DILATIONS AND MARTINGALES

Internat. J. Math. & Math. Sci.

VOL. 16 NO. 4 (1993) 825-827

A

NOTE ON

DILATIONS AND MARTINGALES

825

MARTINL. JONES Mathematics

Department

University of Charleston

66George Street Charleston 29424

South Carolina (Received August 20, 1992)

ABSTRACT. The purpose ofthisnoteistoinvestigate the effect ofdilations onmartingalesandtogive conditions under which a dilatedmartingalewill retainthemartingaleproperty. Thetechniqueofdilating thejointdistributionofasequenceof randomvariableshas applicationsinoptimal stopping theoryand the study of"prophet inequalities"where martingalesplayasignificant^role.

KEY WORDS AND PHRASES. Martingales, dilations, optimal stopping,prophetinequalities.

1991AMS SUBJECTCLASSIFICATION CODE. 60G40.

1. INTRODUCTION.

Martingalesplayasignificant roleinmanyareas ofprobabilityand one such area is in the establishment of"prophet inequalities"inoptimal stopping theory. Prophet inequalitiesarecomparisons between theexpectedmaximumof arewardsequenceand the maximumexpectedrewardof thesame sequencewhen stopped by non-anticipating stoppingtimes. Thename"prophetinequalities"stemsfrom theinterpretation of theexpected maximum of thesequenceastheexpected reward ofaprophet,or observer withcomplete foresight,whiletheordinaryobservermustusenon-anticipating stoppingtimesto decidewhentostop observing thesequence. Inestablishing these inequalitiesit isoftenusefultoascertain the"extremaldistribution" whichattainsornearlyattains theinequality. Oftenthisdistribution is thatofa martingale,buttodiscover this,givendistributionsmustbemanipulatedinsuch awaythat thegainofthe prophetis increased whilethat ofthe ordinary observerisheldconstantordecreased. Thedilationofa jointdistributionofarandomvectorisatechniquewhich isusedto createa new distributionforwhichthe resulting randomvariableshavethesameexpectation, but alargervariancethan the original ones. The purpose ofthisshortnoteistoconsiderthe effect ofdilations onmartingales andtodecide,under what conditionsonthe original martingale,^willthe resulting"dilated" distribution remainthat ofamartingale.

2. MAIN RESULTS.

Forease of reference the definitions of thetwomainobjectsof interest,martingalesand dilations, willbegivenfirst.

DEFINITION1. Let X1,

X2

be integrable randomvariables defined on aprobability space (fL F, P). Foreachk 1,2 let

Fk

^F(X1,

X2

Xk), the sub-borelfieldofF generatedbythe firstkrandomvariables. Thesequence

XI, X2

^isa martingale provided for eachn 2, 3 that E(Xn Fn-l)

Xn-1

almost surely.

Inthis article attention willbe focused initiallyontwo-termmartingalessothe definition of the dilationofajoint distributionwillbe given forapair ofrandom variablesXandY. Thedefinition iseasily extendedtoinclude the jointdistributionof randomvectorsof lengthn> 2.

DEFINITION2. Let XandYbeintegrablerandom variables defined on aprobabilityspace (fl,F,P)andlet a,b,c, and d be realnumberssuch that <a<b<^+,,,, and <c< d <^+o,,.

826 M.L. JONES

b d

DefineX_aandY_casfollows"

(X

Yc

^{(X,Y)ontheset}

^A

^{c [(X,Y) [a,b]x[c,d]}}

(a,c)withprobability

I

^{b-b-aX d-Yd-c dP}

(a,d)withprobability

Ta

Y-c

d--

^dP

(b,c)withprobability

Xb_a-

^a

dd_c-

^{Y dP}

(b,d)withprobability

:

Y-c

d--

^dP.

b d

The distribution of(X_a Y_c willbereferredtoasathejointdilationofthe distributionof (X,Y)overthe

square

[a,b]x[c, d].

Noticethatthe jointdilation ofXandYover [a,b]x[c,d]isthe jointdistribution withthelargest variancethatagreeswiththejointdistributionof

X

andYonthecomplement of thesquare[a,b]x[c, d].

Aproperty ofdilations whichmake themuseful for establishingcertaininequalitiesisthatthe expectedmaximumofthe dilationofadistributionisatleast aslargeasthe expectedmaximum ofthe originaldistribution as isshowninthefollowingproposition.

PROPOSITION1. E[max{X,Y}]^_<

E[max{Xa

^b

Yc

d ^}]"

PROOF.

Let

’(x,y) max{x,y andnotethat

V

^{isconvexinthe}variables x andy. Since

(X, Y)

^{(X,Y) onAc}it will sufficetoshow that the inequality holds when the integration takes place

overthesetAasfollows:

E[max{X,Y}-AI (x,y)

I.t2(dx,dy)

A

x-a+

^b-x

I I _w(b- ab--a,y)12(dx,dy)

A

x-a "b ^b-x

<

I

_A

---a

^,y)+

^{--(a,y) 2(dx,dy)}

x-a ...y-c

+

dc__)

₊^b-x

_dY-C

---a t’’ad-c

^a-c"

---a

^{l’a’ d-c+}

c) I.t2(dx,dy)

A

< x-ay-c

x-a__

c

b-xy-c b-xd-y

b----

d-c

(b,d)

+

(b,c)

⁺

b-Sff

d-cg(a,d) +

b-sff

d-c

(a,c) I.t2(dx,dy)

A

b d

E[max{Xa’ Yc

^}"A],

thusconcluding the proof.

b d

IfX, Yformsatwo-termmartingaleit isnotalwaystruethat

X

_a

Yc

^{willalso beamartingaleas}

the followingexampleindicates.

DILATIONS AND MARTINGALES 827

EXAMPLE1. Let X 1/2, andYbe Bernoulli withp 1/2. EasilyX, Yisseentobemartingale.

The distribution of

X, Y

^{isgivenasfollows:}

(X, Y)

^(1,1),(1,0), (0,1),and(0,0) eachwithprobability 1/4.

Notethat

E(YAI X=

⁰⁾⁼

^P(Y =11X=

⁰⁾⁼1/2 which shows that

X, Y

^isnotamartingale.

Anatural questionisthefollowing: under what restrictionswillthe dilationofatwo-termmartingale X, Yremain amartingale? Apartialanswer

m

thisquestionisafforded bythefollowing.

THEOREM1. LetX,Ybe atwo-termmartingale taking valuesin[a, b]x[c, d] where

b d

<c _< a<b _< d<^+0,,. ThenX_a

Yc

^{ismartingaleifandonlyifX onlytakesvaluesin a,b with}

positive probability.

b d

PROOF. NotethatX_a

Yc

^{(a,c),(a,d), (b,c),(b,d)}almost surely.

Easy

calculations show

b d

thatX_a,

Y

_{c is}martingaleifandonlyif c

d

c

d

b=a)

^a-c

b d-a

P(Y d X_{a d-c}

P(Y =cIX

a=a)-d_c,

cd

^{b b)-d-b P(Ycd=dlXba b) b-c}

P(Y =clX

a=

d-c d-c"

b d

andconditionalprobabilityit Sincef {(X,Y) [a,b]^x[c,d]} thenfrom the definitionofX_a,Y_c

(2.1)

followsthat(2.1)holds ifandonlyifE((X a)(X b)) 0, whereE(XY) E[E(XY X)]

E[X E(Y X)] E(X

2)

isusedtoobtainthe latterexpression. SinceXtakes values in[a, b]itfollowsthat theintegrand,(X a)(X b),mustbezeroalmost surely, thereforeX {a, b}almostsurely.

Theorem shows that for atwo-termmartingale takingvalues in thesquare,the first random variablemustbealready"dilated". The theorem extends easilytothe moregeneralsituationwhere(X, Y) has arangeextending beyond thesquare[a, b]x[c,d]and isrecordedasCorollary below.

COROLLARY 1. LetX, Ybe atwo-termmartingaleand let <c<a<b<d<+*,,,be real

b d

numbers. ThenX_a

Yc

^{ismartingaleifandonlyifon theset (X, Y) [a, b]x[c,d]}, Xonlytakes}

valuesin a,b} withpositive probability.

3. REMARKS. Thereisanaturalextensionof the def’mifion ofa dilationtof’mitesequencesof integrable randomvariables

XI, X2 Xn

^wheren>2. Similarly, thereare extensionsof Proposition and Theorem insuch cases. Inthissetting,ifthe martingale X1,

X2 Xn

^istobe dilated overthesetA

[al,bl]x[a2, b2]x...x[an,bn], thenon

A, Xi

{ai, bi} almostsurely for^all 1, 2 n-1.

REFIRENCES

1. CHOWY., ROBBINS H.,andSIEGMUND D., TheTheory_of Optimal Stopping,Dover Publications,Inc., NewYork, 1991.

2. HILL T.and

KERTZ

R., StopRule Inequalities for Uniformly Bounded

Sequences

ofRandom Variables,Trans. Amer.Math.Sot.,278 (1983) 197-207.

3. JONES M.,Prophet Inequalities forCostof ObservationStopping Problems,J.

Multiva.riat

^Analysis, 34(1990),238-253.

4. KARLIN S.andTAYLOR H.,

A

First

Course

inStochastic

Processes.

Second Ed.,AcademicPress, Inc.,NewYork,1975.