APA lecture 12

Martingales

Def : Let ( r,

F,

P ) be a probability space

A filtration is a sequence of sub - r - fields off( Fh

,n EIN )

such that I Ctu er

,the N

.

Ex : D= [ o,

a ],

F = B ( Co,i ] )

,D= lebesgue measurea for ]

Xn I w ) = nth decimal of co E ( 0,1 ] n 21

Fo = { ¢,

r },

FICK ),

ferch,

ka ) - - . I -- orch -

. x !,

W = 0,17392 - - -

e m

I

Def : Let ( I,

new ) be a filtration on Cr ,F

,P )

A square - int. martingale wit ( In

,new ) is a discrete - time

stochastic process ( Mn,

n EIN ) such that

( i ) E( Mi ) c too th EIN

( ii ) Mn is I - measurable KEN ( adopted process )→ Ciii ) E ( Mine .

Ifn ) = Mn as.

then

Rink : Because EC Mine ,IF ) is Fu - measurable

by definition,

condition Cii ) is actually redundant.

Rink : martingale # Markov process

Example The simple symmetric random walk.

Let ( x,

use ) be iidr.u.rs st.

PCEX -- eel) - PHX = -131=7

So = o,

{ = Xt - - .tl/nhZ1

to - { do,

r },

Fu = Ha,

. . .

,x ) n 21

Then § ,new ) is a sq - int

. martingale iv. rt ( I,

he N )

Proof :

( i ) E (d) = n.EC/h2)=nctosVnz1

( ii ) Sn is Fu - measurable : yes ,as Sext .

-

t Xn ✓

( iii ) E ( Sue ,IF ) = E ( Et Xen IF ) = ECG II ) e EHealth )

= Su t E ( Xue .) = Sn as . #

t

Fu - measurable↳

the,

I I

Remark : If (

Mn,

new ) is a sq - out . martingale ,then

F- C then ) = ECE.ch#IIzD--ECrh)tuerNSo E ( oh ) = E ( Mn - a) =

- . - . = HE (Mo ) th E IN

So a martingale is a process with constant

expectation ; but it is also much more than that !

Ex : The process ( X,

u za ) with X,

i id r.

V.

has constant

expectation ,but it is not a martingale : EHealth ) - Elka ) = o

+ Xn:Illustration for Guin EN )

su;!Ekaanti

sie.

n nee

Property: let ( Mn

,nze ) be a square . integrable martingale

. E ( Muerte ) - F- ( E ( Mutz IIe.) IF ) = Ether IE )

= Mn

- - - - - - - -

IFI Ee .

. E ( Mum II ) = Mn th,

men ,-

i

Remark : martingale t Marka Su .

+martingale property : E c the ,th ) = Mn I nos

. Marka property : B ( { Man EB } Itn ) =P ({ thereBllrh )equivalently : HBEBGR )

Ecg

cnn.it/Tn)--ECgCrh-u)lrh)fgegcuz

)

Sub- and supermanhhgales ( square - integrable )

Defy : A process ( Mn,

new ) is a

✓

SEPT.

} martingaleWar

.

t.

a filtration ( Tn,

new ) if :

( i ) ECT) cease knew

→ ( ii ) Mr is In - measurable Vn E N

( iii ) E ( Mn ,I Tn ) Z Mn as

.

the N

Rem : a ) if M is asub - gnarlingale

,then E ( the . ) Z Eth )

.super - E

2) Not every process is either a sub martingale ,

or a martingale,an asupermartihgale .

Example ( simple asymmetric randan walk )

Soo,Sn=Xnt . . .th with X, use ) i id r

.v. 's with

PC { teen 3) =p = e - P ( { Xi - R) apse

& ,new ) is a safe. } martingale if { PPE

If: E ( See Itn ) = ECS II ) e Eke . II ) = St ECx

Property : If M is a square - integrable martingale⇐& ✓

& Cf : Nsr is a convex function st Ely (oh )%tos

Then @(Mn ),

ne IN ) is a sub martingale .

KEN

PI : E Cyane . ) Itn ) 2 4 CE Cohn.

II )) = 4cm )✓

Jensen

Stopping times & optional stopping theorem

Deff: A random time is a random variable T : R - s N

Let ( I,

ne IN ) be a filtration on Cr,

f,

P ).

Def: A stopping time w.r.t.CI,

n c- IN ) is a random true

such that { WE r : Tla ) = u } E Tu th EN

Exi let ( xn,

ne N ) be a discrete - time stochastic process

I = 5C Xo,

X,

- - -

,X ) new

Then Ta = if { n e N : Anza } is a stopping time

a c- OR W.r. t ( I,

u EN )

In

qAER -

° !

an° .o

°

i

• !⑦ ⑧

8

•.

I n•

Ta

Proof.

To be checked :{Ta -

- n } E I knew

{ Ta -

- n } = { Xue a toe Ken - a & X za }

=L?. .

Kusa } ) n { X za } e E-

-

C- Fec Fn C- I ✓

Def : If ( xn,

new ) is a discrete - time

stochastic process and T is a random time,

then we define

Xtlw.tl/twCw)=eIXlw)1Et=n3lw)Def:We say that a randomtime is

boundedif

there exists N E N such thatTcu ) EN touch

T

( a PIET en } ) =L )

Optional stoppingtheorem

. Let M be a ( square - integrable ) martingaleWent

. a filtration ( Fh,

NEIN )

Let T be a

ftp.p.h.g.h.k.e.w.r.t. ( Fu

,new )

such that F N E IN withE.E III.EItakeThenE (Mo ) = E ( Mt ) = E ( Mn )

T T

. If M is a ( square - integrable ) sub martingale ,

then ( Echo ) E)E ( Mt ) E E ( Mn )- - - - - -

-

pnot proven here

Proof :

EE.

As of Tla ) EN KWEI,

we have

Mtlw ) = Mta ,Cw)=⇐n Malu ) . lEt=n3lw )

-EEE' - ⇐ retainer.⇒÷EE¥¥:¥f÷i!÷

,

= !⇐E(E( Mn - let.us/FnH=IEECMn-1q-=nD--ECMn. let.mg/=ECMn)--ECMo )

Is- - - - -

p

( M = martingale )#

When does the optional stopping theorem fail ?

. If T is not a stopping time :

Ex : T = value of n E { o . .

N } such that Mn ? Mm

V-mcfa.nlThen both E ( Mt ) s Echo ) & ECM

,- I > ECMN)

. If T is not bounded :

Ex : Consider ( X,

n za ) be a sequence of iidr.u.rs

s.t.PK/h--eeH=PGxr--eD=I;Mo--o,Mn=4t.-exn,nnM-martingale , Ta = ihf { ne N : MnZa } a c- IN#

Then E ( Mta) = a so = Echo ) .

= a

Why is it an interesting theorem ?

Consider again the previous example with N -

- simple

Symmetric random walk : Men = hit - -t Xn

.

X 's= can tosses I?FI ⇒ Mn = cumulated gain at time h

AN 21,

E c Mn ) = E ( Mo ) = o

Let I mirinf { n e IN : Mn Z to },

N }

E ( Mt ) = E ( no )- - - -

+ to with high probability↳{ I- x with tiny probability ( when T= N )