Brownian Motion and Stochastic Calculus. Brownian Motion and Stochastic Calculus

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1 Browa Moo Sochasc Calculus Xogzh Che Uversy of Hawa a Maoa earme of Mahemacs Seember, 8 Absrac Ths oe s abou oob decomoso he bascs of Suare egrable margales Coes oob-meyer ecomoso Suare Iegrable Margales 4 Browa Moo Sochasc Calculus Couou Tme Submargales Usually s su ce o oly dscuss submargales by symmery de o echues are he same. oob-meyer ecomoso oob-meyer decomoso clears he obsable for de g sochasc egral ( he somery sraegy) wr suare egrable margales hece s of foudameal morace e o A creasg rocess A s called aural f for every bouded, rgh couous margale fm ; F : < g we have E M s da s E M s da s () (;] (;] for every < < Problem Suose X fx ; F : < g s a rgh couous submargale. Show ha uder ay oe of he followg codos, X s of class L a) X ;a.s for every b) X has he secal form suggesed by oob decomoso. X M + A ; Show also ha f X s a uformly egrable margale, he s of class.

2 Proof. By ooal samlg heorem for bouded sog mes ( oe 4), we have X T fx T >g Also we have P (X T > ) E (X T ) for all a > ; > ; T I a : Therefore X a fx T >g E (X a) su X T dp T I a fx T >g For he secod ar. I su ces o show ha fm g he decomoso s uformly egrable (sce fa g s uformly egrable for all T I a ). Aga from ooal samlg heorem X T E (X a jf T ) for all T I a whch esablshed he eeded uformly egrably. If X s uformly egrable, he X closes X hece X s Problem 3 Le X fx ; F : < g be a couous, o-egave rocess wh X a.s. A fa ; F : < g ay couous, creasg rocess for whch E (X T ) E (A T ) holds for every bouded sog me T of ff g. Iroduce he rocess V max s X s cosder a couous, creasg fuco F o [; ) wh F () de e for < x < : Esablsh he euales for ay sog me T of ff g Proof. e e he sog mes G (x) F (x) + x x u df (u) P [V T ] E (A T ) ; 8 > () P [V T ; A T < ] E ( ^ A T ) ; 8; > (3) E (F (V T )) E (G (A T )) (4) H f f : X g ; S f f : A g (5) T T ^ ^ H

3 (Noce ha H T f V T < s o rue for arbrary e T;.e., f fv T g for some e T he H T ; oherwse, he lef sde s zero sce fv T g s emy for e T he ualy holds aurally) we have P (V T ) E X T fvt g E (XT ) E (A T ) E (A T ) Now by he couy of robably measure we ake a o boh sdes P (V T ) P (V T ^H ) E (A T ) (sce (V T ^H ) (V T ) sce V max s X s moooc o-decreasg o : ) O he oher h, we have P (V T ; A T < ) P (V T ^S ) E (A T ^S) E ( ^ A T ) (6) (sce A T < mles T S ) The we have P (V T ) P (V T ; A T < ) + P (V T ; A T ) E ( ^ A T ) + P (A T ) whch s he rs eualy he corollary ha follows eoe he cdf of V T by F VT (V T ) : By he assumo o F we have E (F (V T )) F (x) F (V T ) df VT (V T ) fxug df (u) fvt ugdf (u) df VT (V T ) (7) df (u) fvt ugdf VT (V T ) P (V T u) df (u) E (u ^ AT ) + P (A T u) df (u) u ( ) E AT fat <ug + P (A T u) df (u) u E F (A T ) + A T df (u) E (G (A T )) A T u Corollary 4 from he above roblem, we have P [V T ] E ( ^ A T ) + P [A T ] E (V T ) E (A T ) ; < < 3

4 Proof. Take F (x) x for < < x : The g (x) x + x u u du x + x u du x x x + x u x x + x x x + x x from he above eualy as desred. E (V T ) E (A T ) Suare Iegrable Margales Oe hg o oce s ha, whe suarg sums of margale cremes akg he execao, o ca eglece h cross-roduc erms. More recsely, f M fm g M s < u < v; he E ((M v M u ) (M M s )) E fe ((M v M u ) (M M s ) jf u )g (8) E f(m M s ) E (M v M u jf u )g Ths fac aled o boh M; M hm we ge E (M v M u ) jf E Mv M u M v + Mu jf E Mv jf E (M u E (M v jf u ) jf ) + E MujF E Mv jf E MujF + E M u jf E M v M ujf E (hmv hm u jf ) (9) sce E (M u M v jf ) E (E ((M u M v ) jf u ) jf ) E ([M u E (M v jf u )] jf ) E M ujf for margale M Thus M v hm v M u hm u (Mv M u ) (hm v hm u ) have zero execaos codoed o F Problem 5 Le fx ; F g be a couous rocess wh he roery ha for each xed > for some > () L V () kk! robably, where L s a rom varable akg values [; ) a.s.. Show ha for > ; kk! V () () robably kk! V () () for < < o fl > g Proof. If L a.s., he we have ohg o rve. So s reasoable o suose P (L > ) > () Le X (; ) su js sj<;s;s [;] fjx s X s j g 4

5 The by he uform couy of X s o he comac se [; ] ; we have! js for ay > : For ay aro su sj<;s;s [;] fjx s X s j g o () [; ] : () () () + () m le k k () max m + () o The s clear ha V () ( ) Xm X () + m X X () + X () X () Xm! X () + su m X X () () + X () + X () X () Coseuely, f > ; he V () ( ) k k! robably. For he secod half, le s assume he corary. The here are cosas ; K > a seuece of aros o () [; ] : () () () + () m such ha for he ses holds ha Thus from s fered ha whch coradcs V () ( ) Xm X () + m X A X () + K su m L > ; V () o ( ) K P (A ) P (L > ) X () X X () () + Xm! X () + su m X () X X () () + X () + X () P V () ( ) P (L > ) > k k! P V () ( ) L > < for all whose kk < () for some () for ay gve ; > : X () Problem 6 Le X be M c ; T be a sog me of ff g. If hx T a.s. he P (X T ^ ; 8 < ) 5

6 Proof. Sce hx s couous odecreasg ^ T we have P (X ^T ; < ) Sce M X hx s a couous margale, by he ooal samlg heorem E (M T ^ ) E (M ^T ) E X^T E (hx ^T ) E X^T whch mles robably for all < : Coseuely For ay [; ); by couy we have Thus as desred. X ^T P X ^T : Q [;) \ fx r^t ga r Q [;) [ P (X r^t 6 ) r Q [;) r ;k #;r ;k Q [;) X ;rk^t X ^T P (X ^T : [; )) P X ;rk^t X ^T : [; ) \ fx r^t ga r ;k #;r ;k Q [;) r Q [;) Problem 7 Show ha for X; Y M c f ; ; ; m g a aro of [; ] robably. kk! k (X k X k ) (Y k Y k ) hx; Y Proof. Le s ake a smarer sraegy wh he followg roadma: by he de o hx; Y (hx + Y hx Y ) 4 he fac ha m X+Y (; ) X [(X k X k ) + (Y k Y k )]! P hx + Y ; V () V () k m X Y (; ) X [(X k X k ) (Y k Y k )]! P hx Y k 6

7 as kk! ; we have oly o show m V () X;Y (; ) X (X k X k ) (Y k Y k )! P 4 V () k X+Y (; k! ) k Noce ha he above scheme he aros used may be very d ere. The Obvously, for he same aro ; V () X;Y (; ) 4 V () X;Y (; )!P hx; Y V () X+Y () (; ) V X Y (; ) V () k X k! Y ; whch yelds he resul. To make hs more formal, we showed ha, sce for ay ; > ; here exss some > ; such ha o V () max P X+Y (; ) hx + Y V () ; P X Y (; ) hx Y < for every wh kk < ; ha, for all such ; holds ha V () X+Y V () X;Y (; ) 4 () (; ) V X Y (; ) he as desred. P V () X;Y (; ) hx; Y < Remark 8 We do have o coe wh he mess by de g k X X k X k ; k Y Y k Y k ; k hx; Y hx; Y k hx; Y k show comue E! (X k X k ) (Y k Y k ) hx; Y k E E! (X k X k ) (Y k Y k ) hx; Y E k k su km ll we re bored [( k X k Y ) k hx; Y ] k o! ( k X k Y ) + ( k hx; Y ) + k X k Y k hx; Y j+ fj k X jg (( X Y ) hx; Y ) (( j X j Y ) j hx; Y ) A su km fj k Y jg E k! k X k Y + su fj k hx; Y jg E km!! ( k hx; Y ) Problem 9 Le X; Y be M c;loc : The here s a sochascally uue adaed, couous rocess of bouded varao hx; Y sasfyg hx; Y a.s., such ha XY hx; Y M c;loc. If X Y we wre hx hx; X hs rocess odecreasg. k 7

8 Proof. Ths roblem d ers from he orgal heorem he exbook ha here he rocessed are oly local margales. Thus s more d cul bu su ces o show X M c;loc ) X hx M c;loc The we ca ake XY hx; Y (X + Y ) hx + Y 4 o (X Y ) hx Y whch s a lear combao of local margales coseuely a local margale. So, le s de e seueces of sog mes fs g ; ft g such ha S ; T. The o o X () X ^S ; Y () Y ^T are ff g-margales (by he ooal samlg heorem). e e O f f : (jx j _ jy j) g se Noe ha R a.s.. Sce ~X () R S ^ T ^ O X ^R ; ~ Y () Y ^R ~X () X ^R X () ^R ; ~ Y () Y ^R Y () ^R hese rocesses are also ff g-margales are M c because hey re bouded. For m > ; ~ X () X (m) ^R (key observao) so ~X () E ~X (m) ^R X (m) ^R ~X (m) E ^R s a margale. Ths mles ~X () E by oob ecomoso. Thus we ca de e E hx ~X () ~X (m) E ^R () whever R be assured ha hx s well-de ed. The rocess hx s adaed, couous, odecreasg sa es hx a.s.. Furhermore E (sce hx ^R ~X () ^R X ^R hx ^R ~X (m)e By Theorem 5.3 he ex, we ca ake ^R ~X ()E ~X () ~X () E hx; Y (hx + Y hx Y ) 4 () ) s a margale for each ; so X hx M c;loc : 8

9 The XY hx; Y M c;loc For he uueess, suose boh A; B sasfy he codos reures of hx; Y. The M XY A B XY B are M c;loc ; so jus as before we ca cosruc a seuece fr g of sog mes wh R such ha M () M ^R N () N ^R are M c : Coseuely M () N () B ^R A ^R M c beg of bouded varao hs rocess mus be decally zero. (see he roof of Theorem 5.3) I follows ha A B Problem Esablsh he followg ) A local margale of class L s a margale ) A oegave lcoal margale s a suermargale ) If M M c;loc S s a sog me of ff g ; he E M S E (hms ) ; where M! M Proof. ) By hyohess, here exss a seuece of sog mes T T + a.s. such ha for each T he soed rocess fx ^T ; F g s a margale. Sce he seuece f ^ T : g s bouded by ; by L roery E (X ^T jf s ) X s^t mles a.s. whch jus ed he margale roery. ) Sce E (X ^T jf s ) E (X jf s ) X s^t X s E (X ^T jf s ) X s^t a.s. for all s for each T ; he (because he s exss a.s.) X s^t X s E (X ^T jf s ) E (X ^T jf s ) by Faou s lemma Lev s lemma, for ay A F s E (X ^T jf s ) dp E (X ^T jf s ) dp Sce A F s s arbrary, he Coseuely (Acually fx g s a margale) ) For A A A X ^T dp E (X ^T jf s ) E (X jf s ) X s E (X jf s ) E (X ^T jf s ) X s^t a.s. for all s for each such T : S s a sog me Esseally, esablsh (locally) uformly egrably E M S E (hms ) A X dp A X ^T dp 9

10 Remark Local margale: for ay s T ; we have ^ T ; s ^ T s E (X jf s ) X s For > T we have So deed s a margale locally. E (X T jf s ) X s Problem Le M fm ; F g M [ M c;loc assume ha s uadrac varao rocess hm s egrable: E (hm ) < : The ) M s a margale, M M are boh uformly egrable; arcular, M M exsa a.s. E M E (hm ) ) we may ake a rgh-couous mod cao of E MjF M ; ; whch s a oeal. Proof. Esseally, esablsh (locally) uformly egrably use mod cao. If M M ; he E M E (hm ) E (hm ) : If M M c;loc ; he E MS E (hms ) E (hm ) : Coseuely fm S g SI uformly egrable (by Holder s Ieualy). Thus M M a.s E (M jf ) M a.s. by Faou s lemma E M E M E M E (hm ) (3) Jesse s eualy yelds M E M jf If follows ha M has a las eleme, e., ha M ; F : egrable (3) holds wh eualy. Leg E MjF s a submargale, whch s uformly gves he reured rocess M Problem 3 Le M M c;loc show ha for ay sog me T of ff g ; P max jm j E ( ^ hm T ) T + P [hm T ] for ay ; > : I arcular, for a seuece M () submargale M ()E! P ) max T T M c;loc we have M () P! P Proof. Take X M A hm use he roblem oe 4. P max jm j P max M T T The secod follows readly by relacg. E ( ^ hm T ) + P [hm T ]