MATH180C: Introduction to Stochastic Processes II

www.math.ucsd.edu/~ynemish/180c

This week:

Homework 7 (due Friday, May 29, 11:59 PM)

Today: Reflection principle> Q&A: May 29

Next: 8.3

O

stoppingtimesandthestrongMarkovpropert.cz)

Def ( Informai ) . Let (Xt )" . be a stochastic process

and Iet Teo be a random variable . We call T

a shopping time if the event

{ Tet }can be détermine d from the knowledge of the

process up to time t ( i.e .

.from { Xs : "et 4)

Exemples 1-et (Xt)to be right - continuons-

:

1. min { tzo : Xt -x } is a stopping time2. INK is a shopping time3. sup ht > o : Xix is not a shopping time

stoppingtimesandthestrongMarkovpropert.ITheoremcnoproofltet(Xt )⇐ o

be a MC,let T be a stop ping time of

(Xt) tzo . Then , conditionat on Toro and Xt =x ,

(Xtttltso

( i ) is independent of { Xs ,OESET }

(ii ) has the same distribution as (A)+» starting from x

Example ( Btttso is Markov.For any xe IR define

tx = min ft : Bt -x } .

Then

• (Btttx - Bia)ois a BM starting from x

• (Btttx - Bix )t > o is Independent of { Bs ,oesetx 4

(independent of what Be was doing before it hit x )

Reflectionprinciplethm.

1-et (Bt)". be a standard BM.Then

for any tzo and xso

P ( max Bu > x ) = Pll Bel > x)oe uet

Prof . 1-et ↳ = min { t : Be -- x }.

Note that Ex is a

shopping time and is unique ly détermined by { Bu ,OEUETX }

From the definition of tx ,max Buzx ⇐ Ex et

.

Theneuet

Pf MaxBuzx ,Bts x ) =P ( Tx Et ,

Bet -a) +ex- Bçç 0 )

eu et

tzpftxçt ) = ÉPÇ.IE#Bu ? " )-

Now Pljnççxbu > x) =P ( B+ > x) -IP (max Buzx , Btcx)OEUEx

⇒ plome.ua#BuEx)=2PlBtsx)=P(lBtI > x) heu

Reflection principle- Bt

Proofwithap-turei.mn?.::EEifBtH (Belt» is a BM

,then ( BI) to is a BM

,where

{ Be ,

t' txB- t =L B"- ( Bt- Bix) ,

tstx

⇒ to each sample path with ç¥ Bu > x and Box we

associated unique path with çç¥ Busx and Bear ,so

Pla?.az#Busx,Btaxl--P(Btsx)--sP(omuax+Busx)=2PlBtsx)

ApplicationoftheRP.distributionofthehittingtimet.cl3g definition ,

Ix et ⇐ max Be ex, so

OEU Et

Pftx et) =P ( max Bt > x ) = 2 P ( Bts x)OEU Et

a

= 2-¥, Îé du § mort ,du = F- dv

*X

- E= VÊ Je Zdv

NE

⇒ p.d.f.at Tx fact) = VÊ e- Ë - Et" = ¥ f-"e- Ë.

Thin. Fait) -_ IÈ §ÈÈdv ,

ta H=Ët"e- ¥

ZerosofBMDenotebyottiti-sjtheprobabilitythatBu-oonlt.tt s )0ft , tts) : =P ( Ba -- O for some uc-H.tt»)

Thx.

For any tissu

O-ltittst-farccos.IE17¥ Compute P( Bu = O for some Ue (t.tt» ) bycondition ing on the value of Bet

.

D- ltitts) = .ËP( Bu -- o for some uc-ltit.SI/Bt--x)pz- e- ËDXA

Define Blu -_ Beau - Bt.

Then

P( Bu -- o on (t.tt» 113f ⇒c) =P ( În -_ - x on lois ) IBT -- x)# *)

symmetryMÎP (Ba = - x on 10,53) =P ( Bu = x ou lois))

ZEROSOFBMPlugging A- * ) into H ) gives

-0ft , txs) = P ( Bu -- x for some ue (ois ] ) ¥, e- Ëdx

= ÏP ( Ba = x for some ut (ois) ) Ë e- dx

+ § P ( Bu = -x for some ut (ois) ) PÉ e - ¥dx

= VÈ [P ( Ba -_ x for some ut (as] ) e- da.

Finaly , Pl Ba - xso for some ut (ois)) =P ( ronçuxsBu > x) - Pltx Es)

(*f- [Etè ( §¥. j" e- ¥ g) dx =¥. ! ! xè(± '

da ) j" dy

ZerosofBM-jae.EEtas. =é¥

⇒ c) = En ! .it?y-j*dy=E!sa#rydyNow use the change of variable E- IÈ

, dy =ztdzs*) = E-Ë¥ ' ztdt -ÊÏ# d- = E- antan ( VÉ )

0

= # arcos (VÉ )[ exorcise

DE

Remarie 1-et To - infftso : Bt --04.Then PIT. -- o) - I

There is a séquence 0f zéros of Btlw) converging to O.

To understand the structure of the set of Zero, → Cantor set

ReflectedBMDef.tt(Bt)⇐ o be a standard BM . The stochastic

Blt ),if Bet ) ? 0process Rt = 113+1 = {

- Bet) ,if BCH ' °

is calle d reflected BM.

Think of a movement in the vicinity of a boundary .

Moments : E- ( Re) = .jo?xpqe-Etdx-- % dx =p¥

Var ( Rt) -- ELBE ) - ( Ef Bell)! t - ¥ E)tTransitiondensity-i.PL Rt EYIR. - x ) =P ( - y ± Bt e- y 1 Bo - x )

¥, e-"

ds ⇒ pays -- Été e-¥7'

)Thx ( Lévy ) Let Mtomçxebu .

Then ( Mt - Belt» is a

reflected BM .