Lecturett Optional stopping theorem
Tim Xn Fn a subMG nezzo ouch that
XnEECXolFn Hn for some X E'L
Then for all stoppingtimes QT suchthat OET
Eared Exo exist and
ECXe zE Xo 2E is
Continuous time
The Xt Fe a right sub MGwith a last element X X EL't Foo
ETH 1 z Xt ft Two Markovtimes
t O a s then Exa EX exist and
ECXdzECXo ZF4Xo a
Note Most common appt is Mg EXEEXO
Proof Discretization argument
X EmFe XaEmFo
Indeed Xt right cont hence progr medsT is stopping time for FttFurther X exists info
Xe Em FeiDiscretizy sie k1ze KELO1,2 ab
GE Fs then Xs if is a subMGWe can apply discrete OST
Te inff sa i Se Ty Ie L I
Oe int Sr she 03 214281 1
Te Oe are g stoppingtimes check
EXE Exa ht
Further Tete Oeta as Goo
By right continuity
Xte X Xq X a s as to
We know Xee Xeneed to show EXz EXeNote that
TIFF 411 7
Z n Xen for nEZzo
gin Fen is a reverse rubMG
E Z NII n Em as fnzi
EL Xe.IF.int iXn
indeed both
Tn Tnt are GE stopping times and
can apply discrete 05T
Z n Z Xe
info EZn info EXT 2 Exo
En is 01 Z nYZ.io
Xen Xe Similarly Xon XoEH EX EX eZEXo
D
I
Gary If Xt is right cont subMG
with lastelement and Tzo MarkovtimesThen
Ethel 7 77 Xo d s
if Ma
If 4 is stopping time
EEXelFo z Xo a s
Proof AE Fai anddefine Markov Tine
OIA 1 TIE IZZ a S
EFX 2 EHz EIXoIatXIaaSince Xe XoEU
ELKEXo 2 120 since Ia EmFoXo E
El tFFX 2 770EmFat
holds for any AEmFo EH170 12 Xo
Rink To check assumption of OST sufficienteither
1XEECH Ft forsome YELKR.FRdifference notassumingYemFo
2 t bounded as Eg if TET forsomedeterministic 1
Can set X e XtatApply OST to It ok because Ithas lastelement xj Xi.coy2 IfCXthzo a right cont subMG
and c is Fe stopping time Then
Xtra ois a right cont subMG
Proof Apply OST tastz AEF
Q flat0z Kate Iat atz.LI
EXozZEXo
ECXeat.IA X atzIAIZECX.catEKX catz X.cat Iac 70 D
Given FG measurablespace statespaceB equivalent IR Polish
A trans probability is a fit
p Xx g Co D et
1 F x e X par is a prob measure
on X Ey2 f BEL Plo B in a mess function
on X GDef A collection of trans prob ftp.tzszois consistent if ft stat
pt.tt BTBPtPFsEgFand pt.tk B IGEB
Pe.tk 8
pg G B 94,13 pardyt
V prob measure on X GPi Pz Pr trans prob
V p pie is theunique prob measure on
Xl
f'tsuchthat
hop PaiCAHA x xA
PfIE 0 130 Pecan x e video
t.tt I.EiIf Pst PI for some family of trans prob
Fethis is a Markov semigroup Paf Peps
An Fe adapted SP is a Markovprocessof trans prob pst if
tttzslPCXtEBlFdps.tLXs.BJ a.s
Homogeneous if Ps Fees is a Markov
semigroup p XeEBlFg PCXeEB Xs