lecture lt ee - stanford university
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Lecturestochasticprocesses collections of iv s Xtindexed by IT on Cr F PIT uncountable lT gb EEDefinitions constructions
Filtrations MGBrownian motion
Howdo we specify ExtCountable case T N X µ takesvalues in IRN D
Bc 0cLR'I t.CA An t tn N Ai AnEB
R'I t.CA Ant LxERV xlt.IE i xHn EAnh
hm Kolmogorov Suppose Hh fun probmeasureon CR Bru st
µn i Aix ir tnxlR fenCA x xAn
Then 7 Pon RN Ba st In
PCR nlt Aa AnD lunlAix xAnl
uRmkWill talk about processes taking valinRGeneralizes to B equiv spaces
Eg Polish spaces completemetricseparabRd
Finite dimens distr fold
Mt tn Probdistr onhBpd
t tn C IT distinct
stent if ft tn.tn TESnV Ai An E B
G Me tnAnX XAn Ut tea Nat Teal
it Mt tn n fAi.xAnHR lue tnlAix xAn
Rink ITER then by cis sufficient to giveµ for taste taii satisfied if
He tn i HIfIjjpAo Ht ti tii.tnHi Aa o
K thm If F IN consistenttold determine
unique IP on Bccan wegenerah.ee s
lRT Lx T ilR EtsittyRt t CA An
BT oC RI t.CA An
Def AE RT has countable repr if 7 Salt tzD C Bc E R'N st
A LxcRT GH HH I EDTlsCX
S base GD representationA Tis D
Lemire B E AERT A hascount tepr
Pref ETB fix S Ct taE S L Tis D D C Be b r
In ECS E B'T
Indeed Tls R'TB'T GRIN Bc meagerbecause
Tls A x xAaxRx E B'TRt Ai An
BTECIR't LA Ant E EClaim he is a o algebraAi E E wt EIAi EEAi Si Di Ai Tsi fDi
S Si Ts proj from Stasi
D YI Ti's si Di C Be
SD is a represent of A Ai j
Xt T F smallest oalgebra suchthatXt Em FX V te IT
ALemina F f X A HEB u
her KelwBte EA
Btw ta XtLw samplepath
The Given fold's Gut tn consistentF R F P and X D R Htt 11
whose fold are given by µPCR't A Ant Mt tn i XAn
Further Pl unique o
Given IP with sane fold'sPC w XCut EB QHw X.tw EBbV B E B'TI know PHw X.ME2t tfAixAulbl QC
Pref r R'T F B Xtcw att
Fix S Cti z can define us onCRNBavia k thinHence THE Els A _IT D DEB
PCA µs D
Well defined A Di Daneed to show Ms Di Ksdkidea look at 5 5 usa
Ak Tif Dx disjoint Wts
PLEIAD CAD
Idea represent An on S EISKA E III Dk Tis 5kEIHe Tls C Ik Be disjoint
PLEIAD iusCEIBA Eius II PLAIDUnfortunately many interesting eventsnot in B'Ttemple IT 10,1 AE IR
A Ht to V t c o.IT E B'TIf it was there would be Ct tz SDE Bc stA f x Nti Ltd IED
Hs Yb either x ye A or xyet A
Take Ct o ft
Htt o ft t yH I
t EIS x cA y A Ksk TslyIn general fold do notdetermine the probof theseeventsExample AFP to D Boo Unif
1 if LawXLlteeo.is Xtcw L I if E w
Fel w L alwaysSame fold
Mt tnAiX xAnl 1hEAa rEAn
PCFezoV t r.PCXt2oVt O
ahh1 t