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