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The Uniform Integrability of Martingales. On a Question by Alexander Cherny∗ 5 1 0 JohannesRuf† 2 DepartmentofMathematics y UniversityCollegeLondon a M May 5,2015 1 ] R Abstract P . h Let X be a progressively measurable, almost surely right-continuous stochastic process such that at Xτ ∈ L1 andE[Xτ] = E[X0] foreachfinite stoppingtimeτ. In2006,ChernyshowedthatX isthen m a uniformlyintegrablemartingaleprovidedthat X is additionallynonnegative. Chernythen posed the question whether this implication also holds even if X is not necessarily nonnegative. We provide an [ examplethatillustratesthatthisimplicationiswrong,ingeneral. If,however,anadditionalintegrability 2 assumptionis made on the limit inferiorof |X| then the implication holds. Finally, we arguethat this v integrabilityassumptionholdsifthestoppingtimesareallowedtoberandomizedinasuitablesense. 2 Keywords: Stoppingtime;Uniformintegrability 2 9 AMS2000SubjectClassifications:60G44 5 0 . 1 Introduction 1 0 5 We fix a filtered probability space (Ω,F,F,P) with expectation operator E[·], where F = (F ) and 1 t t∈[0,∞) F = F ⊂ F. Furthermore, we fix a progressively measurable, almost surely right-continuous : ∞ t∈[0,∞) t v processWX. Wewrite Z ∈ L1 if E[|Z|] < ∞ for some random variable Z. Forsome F–adapted process Y i X and some stopping time η we write Yη to denote the process Y stopped at time η; to wit, Yη = Y for t η∧t r eacht ∈ [0,∞). Allidentifications andstatementsinthefollowingareinthealmost-sure sense. a Weconsider thefollowingfivestatements: (I) X isauniformly integrable martingale. (II) X = lim X exists,X ∈L1 andE[X ]= E[X ]forallstopping timesτ. ∞ t↑∞ t τ τ 0 (III) X ∈ L1 andE[X ]= E[X ]forallfinitestopping timesτ andliminf |X |∈ L1. τ τ 0 t↑∞ t (IV) X ∈ L1 andE[X ]= E[X ]forallfinitestopping timesτ. τ τ 0 ∗I thank Zhenyu Cui for tellingme about the paper Cherny (2006) and Mikhail Urusov for sharing his very helpful insights andhiscareful readingof anearlierversionof thisnote. Iamindebted toBeatriceAcciaio, IoannisKaratzas, KostasKardaras, MartinLarsson,NicolasPerkowski,andPietroSiorpaesformanydiscussionsonthesubjectmatterofthisnote. Iamgratefulto ananonymousrefereeandanassociateeditorfortheirdetailedandhelpfulcomments. Iacknowledgegeneroussupportfromthe Oxford-ManInstituteofQuantitativeFinance,UniversityofOxford, †E-Mail:[email protected] 1 2 (V) X ∈ L1 andE[X ]= E[X ]forallt ∈ [0,∞). t t 0 Theoptionalsamplingtheoremyieldstheimplications(I)⇒(II)⇒(III)⇒(IV)⇒(V).Theimplication (II)⇒(I)followsfromthefollowingsimpleargument. Fixs,t ∈ [0,∞]withs < tandA∈ F . Letτ = s s 1 andτ2 = s1Ac+t1Adenotetwostoppingtimes,whereAc = Ω\A. Then,byassumption,E[Xτ1]= E[Xτ2], whichyieldsE[X 1 ]= E[X 1 ]. Thusweobtainthedesiredimplication(II)⇒(I).However,ifonly(III) s A t A or(IV)isassumed,thisargument onlyyieldsthemartingalepropertyofX butnotitsuniformintegrability. Cherny (2006) now asks the question whether also the implication (IV) ⇒ (I) holds. Before answering thisquestion and discussing theroleof(III),letusfirstbrieflyconsider the statement in(V).Hulley(2009) provides an example of a local martingale X, such that (V) is satisfied but X is not a martingale. Alterna- tively, if X denotes Brownian motion started in 0 then (V) holds but (IV) is not satisfied. To see this, we only need to let τ denote the first hitting time of level 1 by X. Thus, the implication (V) ⇒ (IV) does not holdingeneral. Wenowreturn todiscuss themissing implications, namely whether (III)or, moregenerally (IV),imply (I) (or equivalently, (II)). Cherny (2006) proves that these implications hold if X is nonnegative. The fol- lowingtheoremprovesthattheimplication(III)⇒(I)holdsalways,notonlyifX isnonnegative. However, theexampleofthenextsectionshowsthat(IV)doesnotnecessarily implyanyofthestatements(I)–(III)if thenonnegativity assumption onX is dropped. Yet, asproven inSection 3, ifthe filtered probability space (Ω,F,F,P) allowsforsomeadditional randomization, then these implications hold. Moreprecisely, ifthe filteredprobabilityspace(Ω,F,F,P)allowsfora(0,1)–uniformlydistributedrandomvariable,measurable with respect to F for some finite stopping time η, then the implications (IV) ⇒ (I), (IV) ⇒ (II), and (IV) η ⇒(III)hold. Theorem1. Thestatements(I),(II),and(III)areequivalent. Proof. Weonly need toshow theimplication (III)⇒(I).Westart byarguing that wemayassume, without loss of generality, that F and X(ω) are right-continuous for each ω ∈ Ω. Towards this end, we set F+ = t F and F+ = (F+) . Next, we observe that the F–martingale X is also an F+–martingale due s>t s t t∈[0,∞) TtoExercise 1.5.8 inStroockandVaradhan (2006). Now, Lemma1.1in Fo¨llmer (1972)yields the existence of a right-continuous version of X, which we call again X. Next, we fix a finite F+–stopping time σ and setσ =σ+1,whichisafiniteF–stoppingtimebyTheoremIV.57inDellacherieandMeyer(1978). Then, Xσ satisfies(II)andisthereforeauniformlyintegrableF+–martingale. Theoptionalsamplingtheorembthen also yielbds that Xσb ∈ L1 and E[Xσb] = E[X0]. Thus, (III) also holds for all finite F+–stopping times, and weshallassumefromnowon,throughout thisproof,thatFandX(ω)areright-continuous foreachω ∈ Ω. We now construct a nondecreasing sequence (Tn)n∈N of [0,∞)–valued random variables such that lim |X | = liminf |X | ∈ F . For example, we can choose T as the first time that ||X| − n↑∞ Tn t↑∞ t ∞ n liminf |X || ≤ 1/n. Then T is F –measurable, due to the right-continuity of X, for each n ∈ N t↑∞ t n ∞ (but not necessarily a stopping time). Now we set Y = liminf X ∈ F and note that |Y| = n↑∞ Tn ∞ liminf |X |,thusY ∈L1 byassumption. t↑∞ t Next, let us consider the martingale M given by M = X − E[Y|F ] for each t ∈ [0,∞), where t t t wemay use aright-continuous modification ofthe conditional expectation process thanks toF being right- continuous; see again Lemma 1.1 in Fo¨llmer (1972). Then (III) holds with X replaced by M. We note that liminf |M | = 0 since liminf M = liminf X −Y = 0, thanks to Le´vy’s martingale t↑∞ t n↑∞ Tn n↑∞ Tn convergence theorem andthefactthatY ∈ F . ∞ It is sufficient to show that the martingale M is uniformly integrable, or, equivalently that M ≡ 0. Towardsthisend, weassumethat thereexists ε ∈ (0,1) suchthatP(sup M > ε) > ε. Wethen let t∈[0,1/ε) t σ be the first time that M is greater than or equal to ε and σ the first time after time 1/ε that |M| is less 1 2 3 than orequal toε2/4. Then σ isfinitesince liminf|M | = 0and, withτ = σ ∧σ ,wemayassume that 2 t 1 2 M ∈ L1 andobtain τ 1 ε2 3ε2 ε2 E[M ]= E[M 1 ]+E[M 1 ] ≥ εP σ ≤ − ≥ > ≥ E[|M |] ≥ E[M ], τ σ1 {σ1≤σ2} σ2 {σ1>σ2} (cid:18) 1 ε(cid:19) 4 4 4 σ2 σ2 which contradicts (III) with X replaced by M. Thus M ≤ 0 and in the same manner, we can show that M ≥ 0,whichyieldsthestatement. We briefly remark that if X is nonnegative, then (IV) yields that X is a martingale, thus a nonnegative supermartingale and therefore liminf |X | ∈ L1. Theorem 1 then yields Cherny’s result, namely the t↑∞ t implication(IV)⇒(I)providedthatX isnonnegative. 2 A counterexample Wenowconstructafilteredprobabilityspace(Ω,F,F,P)witharight-continuousmartingaleXthatsatisfies (IV)andhasalimitX =lim X ,butisnotuniformly integrable. ∞ t↑∞ t Example1. WeletΩ = (N∪{∞})×{−1,1}andF thepowersetofΩ. Next,weletPbetheprobability measure on (Ω,F) such that P((n,i)) = 1/(4n2) for all n ∈ N and i ∈ {−1,1} and P((∞,i)) = (1−π2/12)/2foralli∈ {−1,1}. Since 1/(2n2) = π2/12 ∈ (0,1),thisyieldsindeedaprobability n∈N measureon(Ω,F). P Welet σ : Ω → [0,∞], (ω ,ω ) 7→ ω denote the firstcomponent and D : Ω → {−1,1}, (ω ,ω ) 7→ 1 2 1 1 2 ω the second component ofeach scenario (ω ,ω ) ∈ Ω. Then σ andD are independent andP(σ = x) = 2 1 2 1/(2x2)1x∈N forallx ∈ [0,∞],P(σ = ∞) = 1−π2/12,andP(D = −1) = 1/2 = P(D = 1). WenowsetX ≡ Dσ21 andletFdenote thenatural filtrationofX. Towit,X isamartingale that [[σ,∞[[ attimeσ jumpstoeitherσ2 or−σ2 provided thatσ isfinite. Inparticular, wehave X = limX = Dσ21 . ∞ t {σ<∞} t↑∞ Next,weobservethat 1 E[|X |] = E[σ21 ]= n2 = ∞. ∞ {σ<∞} 2n2 nX∈N Hence,X ∈/ L1,andX isnotauniformlyintegrable martingale. ∞ Now,weletτ beanarbitraryfinitestoppingtimeandsetu= τ((∞,−1))∨τ((∞,1)) ∈ [0,∞). Thus, {σ = ∞} ⊂ {τ ≤ u}∈ F ,whichagainyields{σ > u}⊂ {τ ≤ u}since{σ > u}isthesmallesteventin u F thatcontains{σ = ∞}. Thus,since u {τ ∧σ ≤ u} = {σ ≤ u}∪{τ ≤ u}⊃ {σ ≤ u}∪{σ > u}= Ω, thestopping timeτ ∧σ isuniformlybounded byuandweobtainthatX = Xσ = X ∈ L1 and, bythe τ τ τ∧σ optional sampling theorem again, thatE[X ] = E[X ] = 0 = E[X ]. Hence, X satisfies(IV)butnot(I), τ τ∧σ 0 andthereforeneither (II)nor(III). WeremarkthatDellacherie(1970)discusses thefiltrationofasimilarexample. Inordertomotivatethearguments inthenextsection, wenowslightly modifyExample1byextending theunderlying filtration. 4 Example2. Welet(Ω,F,F,P)beanarbitraryprobabilityspacethatsupportsaright-continuous martingale X = Dσ21 with the same distribution as in Example 1 and an independent F –measurable (0,1)– [[σ,∞[[ 0 uniformlydistributed randomvariableU. Ourgoalistoconstruct afinitestopping timeτ suchthatX ∈/ L1. Thus,under thisenlarged filtration, τ thepreviousexampleisnotacounterexample fortheimplication (IV)⇒(I).Indeed, wenotethatτ = 1/U isafinitestopping timeand|X |= σ21 . Therefore, τ {σ≤1/U} ⌊1/U⌋ 1 1 1 1 1 1 1 1 E[|Xτ|] = E σ21{σ≤1/U} = E n22n2 = 2E(cid:20)(cid:22)U(cid:23)(cid:21) ≥ E(cid:20)2U(cid:21)− 2 = Z 2ydy− 2 = ∞, (cid:2) (cid:3) nX=1 0   where⌊·⌋denotes theGaussbrackets, byindependence ofU andX. Example2indicatesthatif“randomizedstopping”ispossible,anon-uniformly integrablemartingaleX willnotsatisfy(IV).Inthenextsection, wewillprovethisassertion. 3 An additonal randomization In this section, we show that the implication (IV) ⇒ (I) holds if we may randomize stopping times. More precisely, weshallmakethefollowingassumption: Thereexistsa(0,1)–uniformly distributed random variableU (R) andafinitestoppingtimeη,suchthatU isF –measurable. η We emphasize that (R) is an assumption on the underlying filtered probability space (Ω,F,F,P), and notonthestochastic process X. Werecall that wealready argued thatX isamartingale if(IV)holds. The conclusion thatX isalsoauniformlyintegrable martingale,ifadditionally (R)holds,followsthenfromthe followingtheorem. Theorem2. If(IV)and (R)holdthensodoes(I);towit,X isthenauniformlyintegrable martingale. Proof. Exactly as in the proof of Theorem 1, we may assume, without loss of generality, that F and X(ω) areright-continuous for each ω ∈ Ω. Lemmata1and 2below then yield that liminf |X | ∈ L1 and the t↑∞ t implication(III)⇒(I),proveninThereom1,yieldstheassertion. Lemma1. AssumethatFandX(ω)areright-continuous foreachω ∈ Ω. IfX ∈ L1forallfinitestopping τ timesτ and (R)holdsthenE[liminf |X ||F ]< ∞. t↑∞ t η Proof. Wedefinetheevent A = E liminf|X | F = ∞ ∈ F . t η η (cid:26) (cid:20) t↑∞ (cid:12) (cid:21) (cid:27) (cid:12) (cid:12) We need to argue that P(A) = 0. Towards this end,(cid:12)we assume that P(A) > 0 and define the function g : [0,1] → [0,∞] by t 7→ 1/P(A∩{U ≤ t}), where U is the uniformly distributed random variable of (R). Wenotethatthefunction1/giscontinuousandnondecreasingandsett = sup{t ∈ [0,1]|g(t) = ∞}. ∞ ThenwehaveP(A∩{U ≤ t }) = 0andP(A∩{U ≤ t}) > 0forallt > t ,whichyields that1 g(U) ∞ ∞ A (with0×∞ = 0)isfinite(almostsurely). 5 WenowletσdenotethefirsttimetafterηsuchthatE[|X ||F ]isgreaterthanorequaltog(U)andnote t η thatσ isastopping time. Then,Fatou’s lemmayields thatσ isfiniteonA. Wenowsetτ = η1 +σ1 , Ω\A A whichisagainafinitestopping time,andobserve 1 E[|X |] ≥ E[1 |X |]= E[1 E[|X ||F ]]≥ E[1 g(U)] ≥ P A∩ P(A∩{U ≤ t}) ≤ = ∞. τ A σ A σ η A t=U (cid:18) (cid:26) n(cid:27)(cid:19) nX∈N Here the last inequality follows from Tonelli’s theorem and the last equality follows from the fact that for each n ≥ 1/P(A) the corresponding term in the sum equals 1/n. To see this, fix n ≥ 1/P(A), let t = sup{t ∈ [0,1]|g(t )≥ n},andusethefactthatP(A∩{U ≤ t }) = 1/n. Thelastdisplaycontradicts n n n theassumption andthusyieldsP(A) = 0. Lemma2. AssumethatFandX(ω)areright-continuous foreachω ∈ Ω. IfX ∈ L1forallfinitestopping τ timesτ andE[liminf |X ||F ]< ∞holdsforsomefinitestopping timeη thenliminf |X |∈ L1. t↑∞ t η t↑∞ t Proof. We let Y = (Y ) denote the right-continuous modification of the finite-valued conditional t t∈[0,∞) expectation process E liminf|X | F . s η∨t (cid:18) (cid:20) s↑∞ (cid:12) (cid:21)(cid:19) (cid:12) t∈[0,∞) (cid:12) Foreachκ> 0theprocess(Y 1 ) isaunif(cid:12)ormlyintegrablemartingaleunderitsnaturalfiltration t {Y0≤κ} t∈[0,∞) and Le´vy’s martingale convergence theorem yields that 1 lim Y = 1 liminf |X |, and {Y0≤κ} t↑∞ t {Y0≤κ} s↑∞ s thuslim Y = liminf |X |. t↑∞ t s↑∞ s Wenowletτ denotethefirsttimeaftertimeηsuchthat|X|isgreaterthanorequaltoY −1,whichisa stoppingtime. Moreover, τ isfinitesinceliminf |X |> lim Y −1.Wethenobtain t↑∞ t t↑∞ t E liminf|X | = E[Y ]≤ 1+E[|X |]< ∞, t τ τ (cid:20) t↑∞ (cid:21) whichyieldsthestatement. References Cherny, A. S. (2006). Some particular problems of martingale theory. In Kabanov, Y., Liptser, R., and Stoyanov,J.,editors,FromStochasticCalculustoMathematicalFinance: TheShiryaevFestschrift,pages 109–124. Springer. Dellacherie, C.(1970). Unexempledelatheóriegeńe´rale desprocessus. InMeyer, P.,editor, Se´minaire de Probabilite´s, IV(LectureNotesinMathematics, Volume124),pages60–70.Springer, Berlin. Dellacherie, C.andMeyer,P.A.(1978). Probabilities andPotential. North-Holland, Amsterdam. Fo¨llmer, H. (1972). The exit measure of a supermartingale. Zeitschrift fu¨r Wahrscheinlichkeitstheorie und VerwandteGebiete,21:154–166. Hulley,H.(2009). StrictLocalMartingalesinContinuousFinancialMarketModels. PhDthesis,University ofSydney,Sydney,Australia. Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprintofthe1997edition.