From 1.3 in KS] ,
optional stopping all have dart
. time counterparts .
martingale . Then
Proof : Let Inc Ine ,
of ( co , -4 n Q ) ult 3
sit . Un Fn i it]. Ht}
( also , ensure te fo ) .
Exercise N! is a martingale v. rat . FL
.
{mqxlnhalsez
9. E nloittlult}
Notice that since X is cant .
ima:*.co.in#iiHlseYaei" " "
T Lahm
Xo :O and ECXI ] so 3
Mick , set
. X has
.
the A+ is P - as . increasing
and Melee *, a martingale
unique . Interested in A ;
Denote 97£ : - At .
Turns out Kelli ,
cont .
q¥ Process Hike.com For a partition Kitto ,t . . . .
.tn }
define the of - variation over
it as
Want to molestand what happens VICK) as Hell -_ may Hit .
- til- O
= 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
(AAAA)
{ Xi - aye - Ki - 43131=0 ( Exercise ) .
Lemmy Keltz , IX. KK Ras
.
proof.EL.si .. .
El ki - Xii. . HI. ] by HA)
'
,
Hae - Xue. .T Effie.Hei - Ai- i I Hed) E K '
El Ei Hea - Xue. . T ) t k " .
• EL E ,
! IfYul 't 2 Hea
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" T E 2K
" t 4K
Lemina If Xen: and bad ( (Xslt k tf s . Ras .)
Then
Proof m
c- mjax Hei - Ai. .IE?Hej-Xq..5
c. max Cllr - Xs ) VI' ut ) s , recut) Ir- SIE 111911
This gives
EU#"" ¥!. " !?!:*!" " 'T )
+ bad .
Hell- 0
ELWIN -KIT ] ' - ELKIE
-
'
→ O as Kkk- O .
in I ⇒ V'Ice)- Hk in probability .
When X not necessarily bend ;
Def in :=ihf{ t> Oilktlsniskz > n ) and X! : = Xtaen is a makingale .
By uniqueness of Doob - Meyer
decamp . LN> t - - Menin
Get VI. ur .x:HH" It as hell- O
Notice x!- Ke Ras .
⇒ ( in PCN can) - Wels. e) 111911-O
c .
LTM
Exercise show in a different ways that for the BM KBehteco.io, we have LB>ee t i
• EICH ''
Exercise show that BM
has infinite length C ! ! ) ,
⇐ "
and then
' ut )
contradiction .