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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 .