Quasi-sure Stochastic Analysis through Aggregation

Transcription

1 E l e c r o n i c J o u r n a l o f P r o b a b i l i y Vol. 16 (211), Paper no. 67, pages Journal URL hp:// Quasi-sure Sochasic Analysis hrough Aggregaion H. Mee SONER Nizar TOUZI Jianfeng ZHANG Absrac This paper is on developing sochasic analysis simulaneously under a general family of probabiliy measures ha are no dominaed by a single probabiliy measure. The ineres in his quesion originaes from he probabilisic represenaions of fully nonlinear parial differenial equaions and applicaions o mahemaical finance. The exising lieraure relies eiher on he capaciy heory (Denis and Marini [5]), or on he underlying nonlinear parial differenial equaion (Peng [13]). In boh approaches, he resuling heory requires cerain smoohness, he so called quasi-sure coninuiy, of he corresponding processes and random variables in erms of he underlying canonical process. In his paper, we invesigae his quesion for a larger class of non-smooh" processes, bu wih a resriced family of non-dominaed probabiliy measures. For smooh processes, our approach leads o similar resuls as in previous lieraure, provided he resriced family saisfies an addiional densiy propery.. Key words: non-dominaed probabiliy measures, weak soluions of SDEs, uncerain volailiy model, quasi-sure sochasic analysis. AMS 21 Subjec Classificaion: Primary 6H1, 6H3. Submied o EJP on March 24, 21, final version acceped Augus 19, 211. ETH (Swiss Federal Insiue of Technology), Zürich and Swiss Finance Insiue, hmsoner@ehz.ch. Research parly suppored by he European Research Council under he gran FiRM. Financial suppor from he Swiss Finance Insiue and he ETH Foundaion are also graefully acknowledged. CMAP, Ecole Polyechnique Paris, nizar.ouzi@polyechnique.edu. Research suppored by he Chair Financial Risks of he Risk Foundaion sponsored by Sociéé Générale, he Chair Derivaives of he Fuure sponsored by he Fédéraion Bancaire Française, and he Chair Finance and Susainable Developmen sponsored by EDF and Calyon. Universiy of Souhern California, Deparmen of Mahemaics, jianfenz@usc.edu. Research suppored in par by NSF gran DMS and DMS

2 1 Inroducion I is well known ha all probabilisic consrucions crucially depend on he underlying probabiliy measure. In paricular, all random variables and sochasic processes are defined up o null ses of his measure. If, however, one needs o develop sochasic analysis simulaneously under a family of probabiliy measures, hen careful consrucions are needed as he null ses of differen measures do no necessarily coincide. Of course, when his family of measures is dominaed by a single measure his quesion rivializes as we can simply work wih he null ses of he dominaing measure. However, we are ineresed exacly in he cases where here is no such dominaing measure. An ineresing example of his siuaion is provided in he sudy of financial markes wih uncerain volailiy. Then, essenially all measures are orhogonal o each oher. Since for each probabiliy measure we have a well developed heory, for simulaneous sochasic analysis, we are naurally led o he following problem of aggregaion. Given a family of random variables or sochasic processes, X P, indexed by probabiliy measures P, can one find an aggregaor X ha saisfies X = X P, P almos surely for every probabiliy measure P? This paper sudies exacly his absrac problem. Once aggregaion is achieved, hen essenially all classical resuls of sochasic analysis generalize as shown in Secion 6 below. This probabilisic quesion is also closely relaed o he heory of second order backward sochasic differenial equaions (2BSDE) inroduced in [3]. These ype of sochasic equaions have several applicaions in sochasic opimal conrol, risk measures and in he Markovian case, hey provide probabilisic represenaions for fully nonlinear parial differenial equaions. A uniqueness resul is also available in he Markovian conex as proved in [3] using he heory of viscosiy soluions. Alhough he definiion given in [3] does no require a special srucure, he non-markovian case, however, is beer undersood only recenly. Indeed, [17] furher develops he heory and proves a general exisence and uniqueness resul by probabilisic echniques. The aggregaion resul is a cenral ool for his resul and in our accompanying papers [15, 16, 17]. Our new approach o 2BSDE is relaed o he quasi sure analysis inroduced by Denis and Marini [5] and he G-sochasic analysis of Peng [13]. These papers are moivaed by he volailiy uncerainy in mahemaical finance. In such financial models he volailiy of he underlying sock process is only known o say beween wo given bounds a < a. Hence, in his conex one needs o define probabilisic objecs simulaneously for all probabiliy measures under which he canonical process B is a square inegrable maringale wih absoluely coninuous quadraic variaion process saisfying ad d B ad. Here d B is he quadraic variaion process of he canonical map B. We denoe he se of all such measures by W, bu wihou requiring he bounds a and a, see subsecion 2.1. As argued above, sochasic analysis under a family of measures naurally leads us o he problem of aggregaion. This quesion, which is also oulined above, is saed precisely in Secion 3, Definiion 3.1. The main difficuly in aggregaion originaes from he fac ha he above family of probabiliy measures are no dominaed by one single probabiliy measure. Hence he classical sochasic analysis ools can no be applied simulaneously under all probabiliy measures in his family. As a specific example, le us consider he case of he sochasic inegrals. Given an appropriae inegrand H, he sochasic inegral I P = H sdb s can be defined classically under each probabiliy measure P. However, hese processes may depend on he underlying probabiliy measure. On he oher hand 1845

3 we are free o redefine his inegral ouside he suppor of P. So, if for example, we have wo probabiliy measures P 1, P 2 ha are orhogonal o each oher, see e.g. Example 2.1, hen he inegrals are immediaely aggregaed since he suppors are disjoin. However, for uncounably many probabiliy measures, condiions on H or probabiliy measures are needed. Indeed, in order o aggregae hese inegrals, we need o consruc a sochasic process I defined on all of he probabiliy space so ha I = I P for all, P almos surely. Under smoohness assumpions on he inegrand H his aggregaion is possible and a poinwise definiion is provided by Karandikar [1] for càdlàg inegrands H. Denis and Marini [5] uses he heory of capaciies and consruc he inegral for quasi-coninuous inegrands, as defined in ha paper. A differen approach based on he underlying parial differenial equaion was inroduced by Peng [13] yielding essenially he same resuls as in [5]. In Secion 6 below, we also provide a consrucion wihou any resricions on H bu in a slighly smaller class han W. For general sochasic processes or random variables, an obvious consisency condiion (see Definiion 3.2, below) is clearly needed for aggregaion. Bu Example 3.3 also shows ha his condiion is in general no sufficien. So o obain aggregaion under his minimal condiion, we have wo alernaives. Firs is o resric he family of processes by requiring smoohness. Indeed he previous resuls of Karandikar [1], Denis-Marini [5], and Peng [13] all belong o his case. A precise saemen is given in Secion 3 below. The second approach is o slighly resric he class of nondominaed measures. The main goal of his paper is o specify hese resricions on he probabiliy measures ha allows us o prove aggregaion under only he consisency condiion (3.4). Our main resul, Theorem 5.1, is proved in Secion 5. For his main aggregaion resul, we assume ha he class of probabiliy measures are consruced from a separable class of diffusion processes as defined in subsecion 4.4, Definiion 4.8. This class of diffusion processes is somehow naural and he condiions are moivaed from sochasic opimal conrol. Several simple examples of such ses are also provided. Indeed, he processes obained by a sraighforward concaenaion of deerminisic piece-wise consan processes forms a separable class. For mos applicaions, his se would be sufficien. However, we believe ha working wih general separable class helps our undersanding of quasi-sure sochasic analysis. The consrucion of a probabiliy measure corresponding o a given diffusion process, however, conains ineresing echnical deails. Indeed, given an F-progressively measurable process α, we would like o consruc a unique measure P α. For such a consrucion, we sar wih he Wiener measure P and assume ha α akes values in S > (symmeric, posiive definie marices) and also d saisfy α s ds < for all, P -almos surely. We hen consider he P sochasic inegral X α := α 1/2 s db s. (1.1) Classically, he quadraic variaion densiy of X α under P is equal o α. We hen se P α S := P (X α ) 1 (here he subscrip S is for he srong formulaion). I is clear ha B under P α S has he same disribuion as X α under P. One can show ha he quadraic variaion densiy of B under P α S is equal o a saisfying a(x α (ω)) = α(ω) (see Lemma 8.1 below for he exisence of such a). Hence, P α S W. Le S W be he collecion of all such local maringale measures P α S. Barlow [1] has observed ha his inclusion is sric. Moreover, his procedure changes he densiy of he quadraic variaion process o he above defined process a. Therefore o be able o specify he quadraic variaion a priori, in subsecion 4.2, we consider he weak soluions of a sochasic differenial 1846

4 equaion ((4.4) below) which is closely relaed o (1.1). This class of measures obained as weak soluions almos provides he necessary srucure for aggregaion. The only addiional srucure we need is he uniqueness of he map from he diffusion process o he corresponding probabiliy measure. Clearly, in general, here is no uniqueness. So we furher resric ourselves ino he class wih uniqueness which we denoe by W. This se and he probabiliy measures generaed by hem, W, are defined in subsecion 4.2. The implicaions of our aggregaion resul for quasi-sure sochasic analysis are given in Secion 6. In paricular, for a separable class of probabiliy measures, we firs consruc a quasi sure sochasic inegral and hen prove all classical resuls such as Kolmogrov coninuiy crierion, maringale represenaion, Io s formula, Doob-Meyer decomposiion and he Girsanov heorem. All of hem are proved as a sraighforward applicaion of our main aggregaion resul. If in addiion he family of probabiliy measures is dense in an appropriae sense, hen our aggregaion approach provides he same resul as he quasi-sure analysis. These ype of resuls, of course, require coninuiy of all he maps in an appropriae sense. The deails of his approach are invesigaed in our paper [16], see also Remark 7.5 in he conex of he applicaion o he hedging problem under uncerain volailiy. Noice ha, in conras wih [5], our approach provides exisence of an opimal hedging sraegy, bu a he price of slighly resricing he family of probabiliy measures. The paper is organized as follows. The local maringale measures W and a universal filraion are sudied in Secion 2. The quesion of aggregaion is defined in Secion 3. In he nex secion, we define W, W and hen he separable class of diffusion processes. The main aggregaion resul, Theorem 5.1, is proved in Secion 5. The nex secion generalizes several classical resuls of sochasic analysis o he quasi-sure seing. Secion 7 sudies he applicaion o he hedging problem under uncerain volailiy. In Secion 8 we invesigae he class S of muually singular measures induced from srong formulaion. Finally, several examples concerning weak soluions and he proofs of several echnical resuls are provided in he Appendix. Noaions. We close his inroducion wih a lis of noaions inroduced in he paper. Ω := {ω C(R +, R d ) : ω() = }, B is he canonical process, P is he Wiener measure on Ω. For a given sochasic process X, F X is he filraion generaed by X. F := F B = { } is he filraion generaed by B. F + := { +, }, where + := + := s> s, P := + P ( + ) and P := + P ( ), where P ( ) := E Ω : here exiss Ẽ such ha E Ẽ and P[Ẽ] =. is he class of polar ses defined in Definiion 2.2. ˆ := P P is he universal filraion defined in (2.3). is he se of all F sopping imes τ aking values in R + { }. ˆ is se of all ˆF sopping imes. 1847

5 B is he universally defined quadraic variaion of B, defined in subsecion 2.1. â is he densiy of he quadraic variaion B, also defined in subsecion 2.1. S d is he se of d d symmeric marices. S > d is he se of posiive definie symmeric marices. W is he se of measures defined in subsecion 2.1. S W is defined in he Inroducion, see also Lemma 8.1. MRP W are he measures wih he maringale represenaion propery, see (2.2). Ses W, S, MRP are defined in subsecion 4.2 and secion 8, as he subses of W, S, MRP wih he addiional requiremen of weak uniqueness. is he se of inegrable, progressively measurable processes wih values in S > d. W := P W W (P) and W (P) is he se of diffusion marices saisfying (4.1). W, S, MRP are defined as above using W, S, MRP, see secion 8. Ses Ωaˆτ, Ωa,b ˆτ and he sopping ime θ ab are defined in subsecion 4.3. Funcion spaces L, L p (P), ˆL p, and he inegrand spaces H, H p (P a ), H 2 loc (Pa ), Ĥ p, Ĥ 2 loc are defined in Secion 6. 2 Non-dominaed muually singular probabiliy measures Le Ω := C(R +, R d ) be as above and F = F B be he filraion generaed by he canonical process B. Then i is well known ha his naural filraion F is lef-coninuous, bu is no righ-coninuous. This paper makes use of he righ-limiing filraion F +, he P compleed filraion F P := { P, }, and he P augmened filraion F P := { P, }, which are all righ coninuous. 2.1 Local maringale measures We say a probabiliy measure P is a local maringale measure if he canonical process B is a local maringale under P. I follows from Karandikar [1] ha here exiss an F progressively measurable process, denoed as B sdb s, which coincides wih he Iô s inegral, P almos surely for all local maringale measure P. In paricular, his provides a pahwise definiion of B := B B T 2 1 B s db s and â := lim ɛ ɛ [ B B ɛ ]. Clearly, B coincides wih he P quadraic variaion of B, P almos surely for all local maringale measure P. 1848

6 Le W denoe he se of all local maringale measures P such ha P-almos surely, B is absoluely coninuous in and â akes values in S > d, (2.1) where S > denoes he space of all d d real valued posiive definie marices. We noe ha, for d differen P 1, P 2 W, in general P 1 and P 2 are muually singular, as we see in he nex simple example. Moreover, here is no dominaing measure for W. Example 2.1. Le d = 1, P 1 := P ( 2B) 1, and Ω i := { B = (1 + i), }, i =, 1. Then, P, P 1 W, P (Ω ) = P 1 (Ω 1 ) = 1, P (Ω 1 ) = P 1 (Ω ) =, and Ω and Ω 1 are disjoin. Tha is, P and P 1 are muually singular. In many applicaions, i is imporan ha P W has maringale represenaion propery (MRP, for shor), i.e. for any (F P, P)-local maringale M, here exiss a unique (P-almos surely) F P - progressively measurable R d valued process H such ha We hus define â 1/2 s H s 2 ds < and M = M + H s db s,, P-almos surely. MRP := P W : B has MRP under P. (2.2) The inclusion MRP W is sric as shown in Example 9.3 below. Anoher ineresing subclass is he se S defined in he Inroducion. Since in his paper i is no direcly used, we pospone is discussion o Secion A universal filraion We now fix an arbirary subse W. By a sligh abuse of erminology, we define he following noions inroduced by Denis and Marini [5]. Definiion 2.2. (i) We say ha a propery holds -quasi-surely, abbreviaed as -q.s., if i holds P-almos surely for all P. (ii) Denoe := P P ( ) and we call -polar ses he elemens of. (iii) A probabiliy measure P is called absoluely coninuous wih respec o if P(E) = for all E. In he sochasic analysis heory, i is usually assumed ha he filered probabiliy space saisfies he usual hypoheses. However, he key issue in he presen paper is o develop sochasic analysis ools simulaneously for non-dominaed muually singular measures. In his case, we do no have a good filraion saisfying he usual hypoheses under all he measures. In his paper, we shall use he following universal filraion ˆF for he muually singular probabiliy measures {P, P }: Moreover, we denoe by (resp. ˆτ) aking values in R + { }. ˆF := { ˆ } where ˆ := P P for. (2.3) ˆ ) he se of all F-sopping imes τ (resp., ˆF -sopping imes 1849

7 Remark 2.3. Noice ha F + F P F P. The reason for he choice of his compleed filraion F P is as follows. If we use he small filraion F +, hen he crucial aggregaion resul of Theorem 5.1 below will no hold rue. On he oher hand, if we use he augmened filraions F P, hen Lemma 5.2 below does no hold. Consequenly, in applicaions one will no be able o check he consisency condiion (5.2) in Theorem 5.1, and hus will no be able o apply he aggregaion resul. See also Remarks 5.3 and 5.6 below. However, his choice of he compleed filraion does no cause any problems in he applicaions. We noe ha ˆF is righ coninuous and all -polar ses are conained in ˆ. Bu ˆF is no complee under each P. However, hanks o he Lemma 2.4 below, all he properies we need sill hold under his filraion. For any sub-σ algebra of and any probabiliy measure P, i is well-known ha an P - measurable random variable X is [ P ( )] measurable if and only if here exiss a - measurable random variable X such ha X = X, P-almos surely. The following resul exends his propery o processes and saes ha one can always consider any process in is F + - progressively measurable version. Since F + ˆF, he F + -progressively measurable version is also ˆF -progressively measurable. This imporan resul will be used hroughou our analysis so as o consider any process in is ˆF -progressively measurable version. However, we emphasize ha he ˆF -progressively measurable version depends on he underlying probabiliy measure P. Lemma 2.4. Le P be an arbirary probabiliy measure on he canonical space (Ω, ), and le X be an F P -progressively measurable process. Then, here exiss a unique (P-almos surely) F + -progressively measurable process X such ha X = X, P almos surely. If, in addiion, X is càdlàg P-almos surely, hen we can choose X o be càdlàg P-almos surely. The proof is raher sandard bu i is provided in Appendix for compleeness. We noe ha, he ideniy X = X, P-almos surely, is equivalen o ha hey are equal d dp-almos surely. However, if boh of hem are càdlàg, hen clearly X = X, 1, P-almos surely. 3 Aggregaion We are now in a posiion o define he problem. Definiion 3.1. Le W, and le {X P, P } be a family of ˆF -progressively measurable processes. An ˆF -progressively measurable process X is called a -aggregaor of he family {X P, P } if X = X P, P-almos surely for every P. Clearly, for any family {X P, P } which can be aggregaed, he following consisency condiion mus hold. Definiion 3.2. We say ha a family {X P, P } saisfies he consisency condiion if, for any P 1, P 2, and ˆτ ˆ saisfying P 1 = P 2 on ˆ we have ˆτ X P 1 = X P 2 on [, ˆτ], P 1 almos surely. (3.4) Example 3.3 below shows ha he above condiion is in general no sufficien. Therefore, we are lef wih following wo alernaives. 185

8 Resric he range of aggregaing processes by requiring ha here exiss a sequence of ˆF - progressively measurable processes {X n } n 1 such ha X n X P, P-almos surely as n for all P. In his case, he -aggregaor is X := lim n X n. Moreover, he class can be aken o be he larges possible class W. We observe ha he aggregaion resuls of Karandikar [1], Denis-Marini [5], and Peng [13] all belong o his case. Under some regulariy on he processes, his condiion holds. Resric he class of muually singular measures so ha he consisency condiion (3.4) is sufficien for he larges possible family of processes {X P, P }. This is he main goal of he presen paper. We close his secion by consrucing an example in which he consisency condiion is no sufficien for aggregaion. Example 3.3. Le d = 2. Firs, for each x, y [1, 2], le P x,y := P ( xb 1, yb 2 ) 1 and Ω x,y := { B 1 = x, B 2 = y, }. Cleary for each (x, y), P x, y W and P x,y [Ω x,y ] = 1. Nex, for each a [1, 2], we define P a [E] := (P a,z [E] + P z,a [E])dz for all E. We claim ha P a W. Indeed, for any 1 < 2 and any bounded 1 -measurable random variable η, we have 2E P a [(B 2 B 1 )η] = 2 1 {E Pa,z [(B 2 B 1 )η] + E Pz,a [(B 2 B 1 )η]}dz =. Hence P a is a maringale measure. Similarly, one can easily show ha I 2 d d B 2I 2 d, P a -almos surely, where I 2 is he 2 2 ideniy marix. For a [1, 2] se Ω a := { B 1 = a, } { B 2 = a, } z [1,2] Ωa,z Ω z,a so ha P a [Ω a ] = 1. Also for a b, we have Ω a Ω b = Ω a,b Ω b,a and hus P a [Ω a Ω b ] = P b [Ω a Ω b ] =. Now le := {P a, a [1, 2]} and se X a (ω) = a for all, ω. Noice ha, for a b, P a and P b disagree on + ˆ. Then he consisency condiion (3.4) holds rivially. However, we claim ha here is no -aggregaor X of he family {X a, a [1, 2]}. Indeed, if here is X such ha X = X a, P a -almos surely for all a [1, 2], hen for any a [1, 2], 1 = P a [X ạ = a] = P a [X. = a] = P a,z [X. = a] + P z,a [X. = a] dz. Le λ n he Lebesgue measure on [1, 2] n for ineger n 1. Then, we have λ 1 {z : P a,z [X. = a] = 1} = λ 1 {z : P z,a [X. = a] = 1} = 1, for all a [1, 2]. Se A 1 := {(a, z) : P a,z [X. = a] = 1}, A 2 := {(z, a) : P z,a [X. = a] = 1} so ha λ 2 (A 1 ) = λ 2 (A 2 ) = 1. Moreover, A 1 A 2 {(a, a) : a (, 1]} and λ 2 (A 1 A 2 ) =. Now we direcly calculae ha 1 λ 2 (A 1 A 2 ) = λ 2 (A 1 ) + λ 2 (A 2 ) λ 2 (A 1 A 2 ) = 2. This conradicion implies ha here is no aggregaor

9 4 Separable classes of muually singular measures The main goal of his secion is o idenify a condiion on he probabiliy measures ha yields aggregaion as defined in he previous secion. I is more convenien o specify his resricion hrough he diffusion processes. However, as we discussed in he Inroducion here are echnical difficulies in he connecion beween he diffusion processes and he probabiliy measures. Therefore, in he firs wo subsecions we will discuss he issue of uniqueness of he mapping from he diffusion process o a maringale measure. The separable class of muually singular measures are defined in subsecion 4.4 afer a shor discussion of he suppors of hese measures in subsecion Classes of diffusion marices Le := a : R + S > d F-progressively measurable and a s ds <, for all. For a given P W, le W (P) := a : a = â, P-almos surely. (4.1) Recall ha â is he densiy of he quadraic variaion of B and is defined poinwise. We also define W := W (P). P W A suble echnical poin is ha W is sricly included in. In fac, he process a := 1 {â 2} + 31 {â <2} is clearly in \ W. For any P W and a W (P), by he Lévy characerizaion, he following Iô s sochasic inegral under P is a P-Brownian moion: W P := â 1/2 s db s = Also since B is he canonical process, a = a(b ) and hus a 1/2 s db s,. P a.s. (4.2) db = a 1/2 (B )dw P, P-almos surely, and W P is a P-Brownian moion. (4.3) 4.2 Characerizaion by diffusion marices In view of (4.3), o consruc a measure wih a given quadraic variaion a W, we consider he sochasic differenial equaion, dx = a 1/2 (X )db, P -almos surely. (4.4) In his generaliy, we consider only weak soluions P which we define nex. Alhough he following definiion is sandard (see for example Sroock & Varadhan [18]), we provide i for specificiy. 1852

10 Definiion 4.1. Le a be an elemen of W. (i) For F sopping imes τ 1 τ 2 and a probabiliy measure P 1 on τ1, we say ha P is a weak soluion of (4.4) on [τ 1, τ 2 ] wih iniial condiion P 1, denoed as P (τ 1, τ 2, P 1, a), if he followings hold: 1. P = P 1 on τ1 ; 2. The canonical process B is a P-local maringale on [τ 1, τ 2 ]; 3. The process W := a 1/2 τ 1 s (B )db s, defined P almos surely for all [τ 1, τ 2 ], is a P-Brownian Moion. (ii) We say ha he equaion (4.4) has weak uniqueness on [τ 1, τ 2 ] wih iniial condiion P 1 if any wo weak soluions P and P in (τ 1, τ 2, P 1, a) saisfy P = P on τ2. (iii) We say ha (4.4) has weak uniqueness if (ii) holds for any τ 1, τ 2 and any iniial condiion P 1 on τ1. We emphasize ha he sopping imes in his definiion are F-sopping imes. Noe ha, for each P W and a W (P), P is a weak soluion of (4.4) on R + wih iniial value P(B = ) = 1. We also need uniqueness of his map o characerize he measure P in erms of he diffusion marix a. Indeed, if (4.4) wih a has weak uniqueness, we le P a W be he unique weak soluion of (4.4) on R + wih iniial condiion P a (B = ) = 1, and define, We also define W := a W : (4.4) has weak uniqueness, W := {P a : a W }. (4.5) For noaional simpliciy, we denoe MRP := MRP W, MRP := {a W : P a MRP }. (4.6) F a := F Pa, F a := F Pa, for all a W. (4.7) I is clear ha, for each P W, he weak uniqueness of he equaion (4.4) may depend on he version of a W (P). This is indeed he case and he following example illusraes his observaion. Example 4.2. Le a () := 1, a 2 () := 2 and a 1 () := E 1 (, ) (), where E := B h B lim 1 h 2hln + ln h 1. Then clearly boh a and a 2 belong o W. Also a 1 = a, P -almos surely and a 1 = a 2, P a 2-almos surely. Hence, a 1 W (P ) W (P a 2). Therefore he equaion (4.4) wih coefficien a 1 has wo weak soluions P and P a 2. Thus a 1 / W. Remark 4.3. In his paper, we shall consider only hose P W W. However, we do no know wheher his inclusion is sric or no. In oher words, given an arbirary P W, can we always find one version a W (P) such ha a W? I is easy o consruc examples in W in he Markovian conex. Below, we provide wo classes of pah dependen diffusion processes in W. These ses are in fac subses of S W, which is defined in (8.11) below. We also consruc some couner-examples in he Appendix. Denoe Q := (, x) :, x C([, ], R d ). (4.8) 1853

11 Example 4.4. (Lipschiz coefficiens) Le a := σ 2 (, B ) where σ : Q S > d is Lebesgue measurable, uniformly Lipschiz coninuous in x under he uniform norm, and σ 2 (, ). Then (4.4) has a unique srong soluion and consequenly a W. Example 4.5. (Piecewise consan coefficiens) Le a = n= a n1 [τn,τ n+1 ) where {τ n } n is a nondecreasing sequence of F sopping imes wih τ =, τ n as n, and a n τn wih values in S > for all n. Again (4.4) has a unique srong soluion and a d W. This example is in fac more involved han i looks like, mainly due o he presence of he sopping imes. We relegae is proof o he Appendix. 4.3 Suppor of P a In his subsecion, we collec some properies of measures ha are consruced in he previous subsecion. We fix a subse W, and denoe by := {P a : a } he corresponding subse of W. In he sequel, we may also say a propery holds quasi surely if i holds quasi surely. For any a and any ˆF sopping ime ˆτ ˆ, le I is clear ha Ω aˆτ := â s ds = n 1 a s ds, for all [, ˆτ + 1 n ]. (4.9) Ω aˆτ ˆ ˆτ, Ωa is non-increasing in, Ωaˆτ+ = Ωaˆτ, and Pa (Ω a ) = 1. (4.1) We nex inroduce he firs disagreemen ime of any a, b, which plays a cenral role in Secion 5: θ a,b := inf : a s ds b s ds, and, for any ˆF sopping ime ˆτ ˆ, he agreemen se of a and b up o ˆτ: Ω a,b ˆτ := {ˆτ < θ a,b } {ˆτ = θ a,b = }. Here we use he convenion ha inf =. I is obvious ha θ a,b ˆ, Ω a,b ˆτ ˆ ˆτ and Ω aˆτ Ωbˆτ Ωa,b ˆτ. (4.11) Remark 4.6. The above noaions can be exended o all diffusion processes a, b. This will be imporan in Lemma 4.12 below. 1854

12 4.4 Separabiliy We are now in a posiion o sae he resricions needed for he main aggregaion resul Theorem 5.1. Definiion 4.7. A subse W is called a generaing class of diffusion coefficiens if (i) saisfies he concaenaion propery: a1 [,) + b1 [, ) for a, b,. (ii) has consan disagreemen imes: for all a, b, θ a,b is a consan or, equivalenly, Ω a,b = or Ω for all. We noe ha he concaenaion propery is sandard in he sochasic conrol heory in order o esablish he dynamic programming principle, see, e.g. page 5 in [14]. The consan disagreemen imes propery is imporan for boh Lemma 5.2 below and he aggregaion resul of Theorem 5.1 below. We will provide wo examples of ses wih hese properies, afer saing he main resricion for he aggregaion resul. Definiion 4.8. We say is a separable class of diffusion coefficiens generaed by if W is a generaing class of diffusion coefficiens and consiss of all processes a of he form, a = a n i 1 E n1 i [τ n,τ n+1 ), (4.12) n= i=1 where (a n i ) i,n, (τ n ) n is nondecreasing wih τ = and inf{n : τ n = } <, τ n < τ n+1 whenever τ n <, and each τ n akes a mos counably many values, for each n, {E n i, i 1} τ n form a pariion of Ω. We emphasize ha in he previous definiion he τ n s are F sopping imes and E n i following are wo examples of generaing classes of diffusion coefficiens. τn. The Example 4.9. Le be he class of all deerminisic mappings. Then clearly W and saisfies boh properies (he concaenaion and he consan disagreemen imes properies) of a generaing class. Example 4.1. Recall he se Q defined in (4.8). Le be a se of deerminisic Lebesgue measurable funcions σ : Q S > saisfying, d - σ is uniformly Lipschiz coninuous in x under L -norm, and σ 2 (, ) and - for each x C(R +, R d ) and differen σ 1, σ 2, he Lebesgue measure of he se A(σ 1, σ 2, x) is equal o, where A(σ 1, σ 2, x) := : σ 1 (, x [,] ) = σ 2 (, x [,] ). Le be he class of all possible concaenaions of, i.e. σ akes he following form: σ(, x) := σ i (, x)1 [i, i+1 )(), (, x) Q, i= for some sequence i and σ i, i. Le := {σ 2 (, B ) : σ }. I is immediae o check ha W and saisfies he concaenaion and he consan disagreemen imes properies. Thus i is also a generaing class. 1855

13 We nex prove several imporan properies of separable classes. Proposiion Le be a separable class of diffusion coefficiens generaed by. Then W, and -quasi surely is equivalen o -quasi surely. Moreover, if MRP, hen MRP. We need he following wo lemmas o prove his resul. The firs one provides a convenien srucure for he elemens of. Lemma Le be a separable class of diffusion coefficiens generaed by. For any a and F-sopping ime τ, here exis τ τ, a sequence {a n, n 1}, and a pariion {E n, n 1} τ of Ω, such ha τ > τ on {τ < } and a = a n ()1 En for all < τ. n 1 In paricular, E n Ω a,a n τ and consequenly n Ω a,a n τ τ τ n, hen one can choose τ τ n+1. = Ω. Moreover, if a akes he form (4.12) and The proof of his lemma is sraighforward, bu wih echnical noaions. Thus we pospone i o he Appendix. We remark ha a his poin we do no know wheher a W. Bu he noaions θ a,a n and Ω a,a n τ are well defined as discussed in Remark 4.6. We recall from Definiion 4.1 ha P (τ 1, τ 2, P 1, a) means P is a weak soluion of (4.4) on [ τ 1, τ 2 ] wih coefficien a and iniial condiion P 1. Lemma Le τ 1, τ 2 wih τ 1 τ 2, and {a i, i 1} W (no necessarily in W ) and le {E i, i 1} τ1 be a pariion of Ω. Le P be a probabiliy measure on τ1 and P i (τ 1, τ 2, P, a i ) for i 1. Define P(E) := P i (E E i ) for all E τ2 and a := a i 1 E i, [τ 1, τ 2 ]. i 1 Then P (τ 1, τ 2, P, a). Proof. Clearly, P = P on τ1. I suffices o show ha boh B and B B T maringales on [τ 1, τ 2 ]. i 1 τ 1 a s ds are P-local By a sandard localizaion argumen, we may assume wihou loss of generaliy ha all he random variables below are inegrable. Now for any τ 1 τ 3 τ 4 τ 2 and any bounded random variable η τ3, we have E P [(B τ4 B τ3 )η] = = E Pi (B τ4 B τ3 )η1 Ei i 1 i 1 E Pi E Pi B τ4 B τ3 τ3 η1 Ei =. Therefore B is a P-local maringale on [τ 1, τ 2 ]. Similarly one can show ha B B T a τ s ds is also 1 a P-local maringale on [τ 1, τ 2 ]. 1856

14 Proof of Proposiion Le a be given as in (4.12). (i) We firs show ha a W. Fix θ 1, θ 2 wih θ 1 θ 2 and a probabiliy measure P on θ1. Se τ := θ 1 and τ n := (τ n θ 1 ) θ 2, n 1. We shall show ha (θ 1, θ 2, P, a) is a singleon, ha is, he (4.4) on [θ 1, θ 2 ] wih coefficien a and iniial condiion P has a unique weak soluion. To do his we prove by inducion on n ha ( τ, τ n, P, a) is a singleon. Firs, le n = 1. We apply Lemma 4.12 wih τ = τ and choose τ = τ 1. Then, a = i 1 a i()1 Ei for all < τ 1, where a i and {E i, i 1} τ form a pariion of Ω. For i 1, le P,i be he unique weak soluion in ( τ, τ 1, P, a i ) and se P,a (E) := P,i (E E i ) for all E τ1. i 1 We use Lemma 4.13 o conclude ha P,a ( τ, τ 1, P, a). On he oher hand, suppose P ( τ, τ 1, P, a) is an arbirary weak soluion. For each i 1, we define P i by P i (E) := P(E E i ) + P,i (E (E i ) c ) for all E τ1. We again use Lemma 4.13 and noice ha a1 Ei + a i 1 (Ei ) c = a i. The resul is ha P i ( τ, τ 1, P, a i ). Now by he uniqueness in ( τ, τ 1, P, a i ) we conclude ha P i = P,i on τ1. This, in urn, implies ha P(E E i ) = P,i (E E i ) for all E τ1 and i 1. Therefore, P(E) = i 1 P,i (E E i ) = P,a (E) for all E τ1. Hence ( τ, τ 1, P, a) is a singleon. We coninue wih he inducion sep. Assume ha ( τ, τ n, P, a) is a singleon, and denoe is unique elemen by P n. Wihou loss of generaliy, we assume τ n < τ n+1. Following he same argumens as above we know ha ( τ n, τ n+1, P n, a) conains a unique weak soluion, denoed by P n+1. Then boh B and B B T a sds are P n+1 -local maringales on [ τ, τ n ] and on [ τ n, τ n+1 ]. This implies ha P n+1 ( τ, τ n+1, P, a). On he oher hand, le P ( τ, τ n+1, P, a) be an arbirary weak soluion. Since we also have P ( τ, τ n, P, a), by he uniqueness in he inducion assumpion we mus have he equaliy P = P n on τn. Therefore, P ( τ n, τ n+1, P n, a). Thus by uniqueness P = P n+1 on τn+1. This proves he inducion claim for n + 1. Finally, noe ha P m (E) = P n (E) for all E τn and m n. Hence, we may define P (E) := P n (E) for E τn. Since inf{n : τ n = } <, hen inf{n : τ n = θ 2 } < and hus θ2 = n 1 τn. So we can uniquely exend P o θ2. Now we direcly check ha P (θ 1, θ 2, P, a) and is unique. (ii) We nex show ha P a (E) = for all polar se E. Once again we apply Lemma 4.12 wih τ =. Therefore a = i 1 a i()1 Ei for all, where {a i, i 1} and {E i, i 1} form a pariion of Ω. Now for any -polar se E, P a (E) = P a (E E i ) = P a i (E E i ) =. i 1 This clearly implies he equivalence beween -quasi surely and -quasi surely. i

15 (iii) We now assume MRP and show ha a MRP. Le M be a P a -local maringale. We prove by inducion on n again ha M has a maringale represenaion on [, τ n ] under P a for each n 1. This, ogeher wih he assumpion ha inf{n : τ n = } <, implies ha M has maringale represenaion on R + under P a, and hus proves ha P a MRP. Since τ =, here is nohing o prove in he case of n =. Assume he resul holds on [, τ n ]. Apply Lemma 4.12 wih τ = τ n and recall ha in his case we can choose he τ o be τ n+1. Hence a = i 1 a i()1 Ei, < τ n+1, where {a i, i 1} and {E i, i 1} τn form a pariion of Ω. For each i 1, define M i := [M τ n+1 M τn ]1 Ei 1 [τn, )() for all. Then one can direcly check ha M i is a P a i-local maringale. Since a i MRP, here exiss H i such ha d M i = H i db, P a i-almos surely. Now define H := i 1 H i 1 E i, τ n < τ n+1. Then we have d M = H db, τ n < τ n+1, P a -almos surely. We close his subsecion by he following imporan example. Example Assume consiss of all deerminisic funcions a : R + S > aking he form d a = n 1 i= a i 1 [i, i+1 ) + a n 1 [n, ) where i Q and a i has raional enries. This is a special case of Example 4.9 and hus W. In his case is counable. Le = {a i } i 1 and define ˆP := i=1 2 i P a i. Then ˆP is a dominaing probabiliy measure of all P a, a, where is he separable class of diffusion coefficiens generaed by. Therefore, -quasi surely is equivalen o ˆP-almos surely. Noice however ha is no counable. 5 Quasi-sure aggregaion In his secion, we fix a separable class of diffusion coefficiens generaed by (5.1) and denoe := {P a, a }. Then we prove he main aggregaion resul of his paper. For his we recall ha he noion of aggregaion is defined in Definiion 3.1 and he noaions θ a,b and Ω a,b ˆτ are inroduced in subsecion 4.3. Theorem 5.1 (Quasi sure aggregaion). For saisfying (5.1), le {X a, a } be a family of ˆF - progressively measurable processes. Then here exiss a unique ( q.s.) -aggregaor X if and only if {X a, a } saisfies he consisency condiion X a = X b, P a almos surely on [, θ a,b ) for any a and b. (5.2) Moreover, if X a is càdlàg P a -almos surely for all a, hen we can choose a -q.s. càdlàg version of he -aggregaor X. We noe ha he consisency condiion (5.2) is slighly differen from he condiion (3.4) before. The condiion (5.2) is more naural in his framework and is more convenien o check in applicaions. Before he proof of he heorem, we firs show ha, for any a, b, he corresponding probabiliy measures P a and P b agree as long as a and b agree. 1858

16 Lemma 5.2. For saisfying (5.1) and a, b, θ a,b is an F-sopping ime aking counably many values and P a (E Ω a,b ˆτ ) = Pb (E Ω a,b ˆτ ) for all ˆτ ˆ and E ˆ ˆτ. (5.3) Proof. (i) We firs show ha θ a,b is an F-sopping ime. Fix an arbirary ime. In view of Lemma 4.12 wih τ =, we assume wihou loss of generaliy ha a = a n ()1 En and b = b n ()1 En for all < τ, n 1 where τ >, a n, b n and {E n, n 1} form a pariion of Ω. Then n 1 {θ a,b } = {θ a n,b n } E n. n By he consan disagreemen imes propery of, θ a n,b n is a consan. This implies ha {θ a n,b n } is equal o eiher or Ω. Since E n, we conclude ha {θ a,b } for all. Tha is, θ a,b is an F-sopping ime. (ii) We nex show ha θ a,b akes only counable many values. In fac, by (i) we may now apply Lemma 4.12 wih τ = θ a,b. So we may wrie a = ã n ()1Ẽn and b = bn ()1Ẽn for all < θ, n 1 where θ > θ a,b or θ = θ a,b =, ã n, b n, and {Ẽ n, n 1} θ a,b form a pariion of Ω. Then i is clear ha θ a,b = θ ãn, b n on Ẽ n, for all n 1. For each n, by he consan disagreemen imes propery of, θ ãn, b n is consan. Hence θ a,b akes only counable many values. n 1 (iii) We now prove (5.3). We firs claim ha, E Ω a,b ˆτ θ a,b Pa ( ) Indeed, for any, for any E ˆ ˆτ. (5.4) E Ω a,b ˆτ {θ a,b } = E {ˆτ < θ a,b } {θ a,b } = E {ˆτ < θ a,b } {ˆτ 1 m } {θ a,b }. m 1 By (i) above, {θ a,b }. For each m 1, E {ˆτ < θ a,b } {ˆτ 1 m } ˆ + Pa ( 1 1 ) Pa ( ), m m and (5.4) follows. By (5.4), here exis E a,i, E b,i θ a,b, i = 1, 2, such ha E a,1 E Ω a,b ˆτ Ea,2, E b,1 E Ω a,b ˆτ E b,2, and P a (E a,2 \E a,1 ) = P b (E b,2 \E b,1 ) =. 1859

17 Define E 1 := E a,1 E b,1 and E 2 := E a,2 E b,2, hen E 1, E 2 θ a,b, E 1 E E 2, and P a (E 2 \E 1 ) = P b (E 2 \E 1 ) =. Thus P a (E Ω a,b ˆτ ) = Pa (E 2 ) and P b (E Ω a,b ˆτ ) = Pb (E 2 ). Finally, since E 2 θ a,b, following he definiion of P a and P b, in paricular he uniqueness of weak soluion of (4.4) on he inerval [, θ a,b ], we conclude ha P a (E 2 ) = P b (E 2 ). This implies (5.3) immediaely. Remark 5.3. The propery (5.3) is crucial for checking he consisency condiions in our aggregaion resul in Theorem 5.1. We noe ha (5.3) does no hold if we replace he compleed σ algebra a τ b τ wih he augmened σ algebra a τ b τ. To see his, le d = 1, a := 1, b := 1+1 [1, ) (). In his case, θ a,b = 1. Le τ :=, E := Ω a 1. One can easily check ha Ωa,b = Ω, P a (E) = 1, P b (E) =. This implies ha E a b and E Ωa,b. However, Pa (E) = 1 = P b (E). See also Remark 2.3. Proof of Theorem 5.1. The uniqueness of aggregaor is immediae. By Lemma 5.2 and he uniqueness of weak soluions of (4.4) on [, θ a,b ], we know P a = P b on θ a,b. Then he exisence of he -aggregaor obviously implies (5.2). We now assume ha he condiion (5.2) holds and prove he exisence of he -aggregaor. We firs claim ha, wihou loss of generaliy, we may assume ha X a is càdlàg. Indeed, suppose ha he heorem holds for càdlàg processes. Then we consruc a -aggregaor for a family {X a, a }, no necessarily càdlàg, as follows: - If X a R for some consan R > and for all a, se Y a := X a s ds. Then, he family {Y a, a } inheris he consisency condiion (5.2). Since Y a is coninuous for every a, his 1 family admis a -aggregaor Y. Define X := lim ɛ ɛ [Y +ɛ Y ]. Then one can verify direcly ha X saisfies all he requiremens. - In he general case, se X R,a := ( R) X a R. By he previous argumens here exiss -aggregaor X R of he family {X R,a, a } and i is immediae ha X := lim R X R saisfies all he requiremens. We now assume ha X a is càdlàg, P a -almos surely for all a. In his case, he consisency condiion (5.2) is equivalen o X a = X b, < θ a,b, P a -almos surely for any a and b. (5.5) Sep 1. We firs inroduce he following quoien ses of. For each, and a, b, we say a b if Ω a,b = Ω (or, equivalenly, he consan disagreemen ime θ a,b ). Then is an equivalence relaionship in. Thus one can form a pariion of based on. Pick an elemen from each pariion se o consruc a quoien se (). Tha is, for any a, here exiss a unique b () such ha Ω a,b = Ω. Recall he noaion Ω a defined in (4.9). By (4.11) and he consan disagreemen imes propery of, we know ha {Ω a, a ()} are disjoin. Sep 2. For fixed R +, define ξ (ω) := a () X a (ω)1 Ω a (ω) for all ω Ω. (5.6) 186

18 The above uncounable sum is well defined because he ses {Ω a, a ()} are disjoin. In his sep, we show ha ξ is ˆ -measurable and ξ = X a, Pa -almos surely for all a. (5.7) We prove his claim in he following hree sub-cases For each a (), by definiion ξ = X a on Ω a. Equivalenly {ξ X a } (Ωa )c. Moreover, by (4.1), P a ((Ω a )c ) =. Since Ω a + and a is complee under P a, ξ is a -measurable and P a (ξ = X a ) = Also, for each a, here exiss a unique b () such ha a b. Then ξ = X b on Ω b. Since Ωa,b = Ω, i follows from Lemma 5.2 ha P a = P b on + and P a (Ω b ) = Pb (Ω b ) = 1. Hence P a (ξ = X b ) = 1. Now by he same argumen as in he firs case, we can prove ha ξ is a -measurable. Moreover, by he consisency condiion (5.8), Pa (X a = X b ) = 1. This implies ha P a (ξ = X a ) = Now consider a. We apply Lemma 4.12 wih τ =. This implies ha here exis a sequence {a j, j 1} such ha Ω = j 1 Ω a,a j. Then {ξ X a } = {ξ X a } Ωa,a j. Now for each j 1, {ξ X a } Ωa,a j j 1 {ξ X a j } Ω a,a j {X a j X a } Ωa,a j. Applying Lemma 5.2 and using he consisency condiion (5.5), we obain P a {X a j X a } Ωa,a j = P a j {X a j X a } Ωa,a j = P a j {X a j X a } { < θ a,a j } Moreover, for a j, by he previous sub-case, {ξ X a j } Pa j ( + D + such ha P a j(d) = and {ξ X a j } D. Therefore {ξ X a j } Ω a,a j D Ω a,a j and P a (D Ω a,a j =. ) = P a j (D Ω a,a j ) =. ). Hence here exiss This means ha {ξ X a j } Ω a,a j Pa ( + ). All of hese ogeher imply ha {ξ X a } Pa ( + ). Therefore, ξ a and P a (ξ = X a ) = 1. Finally, since ξ a for all a, we conclude ha ξ ˆ. This complees he proof of (5.7). Sep 3. For each n 1, se n i := i, i and define n where ξ n i X a,n := X a 1 {} + i=1 X a n i 1 ( n i 1, n i ] for all a and X n := ξ 1 {} + i=1 ξ n i 1 ( n i 1, n i ], is defined by (5.6). Le ˆF n := { ˆ, }. By Sep 2, X a,n, X n are ˆF n -progressively + 1 n measurable and P a (X n = X a,n, ) = 1 for all a. We now define X := lim n X n. 1861

19 Since ˆF n is decreasing o ˆF and ˆF is righ coninuous, X is ˆF -progressively measurable. Moreover, for each a, {X = X a {X, } is càdlàg} {X n = X a,n, } {X a is càdlàg}. n 1 Therefore X = X a and X is càdlàg, P a -almos surely for all a. -quasi surely. In paricular, X is càdlàg, Le ˆτ ˆ and {ξ a, a } be a family of ˆ ˆτ -measurable random variables. We say an ˆ ˆτ - measurable random variable ξ is a -aggregaor of he family {ξ a, a } if ξ = ξ a, P a -almos surely for all a. Noe ha we may idenify any ˆ ˆτ -measurable random variable ξ wih he ˆF -progressively measurable process X := ξ1 [ˆτ, ). Then a direc consequence of Theorem 5.1 is he following. Corollary 5.4. Le be saisfying (5.1) and ˆτ ˆ. Then he family of ˆ ˆτ -measurable random variables {ξ a, a } has a unique ( -q.s.) -aggregaor ξ if and only if he following consisency condiion holds: ξ a = ξ b on Ω a,b ˆτ, Pa -almos surely for any a and b. (5.8) For he nex resul, we recall ha he P-Brownian moion W P is defined in (4.2). consequence of Theorem 5.1, he following resul defines he -Brownian moion. As a direc Corollary 5.5. For saisfying (5.1), he family {W Pa, a } admis a unique -aggregaor W. Since W Pa is a P a -Brownian moion for every a, we call W a -universal Brownian moion. Proof. Le a, b. For each n, denoe τ n := inf : â s ds n θ a,b. Then B τn is a P b -square inegrable maringale. By sandard consrucion of sochasic inegral, see e.g. [11] Proposiion 2.6, here exis F-adaped simple processes β b,m such ha Define he universal process lim EPb m τn 1 2 âs (β b,m s â 1 2 s W b,m := β b,m s db s. ) 2 ds =. (5.9) Then lim EPb sup m W b,m τ n W Pb 2 =. (5.1) 1862

20 By Lemma 2.4, all he processes in (5.9) and (5.1) can be viewed as F-adaped. Since τ n θ a,b, applying Lemma 5.2 we obain from (5.9) and (5.1) ha lim EPa m τn 1 2 âs (β b,m s â 1 2 s The firs limi above implies ha lim EPa sup m ) 2 ds =, lim EPa sup m W b,m τ n W Pa which, ogeher wih he second limi above, in urn leads o W Pa Clearly τ n θ a,b as n. Then W Pa W b,m τ n 2 =, = W Pb, τ n, P a a.s. = W Pb, < θ a,b, P a a.s. W Pb 2 =. Tha is, he family {W Pa, a } saisfies he consisency condiion (5.2). We hen apply Theorem 5.1 direcly o obain he aggregaor W. The Brownian moion W is our firs example of a sochasic inegral defined simulaneously under all P a, a : W = â 1/2 s db s,, q.s. (5.11) We will invesigae in deail he universal inegraion in Secion 6. Remark 5.6. Alhough a and W Pa are F-progressively measurable, from Theorem 5.1 we can only deduce ha â and W are ˆF -progressively measurable. On he oher hand, if we ake a version of W Pa ha is progressively measurable o he augmened filraion F a, hen in general he consisency condiion (5.2) does no hold. For example, le d = 1, a := 1, and b := [1, ) (),, as in Remark 5.3. Se W Pa (ω) := B (ω) + 1 (Ω a 1 Pb )c(ω) and W (ω) := B (ω) + [B (ω) B 1 (ω)]1 [1, ) (). Then boh W Pa and W Pb are F a F b -progressively measurable. However, θ a,b = 1, bu P b (W Pa = W Pb ) = Pb (Ω a 1 ) =, so we do no have W Pa = W Pb, P b -almos surely on [, 1]. 6 Quasi-sure sochasic analysis In his secion, we fix again a separable class of diffusion coefficiens generaed by, and se := {P a : a }. We shall develop he -quasi sure sochasic analysis. We emphasize again 1863

21 ha, when a probabiliy measure P is fixed, by Lemma 2.4 here is no need o disinguish he filraions F +, F P, and F P. We firs inroduce several spaces. Denoe by L he collecion of all ˆ -measurable random variables wih appropriae dimension. For each p [1, ] and P, we denoe by L p (P) he corresponding L p space under he measure P and ˆL p := P L p (P). Similarly, H := H (R d ) denoes he collecion of all R d valued ˆF -progressively measurable processes. H p (P a ) is he subse of all H H saisfying H p T,H p (P a ) := EPa T and H 2 loc (Pa ) is he subse of H whose elemens saisfy T all T. Finally, we define Ĥ p := P p/2 a 1/2 s H s 2 ds < for all T >, a1/2 s H p (P) and Ĥ 2 loc := P H s 2 ds <, P a -almos surely, for H 2 loc (P). The following wo resuls are direc applicaions of Theorem 5.1. Similar resuls were also proved in [5, 6], see e.g. Theorem 2.1 in [5], Theorem 36 in [6] and he Kolmogorov crierion of Theorem 31 in [6]. Proposiion 6.1 (Compleeness). Fix p 1, and le be saisfying (5.1). (i) Le (X n ) n ˆL p be a Cauchy sequence under each P a, a. Then here exiss a unique random variable X ˆL p such ha X n X in L p (P a, ˆ ) for every a. (ii) Le (X n ) n Ĥ p be a Cauchy sequence under he norm T,H p (P a ) for all T and a. Then here exiss a unique process X Ĥ p such ha X n X under he norm T,H p (P a ) for all T and a. Proof. (i) By he compleeness of L p (P a, ˆ ), we may find X a L p (P a, ˆ ) such ha X n X a in L p (P a, ˆ ). The consisency condiion of Theorem 5.1 is obviously saisfied by he family {X a, a }, and he resul follows. (ii) can be proved by a similar argumen. Proposiion 6.2 (Kolmogorov coninuiy crieria). Le be saisfying (5.1), and X be an ˆF - progressively measurable process wih values in R n. We furher assume ha for some p > 1, X ˆL p for all and saisfy E Pa X X s p c a s n+ɛ a for some consans c a, ɛ a >. Then X admis an ˆF -progressively measurable version X which is Hölder coninuous, -q.s. (wih Hölder exponen α a < ɛ a /p, P a -almos surely for every a ). 1864

22 Proof. We apply he Kolmogorov coninuiy crierion under each P a, a. This yields a family of F Pa -progressively measurable processes {X a, a } such ha X a = X, P a -almos surely, and X a is Hölder coninuous wih Hölder exponen α a < ɛ a /p, P a -almos surely for every a. Also in view of Lemma 2.4, we may assume wihou loss of generaliy ha X a is ˆF -progressively measurable for every a. Since each X a is a P a -modificaion of X for every a, he consisency condiion of Theorem 5.1 is immediaely saisfied by he family {X a, a }. Then, he aggregaed process X consruced in ha heorem has he desired properies. Remark 6.3. The saemens of Proposiions 6.1 and 6.2 can be weakened furher by allowing p o depend on a. We nex consruc he sochasic inegral wih respec o he canonical process B which is simulaneously defined under all he muually singular measures P a, a. Such consrucions have been given in he lieraure bu under regulariy assumpions on he inegrand. Here we only place sandard condiions on he inegrand bu no regulariy. Theorem 6.4 (Sochasic inegraion). For saisfying (5.1), le H Ĥ 2 be given. Then, here loc exiss a unique ( -q.s.) ˆF -progressively measurable process M such ha M is a local maringale under each P a and M = H s db s,, P a -almos surely for all a. If in addiion H Ĥ 2, hen for every a, M is a square inegrable P a -maringale. Moreover, E Pa [M 2 ] = EPa [ a1/2 s H s 2 ds] for all. Proof. For every a, he sochasic inegral M a := H sdb s is well-defined P a -almos surely as a F Pa -progressively measurable process. By Lemma 2.4, we may assume wihou loss of generaliy ha M a is ˆF -adaped. Following he argumens in Corollary 5.5, in paricular by applying Lemma 5.2, i is clear ha he consisency condiion (5.2) of Theorem 5.1 is saisfied by he family {M a, a }. Hence, here exiss an aggregaing process M. The remaining saemens in he heorem follows from classical resuls for sandard sochasic inegraion under each P a. We nex sudy he maringale represenaion. Theorem 6.5 (Maringale represenaion). Le be a separable class of diffusion coefficiens generaed by MRP. Le M be an ˆF -progressively measurable process which is a quasi sure local maringale, ha is, M is a local maringale under P for all P. Then here exiss a unique ( -q.s.) process H Ĥ 2 such ha loc M = M + H s db s,, q.s.. Proof. By Proposiion 4.11, MRP. Then for each P, all P maringales can be represened as sochasic inegrals wih respec o he canonical process. Hence, here exiss unique (P almos surely) process H P H 2 (P) such ha loc M = M + H P s db s,, P-almos surely. 1865