2 (The Law of Large Numbers for Browia Moio). Le {B } [0,) be a Browia moio. The B lim = 0, a.s. Proof. Usig he (discree-ime) Law of Large umbers, we have B lim 1 = lim k B k 1 ) = 0, a.s. k=1(b The idea is o show ha he rajecory of B cao deviae oo much from B o [, + 1]. Ideed, if we ca prove ha [ ] P sup B B 2/3 <, (15.1) =0 [,+1] Las Updaed: February 16, 2015

2 Lecure 15: Firs Properies 2 of 8 he Borel-Caelli heorem will fiish he job for us (why?). Therefore, all we eed o do is esimae he probabiliy i (15.1). We use Kolmogorov s iequaliy (a specializaio of he maximal iequaliy for submarigales) applied o he discree radom walk (B +k2 m B ) k N0 o coclude ha P[ sup 0<k 2 m B +k2 m B 2/3 ] 1 4/3 E[(B +1 B ) 2 ] = 1 4/3. However, by he coiuiy of rajecories of B, we have { B B 2/3} = { sup [,+1] m N ad he uio is icreasig. Therefore, [ P ad (15.1) follows. sup B +k2 m B 2/3 0<k 2 m ] sup B B 2/3 1, 4/3 [,+1] }, Proposiio 15.3 (Log-erm behavior of rajecories). Le {B } [0,) be a Browia moio. The, Proof. For K (0, ), we have B =, ad lim if B =, a.s. P[B > K i.o.] = P[ N m {B m > K m}] = lim P[ m {B m > K m}] lim P[B > K ] = P[B 1 > K] > 0. (15.2) We ca view B as a sum B = k=1 ξ k, where ξ k = B k B k 1 of idepede radom variables ad oe ha for ay 0 N, B / > K if ad oly if (B B 0 )/ > K. By Kolmogorov s 0-1 law 1 we have 1 ha is he oe ha says ha he σ- algebra N σ(ξ k ; k ) is rivial whe P[B {ξ } N are idepede. > K i.o.] = 1, so ha B K, a.s., for all K > 0. Therefore, B B = +, a.s., The saeme abou he lim if follows from he fac ha { B } [0,) is also a Browia moio. A ice corollary of he above resul is ha he Browia moio visis each poi a R i each ime-ierval [, ), 0. Las Updaed: February 16, 2015

3 Lecure 15: Firs Properies 3 of 8 Corollary 15.4 (Recurrece of he Browia moio). The Browia moio is recurre, i.e., for each a R, he (radom) se L a (ω) = { [0, ) : B (ω) = a} is ubouded, a.s. The quadraic variaio of he Browia moio We sar by iroducig some space-savig oaio relaed o pariios. Give > 0, a sequece 0 = 0 < 1 < < k = is called a pariio of [0, ] ad he se of all pariios of [0, ] is deoed by P [0,]. The elemes 0, 1,... of a pariio are referred o as is odes. For = { 0,..., k } P [0,], he mesh [0,] of is defied by [0,] = sup i {1,...,k} i i 1. A sequece { } N i P [0,] is said o coverge o ideiy, deoed by Id, if [0,] 0, for each 0. Addiioally, we say ha he covergece is fas, deoed by sum Id, if [0,] <. For = { 0,..., k } P [0,], a real fucio f : [0, ] R (or, more geerally, f : [0, ) R) ad p 1, we defie he p-variaio Var p ( f ; ) of f alog by Var p ( f ; ) = f (0) p + k i=1 f ( i) f ( i 1 ) p. The oal p-variaio Var p ( f ; [0, ]) of f is give by Var p ( f ; [0, ]) = sup P [0,] Var p ( f ; ), ad he fucio f is said o be of fiie p-variaio if Var p ( f ; [0, ]) <, for all 0. Whe p = 1 we simplify he oaio by wriig Var for Var 1 ad refer o he fucios of fiie 1-variaio simply as fucios of fiie variaio or recifiable fucios. For a sochasic process {X} [0,), ad P [0,], he expressio Var p (X; ) refers o he radom variable, whose value o ω Ω is Var p (X (ω); ). A similar ierpreaio ca be applied for he oal variaio Var p (X; [0, ]). Proposiio 15.5 (The quadraic variaio of he Browia Moio). Le {B } [0,) be a Browia moio, ad, for 0, le { } N be a sequece i P [0,] wih [0,] 0. The lim Var 2 (B; ) =, i L 2. (15.3) If sum Id, he he covergece i (15.3) also holds i he a.s.-sese. Las Updaed: February 16, 2015

4 Lecure 15: Firs Properies 4 of 8 Proof. We sar wih wo very simple ideiies, valid for all 0 r s u v: [ ( ) ] 2 E (B s B r ) 2 (s r) = 2(s r) 2, ad [( )( )] E (B s B r ) 2 (s r) (B v B u ) 2 (v u) = 0. I follows ha, for ay pariio P [0,], we have [ ( ) ] 2 E Var 2 (B; ) = 2 Var 2 (Id; ), where Id deoes he ideiy fucio s s. The firs claim is ow a cosequece of he esimae Var 2 (Id, ) = k i i 1 2 [0,]. i=1 The secod oe follows from he firs oe ad a applicaio of he Borel-Caelli lemma. Remark Noe ha Proposiio 15.5 does o imply ha he pahs of Browia moio have fiie 2-variaio, a.s. I fac, i ca be proved ha, for each > 0, Var 2 (B; [0, ]) =, a.s. Corollary 15.7 (No-recifiabiliy of Browia pahs). Pahs of he Browia Moio have ifiie variaio o [0, ], for all 0, a.s., i.e. > 0, Var 1 (B; [0, ]) =, a.s. Proof. For a pariio = { 0,..., k } P [0,], we clearly have Var 2 (B; ) δ(b; ) Var 1 (B, ) where δ(b, ) = sup B i B i 1. 1 i k If { } N is a sequece of pariios i P [0,] wih sum Id, Proposiio 15.5 implies ha Var 2 (B, ), a.s. O he oher had, he coiuiy (ad herefore uiform coiuiy o compacs) of he pahs of he Browia moio allows us o coclude ha δ(b, ) 0, a.s. I follows ha, ecessarily, Var 1 (B, ), ad, a foriori, ha Var 1 (B, [0, ]) = sup Var 1 (B; ) sup Var 1 (B; ) =, a.s. P [0,] Logarihmic laws (*) We fiish wih some fie regulariy properies of Browia pahs. Las Updaed: February 16, 2015

5 Lecure 15: Firs Properies 5 of 8 Defiiio 15.8 (Uiform local modulus of coiuiy). The map δ, defied i a righ eighborhood of 0, is called he uiform local modulus of coiuiy for he fucio f : [0, 1] R, if here exiss h 0 > 0 such ha, for each [0, 1], f ( + h) f () δ(h), for all 0 < h < h 0 wih + h 1. The esimaes from he Lévy-Ciesielski cosrucio above, lead relaively direcly o he followig resul: Proposiio 15.9 (A uiform local modulus of coiuiy for he Browia moio). There exiss a cosa C such ha, for almos all ω δ(h) = C h log(1/h) is he local uiform modulus of coiuiy of he rajecory B (ω). Proof. We use he oaio from he Lévy-Ciesielski cosrucio ad recall ha B () = () B, uiformly over [0, 1], a.s. k=1 I he course of he proof of Proposiio 14.19, we esablished ha he followig boud [ ] P sup () C 1 2 ev. = 1, (15.4) [0,1] holds for all C 1 large eough, say C 1 = 2. Thaks o he piecewiseliear srucure of (), ad he fac ha he iervals of lieariy are of he size 2, we clearly have P [ sup d d () [0,1] ] C 1 2 ev. = 1. (15.5) Therefore, here exiss a radom variable L wih values i N such ha sup () C 1 2 ad sup d d () C 1 2, for all L, a.s. [0,1] [0,1] For ay radom variable K N wih K L, [0, 1], ad h > 0, we have B +h B () ( + h) () () (I) + (II) + (III), where (I) = h =0 L =0 sup d d () [0,1], (II) = h K =L+1 C 1 2, Las Updaed: February 16, 2015

6 Lecure 15: Firs Properies 6 of 8 ad (III) = 2 =K+1 C 1 2. Firs, we pick (a radom) h 0 > 0 so small ha (I) h log(1/h), for 0 < h h 0. The, give 0 < h < h 0, we cosruc a N-valued radom variable K such ha 2 K 1 h < 2 K, ad se K = max(k, L + 1). Usig he esimae 2 = N2 N =N =N = N2 N k=0 N 2 ( N) (1 + k N )2 k C 2 N2 N, where C 2 = k 0 (1 + k)2 k <, we coclude ha Fially, (III) 2C 1 (II) hc 1 K C 1 =1 k=k =K +1 4C 1 C 2 h log(1/h). 2 2C 1 C 2 (1 + K )2 K 1 K 2 C 1 2 ( 2K ) 2K = C 1 =1 k=k (k K )2 k k2 k C 1 C 2 K 2 K 2C 1 C 2 h log(1/h). All i all, B +h B (1 + 6C 1 C 2 ) h log(1/h), for h h 0. The full resul of Lévy (whose proof we omi) is ha he fucio δ of Proposiio 15.9 is opimal, ad ha he he cosa C ca be chose o be equal o 2. Theorem (Lévy s modulus of coiuiy). sup [0,1] h 0 B +h B 2h log(1/h) = 1, a.s. If oe focuses o he flucuaios of he Browia moio aroud a sigle, fixed, poi 0, oe ges a slighly igher esimae. Theorem (Law of ieraed logarihm). For each 0, we have h 0 B +h B 2h log(log(1/h)) = 1, a.s. Las Updaed: February 16, 2015

7 Lecure 15: Firs Properies 7 of 8 Addiioal Problems Problem 15.1 (p-variaios of fucios). 1. Show ha, for a fucio f : [0, ] R, Var p ( f ; [0, ]) < implies Var q ( f ; [0, ]) < for q > p > For each q > 1, fid a example of a fucio f : [0, 1] R wih he propery ha Var p ( f ; [0, 1]) <, p > q, Var p ( f ; [0, 1]) =, p q, 3. For each q 1, fid a example of a fucio f : [0, 1] R wih he propery ha Var p ( f ; [0, 1]) <, p q, Var p ( f ; [0, 1]) =, p < q. 4. For a fucio f : [0, ) R of fiie variaio (remember, ha meas ha Var 1 ( f ; [0, ]) < for all > 0), defie F : [0, ) R by F() = Var( f ; [0, ]), 0. Show ha f () f (s) F() F(s), for all, s [0, ). Deduce ha f ca be wrie as a differece of wo moooe fucios, ad, more geerally, he f is of fiie variaio if ad oly if i ca be wrie as a differece of wo moooe fucios. Problem 15.2 (Quadraic covariaio of idepede Browia moios). Le he sochasic processes {X } [0,) ad {Y } [0,) be defied o he same probabiliy space. For a pariio P [0,], we defie he quadraic covariaio of X ad Y alog by Var 2 (X, Y; ) = k i=1 (X i X i 1 )(Y i Y i 1 ), where = {0 = 0,..., k = }. If X ad Y are wo idepede Browia moios, i.e., such ha he σ-algebras σ(x ; 0) ad σ(y ; 0) are idepede, show ha Var 2 (X, Y; () ) 0 i L 2, for each sequece { () } N i P [0,] wih () Id. Problem 15.3 (Higher-dimesioal Browia moio). For d N, a vecor-valued sochasic process (B (1),..., B (d) ) 0, wih values i R d, is said o be a d-dimesioal Browia moio if is compoes B (1),..., B (d) are idepede Browia moios. Give N, fid ecessary ad sufficie codiios o he R d -marix H such ha he Las Updaed: February 16, 2015

8 Lecure 15: Firs Properies 8 of 8 process W (1) ṭ W =. where W = HB ad B = W () is a -dimesioal Browia moio. B (1) ṭ. B (d) 0 Problem 15.4 (Moooiciy ad maxima of he Browia pah). Prove he followig saemes for a Browia moio B: 1. B is moooe o o ierval of he form [r, s], 0 r < s, a.s. 2. For each p > 0, he disribuio of he radom variable M = sup s B s is diffuse, i.e. P[M = a] = 0, for all a R. Hi: Argue, firs, ha i is eough o assume ha = 1. Le M 1 be a idepede radom variable wih he same disribuio as M 1. Show ha 2M 1 ad max(m 1, W 1 + M 1 ) have he same disribuio. Deduce ha he oly possible aom for M 1 is 0. The show ha P[M 1 > 0] = B aais differe maxima o ay wo o-overlappig iervals (r 1, s 1 ) ad (r 2, s 2 ), a.s. 4. Each local maximum of B is a sric local maximum, a.s. 5. B achieves is global maximum o [0, 1] i exacly oe poi, a.s. Problem 15.5 (No-differeiabiliy of Browia pahs). 1. Show ha if f : [0, 1) R is differeiable a [0, 1) (he righ derivaive is cosidered a he = 0), he here exiss l, 0 N such ha f ( j ) f ( j 1 ) l, for all 0 ad all i < j i + 3, where i = + 1 ad x deoes he larges ieger o larger ha x. 2. Le {B } [0,) be a Browia moio. For s [0, 1) we defie D s as he se of all ω Ω such ha he rajecory B (ω) is differeiable a s. Show ha s [0,1) D s Γ, where Γ = 3. (*) Show ha P[Γ] = 0. l 1 +1 lim if i+3 i=1 j=i+1 { Bj/ B (j 1)/ l }. Las Updaed: February 16, 2015