Martingales part 2 - continuous case-

From 1.3 in KS],

Doob's inequality , Doob . Meyer decamp,Doob's up- crossing ,

optional stoppingall have dart

.time counterparts .

The ( Doob's maximal cheat . I

Assume GXelec.com, is a cent.

martingale . Then

PC supHsl Sl )

a. set

e t ) HeldDef ELIXTD187%1141

Proof : Let Inc Ine ,

be a sequenceof finite subsets

of ( co , -4 n Q ) ult 3

sit. Un Fn i it]. ④ Ht}

( also , ensure te fo ) .

Def N! - Xe.

.teeth

In = Fen.

where he is an enumeration of tin

Exercise N! is a martingalev. rat . FL

.

So

lpcmnax IN! 1st )

t S IN.÷¥¥.me/dl?SlXeldP{ may IN! Ise }

{mqxlnhalsez

Let n- no,

notice that

l PC max lxql Sl )

9. E nloittlult}

E S Htt DP .

Imax Hal > l }9- e Nats )u4t4

Notice that since X is cant.

ima:*.co.in#iiHlseYaei" " "

T Lahm

Exercise

squarecntegrablemarkagateshz-LHtfte.com: X is a martingale

Xo :O and ECXI ] so 3

Mick,set

.X has

cont . sample puts .

Take Xt Mz,I is asubmakongale

.

From Doob - Meyer decomposition;I 4Ae$eIo .co , sit .

the A+ is P- as . increasing

and Melee*,a martingale

sit.

I Ate MeMoreover M and A are Peas .

unique . Interested in A ;

uniquely defined sit .

XI - Ae is a martingale .

Denote 97£ : - At.

Turns out Kelli,

then

the Hk is P - as.

cont.

Exercise Prove that

ELXI-xih-d-EKXZ-44.RS)

Call CX) the quadratic variation .of X .

q¥ Process Hike.comFor a partitionKitto ,t . . . .

.tn }

Otto Et , c. .. . c. Em Et

define the of - variation over

it as

V! htt j,

Hei - Xej . . It .

Want to molestand what happensVICK) as Hell -_ may Hit .

- til- O

Thnx for Kelli ,then

VICK ) - (Xz in probability.

Prelim .

H) Of scteucv

EUN - Xu) ( Xt - Xs))- ELENA- Xu)Ht-WIT)

= EL EL Xv - Xu Hu]Ht - Xs) ] = O

d.) ELK. -WIFE]= ELXi - IX. Xue Xi Ift ]= Exit Xi - 2ELXuxuhi.at Ife= ECXE-ixi-2-krlfut.la IIe ]he

-- Xu

= ELXE - XI HIT

HAA)

EL Xi -Wu - HE - Wa) Hito( Exercise )

(AAAA)

Ek Xi - Ha -Hi - uh .

{ Xi - aye - Ki - 43131=0( Exercise ) .

Lemmy Keltz, IX. KK Ras

.

for some Ks.

For anyone of it],

EL CHICK) 5 ] E G K" .

proof.EL.si.. .

Kei .¥. . 5 IE. I

= Ema.

ElHei - X.ci. . I IIe. ]

I E."

El ki - Xii. . HI. ]by HA)

= EL Ein. . Kai - Hii I fee )= EL KI HEa] E K

'

• ELI.I...

Hei - Ai . . Ike.- tea.5 ]

-

- ELELE.E...

Hei - Ai . . I Hea- Xen. .TED= EL

,

Hae - Xue. .T Effie.Hei - Ai- i I Hed)E K'

El Ei Hea - Xue. . T ) t k".

• EL E,

Hea- Xena , I ]

'- EL mua x ( Xue

- ku..I He. - Ha..ITw(t m

! IfYul't 2 Hea

.T )

E 4k' ECE

,

Hee - Ha-I ]

E 4k' K2 = 4 K

4.

Finally

ECCE' If = EfcE.Hei - Ho. . 51]= EL Ee , Hei - Aj. . IHei - Ai. . T)= EL ¥,Hei - Aj. . IHei - Ai. . T)+ EL iii. He; - 1¥14 ]

= EL 2 Kang - Kei - if Hei - Ai -c T)

+ EL Ei , Hei Hei . . I" TE 2K

"t 4K

"s 6 K

"em

Lemina If Xen:and bad ( (Xslt k tf s . Ras .)

Then

Lim EL V'Ice )] = OKEUN 0

Proofm

V.TK#g=ECXtj-Xtj..l4jet

c- mjax Hei- Ai. .IE?Hej-Xq..5

c. max Cllr - Xs ) VI' ut )s , recut)Ir- SIE 111911

This gives

EU#"" ¥!." !?!:*!" " 'T )

".

CERVI 531k

← ' El !?.si?...Hnxs5IIhr' K'

was

Kakao of since K cant.

+ bad .

Exercise Had

Proof of thinnm

want to show ; t e > O

Lim PU VMM -Mels. 4=0

Hell- 0

Assume first X and 44 are bnd.

( i.e . I k Ksk K . 3k. K ) .

ELWIN -KIT ]'- ELKIE

,Hei- Ai. . I - 4%-44*13)

-

- E. ELKA; - xei.it -XXI, - 41¥15)c. 2£, ELCA; .- kg - it

'

] -' ERM. - 4¥11

⇐ SELVI''ve) ]t2ELs;pKX¥. -K¥1 .

EH.

-Wen) )

⇐ IELV.%e5ft2KELS.pk/l7r-LX2sl)site catsIr-SIE HEH

→ O as Kkk- O .

In particular,since VI Kkk

in I ⇒ V'Ice)- Hkin probability .

When X not necessarily bend ;

Def in :=ihf{ t> Oilktlsniskz > n )andX! : = Xtaen is a makingale .

By uniqueness of Doob - Meyer

decamp . LN>t-- Menin

( la . since Kenan - hkttntnis a martingale ) .

Get VI. ur .x:HH" Itas hell- O

Notice x!- Ke Ras.

since pcqmwxi-ket-plnlisn.int)x -

- O

Exercise

Let a > 0 .

Palika ) -Welte )

-- Paktika ) -Wels. e. East )

t

PCIV.EE#I-CX7els.e.tnzt )

c. Please ) -1 PCN.7ur.li ) - Wiehe)-→ O as Hell -eO

⇒( in PCN can) - Wels. e)111911-O

c.

Pctnct ) t n

send n- as⇒ the result .

LTM

Exercise show in a different waysthat for the BM KBehteco.io,we have LB>ee t i

• EICH''

I - ti ]- O

• EL BI - t Ifs] - B? - s

Exercise show that BM

has infinite length C ! ! ),

i.e.

VI' ut ) -60 as Hell-O

⇐"

l Bt; - Bes . . I"

- length over

-wit)

"

im-

¥Hint Assume V' 'fue) -6 Ke

and then

V'Ice ) E mgx Hei- Boo . . I V 't

'ut )

send HRH- O gives a

contradiction.