Upload
others
View
11
Download
0
Embed Size (px)
Citation preview
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.