white strictly far ) fnf regular [ with f fcs)8( hs ) as function Preliminary question jointly speaking does not exist! Brownian : APA Lecture 1.

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1 Am : APA Lecure 13 Brownin moion Preliminry quesion : Wh is he equivlen in coninuous ime of sequence of? iid Ncqe rndom vribles ( n nzn noise ( 4 e Re whie ( ie se every fm ( xh o + nd covrince E ( xrxs 'S( ± s if { o oherwise bu disribuion [ wih f fcs8( hs s fr fnf regulr ] Answer : formlly Gussin of rndom vribles is joinly Gussin wih men E (g Problem : SCT is no funcion speking does no exis! so Whie noise sricly

2 Brownin moion : d ( n Ex Formlly le us define Br Xsds o Re By lineriy (Br en is lso Gussin E ( B E ( Br Such E ( fksds f Elxs ds o E ( fxudufsxrdr min ( s : As is well defined Gussin fdufsdr Ekuxr o e eues nion If we dd he ssumpion h his hs coninuous rjecories his defines he Brownin

3 Here is how he rjecories of hese look like : es * ( whie noise womn ivwfgwrnmmms

4 2 B B Zss ZECBRBDTE Properies : E ( Bf so ^ E(sin ( oc±rn Is BOC±r lso : F ( B so Br o for zszrzo ~ NCO EKBR Ks E((Br 2 E(Br2 his Br r ro 432 s so Br & Br re uncorreled nd joinly Gussin so hey re independen so Br I FsB5( Br oeres

5 s The wo properies underlined in on he ls green pge wih he ogeher fc h Boo nd h he hs rjecories provide n lerne definiion B of Brownin moion Remrk : coninuous compred lrge o he inervl lengh sn For for for smll zs S Br mpliude smll his on 0 ( if is he sying This hs 2 inerpreions: h B hs his is conrry sying h Br is huge compred o he B hs herefore shor ime scle rough rjecories n

6 i link Brownin moion c Rndom wlk : Le ( Sn new be he simple symmeric rndom wlk nd ( Y er+ be he broken he Connecing he dos : + + o " z slope # A Define now BCI ' Yp q or %g i > ER+ ; y ^ " Yn Bln I i i ' > e R+ " " " \/ 4h slope en! wih sldpe o in' The CLT shows h IF" " Bin ' d Br~N(o h os ( dqvhve ( Br e is herefore nhwus

7 T Mringle propery The Brownin Mohn is coninuous ime mringles ie E ( B IFSB zszo ( recll h FSB Proof : E ( B Is + E ( Br ^ d ME FE! # I ( + B To or 's # wwwmhg?gyfbj# ' er

8 Prove k Edg Exercises 1 Le p : R R be convex funcions Prove ( B (B 1 oo er+ is sub mringle he wih respec 2 Prove h ( Mr wih respec R o ( FBT erre BE o (FBT e Re e is mringle 3 h ( N ego ( Br en is mringle wih respec o ( FBT e R+ Anlogy 52 n new exp &{ (Sn o log ooshci :( ; nen re lso mringles

9 ( n ( Levy 's heorem The Brownin moion ( Br ( ( Br Bf wih e Re is he coninuous rjecories er+ is mringle we e Re is mringle w unique such h Boo nd IB FP o R re R [ equivlenly < B > r he R+ : see nex slide ] This heorem provides hird possible definiion of he Brownin moion

10 < j Qudric vriion We define < B > nhws IF Bbn? where o o < c n is priion of he inervl [ q ] such h D j s e nf±n ( Proposiion: < B > p s Proof : By By ~N(o yj so [ oo nk o E( (B P By rj nd VrC@rjBgl22l;D

11 ( cr 'd : So Proof E ( IF nd Vr ( E ( By IF 2 ( j Bm Buffy ( o o Is r((bn5bbn2 j ' E ZD ( j ; he 2Do This implies h E ( By By 2 Is ( by Chebyshu n s nd herefore he resul # Remrk : I is no h BP coincidence mringle is

12 ' ( Consequence : Innes Bnj Bj + s s no differenible Proof : Assume This would Is ( By imply # Isf sk< + IF 113g I cos By > o Fun h probbiliy ' E e lbh5b I B Bynl %9! ' becuse es B is coninuous wih posiive Brin Ffs pmwe probbiliy conrdicion! #

13 fpl on Remrks : Any " resonble " ( ie coninuously differenible curve f : R + R sisfies : hug If I ; E C ( o < him e < + s b n hs Ff ( fpflbi < him Es E CZ ( j hs j 2 o < CLD ( j The Brownin moion insed drws curves of infinie vriion ( & herefore infinie lengh # nd non Zero qudric vriion on finie inervl [ ]!

14 iregrble Generlizion : Le R+ be squre coninuous ime if mringle wih coninuous rjecories n > h ge(m5nny :c FELT dd :B mshed < M > Proposing: r is he unique dped wih coninuous rjecories such h (M < My e Re is lso mringle Felly le s menion h ll heorems seen for in discree ime hold mringles for coninuous ime