MainSteps_ (1)

B00wnianMotion_pastI_

^•

construction

•

Nowhere differentiability

Definition A real valued stochastic process { Blt ;t>o }

is called a Brownian motion with starting point ✗ ER if ti ) 1310 ) = x

tie ) For any MEN and Otto < tis

^{. . -}

< tn the increments

B. ( tn )

^-

Bltn

^-

a) , Bltn

^.

, )

^-

Bltn

^-

a) ,

^{. . . .}

, Blt , )

^-

Blto ) , BAD

are jointly independent

^.

Erie ) Blt -1h )

^-

Blt ) =D NCO ,h )

Civ ) t → Blt ) is continuous , almost surely

^.

Specialise 2=0 <→ standard Brownian motion

^.

Goat : Need to show that Brownian motion exists

^.

MainSteps_ (1) Canonical construction ( using Kolmogorov 's

extension theorem )

This will allow us to deduce that a continuous time

stochastic process exists with properties Ii )

^-

Liii )

^.

(2) Continuous modification

This cont will allow us to modify the sample paths of the

.

time 5. P

^.

constructed in step 1 so that sample paths of the modified 5. P

^.

are continuous with probability one

^.

( i.e

^.

it satisfies property Civ ) )

canonicakconstruction.DE/ini1ions- (a) Let ER , Fe , B) be a probability space

^.

If TCR is an interval , then the collection of r.rs { Xt , 1- c- T ) is called a continuous time stochastic process

^.

(b) The collection of probability measures { flt , ,

^{. .}

.tn , n c- IN t ; c- T i

^-

-1 ,

..

.

n ]

are said to finite dimensional distributions ( f. d. d.) of txt , t c- T ) if

flt , ,

^{. .}

.tn (B) = PC ( ✗ ti ,

^{. . . .}

,Xtn ) c- B) FB c- B( Rn )

^.

G) A collection of f. d. d. is said to be consistent if

* Mtn ,

^{. . .}

.tn ( Bn ✗

^{" -}

✗ Bn ) = µ tan ,

,

^{. . .}

. + * in , ( B a- a) ✗

^{" '}

✗ Bored

if for all permutations ñ ; Bre B.tn distinct

^.

* flt , ,

^{.. .}

.tn ( Bn ✗

^{- - -}

✗ Bn

^-

n ✗ IR ) = µ ,

^{. ..}

.tn

^.

. ( B , ✗

^{. - -}

✗ Bn

^-

1)

^.

Theorem

-

For any consistent collection of f- d. d. I a probability space ( rite , P ) , and a S

^-

P

^.

I with Xtcw ) , TET } on it , whose f. d. d. are in

agreement with the given collection

^.

Further The restriction

of by P the to Fix =o(hXt;ttT } ) is uniquely determined

specified f. d. d-

A-ppliak-mremi-Y-o-Brownian-mo.cn

Fix NEIN ; I , ctt , <

^{. - -}

< tn so

^.

Then the joint law of ( BAD , BAD

^-

BLED ,

^{. ..}

. Bltn )

^-

Bctn

^-

D)

is to be given by product of independent NGS -4 ) , NCD.to

^-

Ed ,

^..

- . ^.

> & N lo , tn

^-

tn→ )

^{. -}

Blts )

Note ( Blt , ) ,

^{. . .}

, BC.tn ) ) = A / 1- BAD Bctn

^-

: ) Blt

^-

Bltn . )

^-

) It

for some deterministic matrix A

^.

One to determine can therefore the apply the change of variable formula

joint density

^.

It turns out to be a multivariate normal density

with mean vector

µ and covariance matrix ICE )

µ = ( x ,

^{. . . .}

x ) because E[ Blt )]=E[ Blt ) +

^-

1310 BID ) ]

El Bti Btj ] = E- [tBtj

^-

Be :) Bt :] + E[ Bt , ? ] = "

^'

= E[ Btj

^-

Bt ;] E[ Bt ;] + ti + x2 ( Bt ; -1 NG , ti ) )

tin Ltj indie

increment = t ; + x2

( Btj

^-

Bt ; =D N / 0 , tj

^-

ti )

Cov ( Bti , Bt ;) = E[Bt;Btj ]

^-

E [ Bt ;] E- [ Btj ]

= Lt ; + xD

^-

at = ti = tix tj

Thus I

;] (E) = tin tj

Me , ,

^{. .}

.tn = MVN ( ✗ 1- , ICE ) )

^.

Easy to convince yourself that the collection

{ 1kt , ,

^{. ..}

,tn ; NEIN , t , ,

^{. .}

.tn 30 ] is a consistent collection

of f. d. d- Therefore , by the theorem above , a cont

^.

time 5. P

^.

with

'

properties 15 )

^-

fiii ) mentioned

^•

in the definition of Brownian motion exist

^.

Baektotheproofofheorem

D= RT = collection of all function x : T → R

Fo = collection of all finite dimensional measurable

rectangles

= s.to/t=&xc- collection RT : of alt all ;) c- A B ; C- RT 5=1 ,

^{. . ..}

n ] for some n c- IN

& Ba ,

^{. . .}

Bn C- B.

Set BT = oligo )

^.

he = collection of all sets that admit a countable

representation

= all AE IRT S.t. F C C T countable ID c- Bc

A-

^-

txt RT : ( xlt , ) , xlta ,

^{. . .}

) B. c- c- D }

-

Borel 8- algebra

Lemma 03 " =L

^.

induced by co

^-

ordinates

- in C

Boofofk-mmaltoproveeec.TT ' -4 , tz ,

^{.. ..}

}

Fix any A c- le F C CT countable s

^.

t

^.

A = { ✗ C- RT : ( xcte )

,

xctz ) ,

^{. . .}

) E D }

for some DE Bo

^.

Let p : LRT , BT ) → ( R ' ,B° ) be the projection map

{ xlt ) ) te , → (21-4) , xltz ,

^{. . .}

)

Observe A = p

^-

' (D) , by definition

^.

Claim : P is measurable

^.

To see this claim let Leo = { I c- B. ° : p

^-

1- (5) c- BT )

^.

If Ñ is a finite dimensional measurable rectangle then

so is p

^-

' (5) Flo

Thus leo contains fall finite dimensional measurable rectangles

^.

Also , Up np-i-CD~rj-p-tcn-D.DK

^-

' ( In ) = f- ' ( 4 In ) & K K

Ceo is a monotone class Ceo 3 Bc

^.

As A =p

^-

' (D) , by our claim A c- BT

ftp.soveos#/-

Observe Teo Cle

^.

Also not 6 is a o

^-

algebra

* Ieee

* A c- ee A = { ✗ C- RT : ( xltn ) , xltz ) ,

^{. . .}

) c- D } For

some ☐ c- Be

A' = { HERT : Gilt , ) , xltsl ,

^{.. .}

) c- D ' ]

A ' c- le Tye = { act } tea

* Are c- le

.

Km ; An = { KEPT / alone Die ] Dk c- Bck

need to show UAK C- Ce

.

Kit

C = UCK T c # Ck Pcr ,

= tripe

K91 -7

for

Are poi ' ( Dn ) = to

^-

top ( Da )

¥ , Ar , = pit ( § , pitch ) =p .

^-

'

^-

(D) c- ee

^.

Recall from above pit ( Dn ) c- Be § , fitch c- Be

11

D

BaeK1otheproofofTheorem_ ( application extension of Kolmogorov theorem ) 's Fix A c- BT

^.

Need to define PCA )

^.

By Lemma F C = { ta , ta ,

^{. -}

] CT countable & D E- Bc

s.li/t--po-1- (D)

Observe the sub collection { yet , , ↳ ,

^{. . .}

.tn , n c- IN ti e- C } is

also a consistent collection of f. d. d.

By Kolmogorov 's extension theorem 7 a prob

^.

measure

Pc on ( IRC , B) st

^.

the finite dimensional marginals

are given by the f. d. d.

Define P( A) = Pc (D)

^.

Need prob

.

to measure argue that on ART this , BT is )

^.

well defined & is indeed a

If AE Bi =L has two countable representations , say

A-

^-

poi ' (D) = poi ' ( Dz )

^.

Need Pc , ( Dn ) = Pczl Dz )

^.

1- c C , T # c # cz

Set c-

^-

Gu Cz -

Pcz A-

^-

poi ' (D) = pi ' ( p ,

^-

1- ( DD ) =p ( pi ' CDs )

D= pit ( Dn ) = pit ( Dz )

Since C o C1 and all finite dimensional marginals of Pc and Pc , agree by Kolmogorov 's extension theorem

Pc , = Poop ? Pc , ↳ D= Pc ( D)

Similarly , Pez CDs ) = POCD )

^.

This implies Pt ) is well

defined

^.

Remains to show Pt ) is countably additive

^.

Fix a countable

FCK CT countable & Dk e BCK g. +

^.

collection of disjoint

sets { Ari , AKEBT

AE por ?( DK ) An = pit ( In ) where C-

^-

Ucr K ? 1

•

where { ÑKY are disjoint and are in Bc

- )

UAK F- I =p ( ¥ .DK

P( ¥ , AK ) = Pc ( U Ik ) = I PCCIK ) = -2 PLAN

^.

KM Kzn Kzn

Uniqnenessontl say pic ) and 1134

If there S

^-

t

^.

Pal exists A) =) PIA two )

^.

such prob

^.

measures / Then FA c- 7×

A. c- Fix =) A = { w/ ( ✗ t.co ) , Xtzlwl ,

^{. ..}

) C- D)

for theorem some 7 CCT a countable & DEB ! By Kolmogorov 's extension

unique prob

^.

measure on ochxtiyiz , ) such

that all finite dimensional marginals agree with the given f- d. d. Since 1174911317 , restricted to sthxt ;] in ) are two such

measures we get a contradiction

^.

cooiinuousmodification-h.lt , 1- c- I } and { Yt , 1- c- I } be

two 5. P

.

Then { Yt , te I } is said to be a continuous

modification of { ✗ t.tt I } if

t1→ Ytlw ) is continuous for all wer &

p( ✗ +1--4+1--0

V-tc-I.kolmogorov-censov-heore.me ICR be a compact

interval

^.

{ ✗ t , to I ] be a cont

^.

time 5. P

^.

Assume that

72,13 , C) 0 5. t

^.

E- [ / Xt

^-

Xs / d) £ clt-si-PV-s.tt I

^.

Then I a continuo s modification of { ✗ t , te I ]

which is locally or

^-

Holder continuous to c- Co , Pla )

^.

Recall f is locally 8- Holder continuous

( Hu c- I I Cuca , the > 0 s

^-

t

^'

sup

It

^-

uifhu H%É / f Cw

^.

5=1 C- Is

^-

ul

Application

of-kotm-g-enso.Theoreminth-seltingofBMT.tv

previous step already shows 7 a cont

^.

lime

5. P

^.

{ I Iti t c- [ 0,141 ] , any MCN , for which required

^-

distributional assumption holds

^.

E- [ ( Ict )

^-

Bts ) ) " ] =3 It

^-

si

Obsey , , Hd

* win -5

The requirement for KCT holds with 2=4,13--1

^.

7 a cont

^.

modification { Blt ) ; t c- [ 0,14 ] } that

is ✗ c- ( 0,4g ) holder continuous

^.

( Need additional argument to show the existence of

{ Blt ] , -170 ] ) Ctaim : { Blt ) , telco , D) r

^-

holder continuous for

any ✗ c- 10,42 )

^.

Use that E- [ ( Be

^-

Bs ) " ]

⁼

Ck Ct

^-

s ) " for any K > 1

and some Ck < 0

^.

we can take ✗ = 2k I 13 = K

^-

1

^.

in KCT

^.

Thus , for any 8<112 I K* s.t.rs EE 2K #

^.

Work with ✗ = 2K¥ & P=k* -1

^.

"%¥a%:÷:¥÷

Otto , , ] = set of all dyadic rationals in 10,1 ]

= { % ;j=0 , 1 ,

^{. .. .}

, 241 LEIN }

By Markov inequality

P( txt

^-

Xsl > e) £ E- ✗ Edit

^-

Xsld ]

f C E- ✗ It

^-

s / " P

Fix 811312 ; set y =p

^-

ar > 0

t= 5¥ ; size ; e- 2- re

Pl 111¥

^-

✗ size / 72

^-

• e) ecz 're ⇐ ej :P

By the union bound

P(mjÉÉ / Xi ;

^-

✗ %) zz

^-

• e) ← = Canez C z

^-

Te

^-

Pr

Thus ¥ , plmii.in/j1Xi-:-.-X%e1zz- " ) < a

By BCLI

in "aÉ / X ;÷cw )

^-

Xjzelw > / < 2- re

j=o He > nrlw )

w 4- Nr where P( Nr ) -0

^{- .}

Claim : Outside Ny ttt ✗ +1W ) is locally r

^-

Holder

continuous on [ on ]

^.

That is : t wet Mr

/ Xtlw )

^-

✗ stole Colt

^-

sp

whenever t.se Ohio

, , It

^-

sl < Brew ) = 2- now >

Cy = 2/(1-20)

^.

troofofcaim Fix t.se To > ☐ sit

^.

It

^-

s ) < fylw )

^--

2- " " "

F MEIN s

^.

-1

^.

2- ME / t

^-

s / < 2. 2- m m > nooo )

i¥¥=:m

"

Im

as t

, SE Chio , , , 3 re > re

^-

i >

^{. . -}

> on > m & qj > qj , >

^{- . .}

> 9in

g. t

^.

t= ifm + Yzrn +

^{. - -}

+ Yzre ; S=i÷m

^-

Iq ,

^{- . . . -}

Yzaj

te-iyzm-Yzri-1-ii-z-re.nl/tlw)-XslwHtIXtlw)-Xi/zmlw1/-/Xi/zmh )

^-

✗

i-j.CN/#Xi-gmlws-XslnHfIXtlw)-Xtelw)/+1Xtelw ) - Xigmiw )§ 2- rm - f 2- Me + I Xtelw ) - Xigmlwl Game argument -22-01

: l > m

: continue

t §m 2- re

/ Xtlw )

^-

Xslwll £2 -4,2

^-

M = Egg 2- Me ?=→ It

^-

sir

since the above argument holds for any 2C Pla ; considering

a Leg

^.

8kt Pk we obtain that 3 null sets Nk sit

^.

outside N= ¥ , NK ; for any K > 1 , s - t c- Onto > ☐

/ ✗ + Lw )

^-

Xscw ) / E C. EDIT

^-

sl " if It

^-

Sl threw )

^{- - .}

c* )

set Ñscw ) = Xslw ) I lw¢N ) se che ? ,☐

= him n→N ✗ snood )ICw¢N ) se Il Ohio ?☐

where { Sny ng ,

C- Ohio , ☐ s.t.sn → s

^.

The limit exists because for w¢N

/ ✗ go.tw?--Xsmlw7/fCklw)lsn-sm/ " for all N large

, m

4 hence Cauchy

^{. .}

( by # )

The the sequence limit does hbn } not

.

defend also on the choice of

If { Jn ] c- 9% , , is another seq

^.

s.t.sn → s

/ Xsnlw )

^-

✗ In LW ) / f Cklw > Isn

^-

In / " for all Kogen

^.

Again , due to 1*1 we find that

/ It cut

^-

Is lw ) / f crew ) / t -40K if It -51£ honors

V-w.ch , trig 1 V-t.SE I

^.

{ It , t c- I ) is a continuous time S

^.

P

^.

with continuous

sample path

^.

Finally , need to show that { Ñtjtt I ] is a modification

of hxeitc-IY-iie.pl/t=Xt)--OV-ttI

If b- c- Chief , , ] then P(Xt=Ñt ) f PCN ) = 0

If 1- c- It Ohio , , then for w¢N ÑTCW ) = n him → a Xtncw )

( where t.tn ] c- OÑ[o,☐ is some Seg

^.

s.t.tn → t ) µ Enough a show mill outside to set

By our assumption

^.

✗ tlw )

P( I Xt

^-

Xtn / 7£ ) t nd ELIXT

^-

Xtnl 'T

f n ✗ € It .tn/ " I

choose htn } c- Chio , ☐ so that

I PCIXT

^-

Xtn / In ) so

NZ 9

By BCL I P( tinny , Xtn = ✗ t ) = 1

P( ✗ + =/ It ) £ PCN ) + PC Kyo Xtn # Xt ) = 0

Extensionfromacompaetintervatoz

Lemma- Suppose { Xt , b- 70 ) is a continuous time S

^.

P

^.

Such that it has locally 8- Holder continuous modifications

on [ 0

, Tn ] , th , where Tn TN

^.

Then it has also such

modifications for the entire range food

^.

proofofkmma-hethxtlm.tt Loin ] ) modification be the continuous of txt ; tells Tn ] }

^.

Define It = It " if -1£ Tn ( outside a null set )

Need to show this is well defined ; locally 8- Holder continuous

modification

^.

An __ { w : ÑÑw ) = Xtlw ) t te lo , Trina } PLAN )=1

^.

A- If , An P( A) =L

^.

On A II " Lw ) = It " Lw ) t tech n [ 0 , -1nA -1m ]

Since wi-X~t.tn ) ( W ) , Ñtc " 1W ) are continuous we have that

I W ) = II " Lw ) two A and t c- [ 0 , Tna -1m ]

^.

{ It ; -130 ) is well defined

^.

Locally 8- Holder continuity

follows from definitions of { Xt " ; t c- coin ] ]

^.

caveatwiththecanonicalconstrudion-Noneofthe.it nice " sets that would of interests are in

B. ?

Exam# let -1=10,1 ] and

A- = { * c- RT : alt ) I r t TET } , where TER

^.

Then A ¢ BT

^.

This shows that sup ✗ It is not measurable TET w.r.to

^.

BT

.

Proof : If A c- BT then there exists

a countable C C T and a DEBC

A = { ✗ c- RT : ( Itta ) , ✗ ( ta ) ,

^{. . .}

) c- D }

^{. . - -}

( * )

consider any t* c- Tlc and some do C- A.

Set Iott ) = xoct ) tt=t*

= Holt ) -11 1- = 1- *

By construction Io ¢ A ( because ✗ c- A sup acts 8)

On the by c* ) ÑOEA

^.

Hence we arrive at a TET contradiction

.

Same argument shows that CCT )¢BT

^.

This is problematic because we will then miss out on

many nice properties of cont

^.

junction

^.

Solution : Ket T be a compact interval

^.

( cc -11,11

^.

Ibn ) is a complete metric space

^.

Hallo = Sup t / Xlt ) / ✗ c- CCT )

c- T

Bcci , = Borel 0

^-

algebra on CCT ) induced by open balls

Then Bcct ) = IT = { An CCT ) : A EBT }

Proof : Bcci ] C Fi

Fix any 070 ; x c- CCT ) ; Blair ) := { y c- CCD : Ily

^-

allow ]

={ YECCT ) : sup tout ) -212-31<0 ]

1- c- Got

= CCT ) n { YEIRT : sup txcttyctl

tears -1 < ry

T

this set has a

countable representation

c- 77

^.

Observe 7 is a G- algebra ② cc Ct

^.

To prove the converse note

F. = 0170 ) A CCT ) Fso = all

^-

finite dimensional

measurable rectangles

= 6 ACCT )

to = finite dimensional open

each set in Ion ECT ) intervals

^.

is open in CCT )

In CCT ) e Bcci ) 77C Bcci

Rem To extend the above result for -1=[0-0]

simply consider P to be the metric for the uniform

convergence as above

^.

on compact sets & then proceed similarly

A 11 x

^-

yllj That is play ):= I

j= , É ¥Fyyj 11£ "j= b- Sup c- [ o ,j ] 12-1+4

My c- CCT ]

^.

Cogotary : There exists a continuous time S.P.

B. c. ¥4 EBA ) ; t > OY that satisfies all four properties of

the Brownian motion & measurable w.rs

^.

E. ② act]

^.

theorem ( Paley

^-

Wiener

^-

Zygmeend ) Let { Blts , Ezo } be

a Brownian Motion

^.

Almost surely all paths of a BM are

nowhere differentiable

^.

Proof- : Fix K > 1 ; set BCK > ( t ) : = Fa Bckt )

Fact : { Blk > Lt ) ; teco.it?i:sB%fBM

^.

( Cheek )

Claim : Enough to prove nowhere differentiability on [ 0,1 ]

^.

( due to the fact )

BKC ) nowhere diffbk on [ 0,1 ] BC ) nowhere diffbk

w.fr

^.

1 on [ ok ]

W.p.

^.

1

Now take a countable union

^.

{ Blt ) ; t 70 ] has nowhere differentiable paths up

^.

1

^.

Turning to prove non

^-

differentiability of BM on [ 0 . D:

If f differentiable at some t c- IR

then limo fl-t-hn-fftc.ae

^.

F s.to/fLs)-fCt)/sK1s-t1 M , K < a

whenever Is

^-

tis yµ Set N=2M in > 2N

t c- [ in , 5¥ ]

←+!É÷¥#

E- In ÷

Thus : f- differentiable at t

If 1in )

^-

fi 31 £ / fun )

^-

fast

+ Ifct )

^-

f- ( ¥31

t.tk/n-t2Kfn--3K/n

^.

Similarly If(i±n )

^-

fo¥ ) / ; If(i¥ )

^-

flint £3k /n

Set of w sit

^.

✗ Blahs a differentiable for some te [ 0 , is

c U V M Ñ " { w : 1B¥ ) ( w )

^-

B ( %) Inst ;

N c- 21N KT NZN 5=1

/ Blin )lw )

^-

130¥ 11h01 ,

I BCi¥)cw ]

^-

Blitz ) ( w ) / { 3¥ }

- Cri , K, n

Enough to show that p( n

n

^-

'

µ ¥ , Ci , K ,n ) = 0

^.

Then from we below can

^.

use continuity

P( ci.mn ) = PLINIO , Mn ) / EBI ) }

= PLINIO , DIE 3÷, ) 3 I (f) 3

É g- =\ ? IPCCi.mn ) I n.L-s.cz → 0

^.

This completes the proof

^.