On R d -valued peacocks

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1 On R d -valued peacocks Francis HIRSCH 1), Bernard ROYNETTE 2) July 26, 211 1) Laboraoire d Analyse e Probabiliés, Universié d Évry - Val d Essonne, Boulevard F. Mierrand, F-9125 Évry Cedex francis.hirsch@univ-evry.fr 2) Insiu Elie Caran, Universié Henri Poincaré, B.P. 239, F-5456 Vandœuvre-lès-Nancy Cedex bernard.roynee@iecn.u-nancy.fr Absrac: In his paper, we consider R d -valued inegrable processes which are increasing in he convex order, i.e. R d -valued peacocks in our erminology. Afer he presenaion of some examples, we show ha an R d -valued process is a peacock if and only if i has he same one-dimensional marginals as an R d -valued maringale. This exends former resuls, obained noably by V. Srassen 1965), J.L. Doob 1968) and H. Kellerer 1972). Key words: convex order; maringale; 1-maringale; peacock. 2 MSC: Primary: 6E15, 6G44. Secondary: 6G15, 6G48. 1 Inroducion 1.1 Terminology Firs we fix he erminology. In he sequel, d denoes a fixed ineger and R d is equipped wih a norm which is denoed by. We say ha wo R d -valued processes: X, ) and Y, ) are associaed, if hey have he same one-dimensional marginals, i.e. if:, X law) = Y. A process which is associaed wih a maringale is called a 1-maringale. An R d -valued process X, ) will be called a peacock if: 1

2 i) i is inegrable, ha is:, E[ X ] < ii) i increases in he convex order, meaning ha, for every convex funcion ψ : R d R, he map: is increasing. E[ψX )], + ] This erminology was inroduced in [HPRY]. We refer he reader o his monograph for an explanaion of he origin of he erm: peacock, as well as for a comprehensive sudy of his noion in he case d = 1. Acually, i may be noed ha, in he definiion of a peacock, only he family µ, ) of is one-dimensional marginals is involved. This makes i naural, in he following, o also call a peacock, a family µ, ) of probabiliy measures on R d such ha: i), x µ dx) <, ii) for every convex funcion ψ : R d R, he map: ψx) µ dx), + ] is increasing. Likewise, a family µ, ) of probabiliy measures on R d and an R d - valued process Y, ) will be said o be associaed if, for every, he law of Y is µ, i.e. if µ, ) is he family of he one-dimensional marginals of Y, ). Obviously, he above noions also are meaningful if one considers processes and families of measures indexed by a subse of R + for example N) insead of R +. I is an easy consequence of Jensen s inequaliy ha an R d -valued process which is a 1-maringale, is a peacock. So, a naural quesion is wheher he converse holds. 2

3 1.2 Case d = 1 A remarkable resul due o H. Kellerer [K], 1972) saes ha, acually, any R-valued process which is a peacock, is a 1-maringale. More precisely, Kellerer s resul saes ha any R-valued peacock admis an associaed maringale which is Markovian. Two more recen resuls now complee Kellerer s heorem. i) G. Lowher [L], 28) saes ha if µ, ) is an R-valued peacock such ha he map: µ is weakly coninuous i.e. for any R-valued, bounded and coninuous funcion f on R, he map: fx) µ dx) is coninuous), hen µ, ) is associaed wih a srongly Markovian maringale which moreover is almos-coninuous see [L] for he definiion). ii) In a previous paper [HR], 211), we presened a new proof of he above menioned heorem of H. Kellerer. Our mehod, which is inspired from he Fokker-Planck Equaion Mehod [HPRY, Secion 6.2, p.229]), hen appears as a new applicaion of M. Pierre s uniqueness heorem for a Fokker-Planck equaion [HPRY, Theorem 6.1, p.223]). Thus, we show ha a maringale which is associaed o an R-valued peacock, may be obained as a limi of soluions of sochasic differenial equaions. However, we do no obain ha such a maringale is Markovian. 1.3 Case d 1 Concerning he case R d wih d 1, and even much more general spaces, we would like o menion he following hree imporan papers. i) In [CFM] 1964), P. Carier, J.M.G. Fell and P.-A. Meyer sudy he case of wo probabiliy measures µ 1, µ 2 ) on a merizable convex compac K of a locally convex space. They prove, using he Hahn- Banach heorem, ha, if µ 1, µ 2 ) is a K-valued peacock indexed by {1, 2}), hen here exiss a Markovian kernel P on K such ha: θdx 1, dx 2 ) := µ 1 dx 1 ) P x 1, dx 2 ) is he law of a K-valued maringale Y 1, Y 2 ) associaed o µ 1, µ 2 ). ii) In [S] 1965), V. Srassen exends he Carier-Fell-Meyer resul o R d - valued peacocks wihou making he assumpion of compac suppor. Then he proves ha, if µ n, n ) is an R d -valued peacock indexed by N), here exiss an associaed maringale which is obained as a Markov chain. 3

4 iii) In [D] 1968), J.L. Doob sudies, in a very general exended framework, peacocks indexed by R + and aking heir values in a fixed compac se. In paricular, he proves ha hey admi associaed maringales. Noe ha in [D], he Markovian characer of he associaed maringales is no considered. 1.4 Organizaion The remainder of his paper is organised as follows: In Secion 2, we presen some basic facs concerning he R d -valued peacocks and we describe some examples, hus exending resuls of [HPRY]. In Secion 3, saring from Srassen s heorem, we prove ha a family µ, ) of probabiliy measures on R d, is associaed o a righconinuous maringale, if and only if, µ, ) is a peacock such ha he map: µ is weakly righ-coninuous on R +. In Secion 4, by approximaion from he previous resul, we exend his resul o he case of general R d -valued peacocks. 2 Generaliies, Examples 2.1 Noaion In he sequel, d denoes a fixed ineger, R d is equipped wih a norm which is denoed by, and we adop he erminology of Subsecion 1.1. We also denoe by M he se of probabiliy measures on R d, equipped wih he opology of weak convergence wih respec o he space C b R d ) of R-valued, bounded, coninuous funcions on R d ). We denoe by M f he subse of M consising of measures µ M such ha x µdx) <. M f is also equipped wih he opology of weak convergence. C c R d ) denoes he space of R-valued coninuous funcions on R d wih compac suppor, and C c + R d ) is he subspace consising of all he nonnegaive funcions in C c R d ). 4

5 2.2 Basic facs Proposiion 2.1 Le X, ) be an R d -valued inegrable process. Then X, ) is a peacock if and only if) he map: E[ψX )] is increasing, for every funcion ψ : R d R which is convex, of C class and such ha he derivaive ψ is bounded on R d. Proof Le ψ : R d R be a convex funcion. For every a R d, here exiss an affine funcion h a such ha: x R d, ψx) h a x) and ψa) = h a a). Le {a n ; n 1} be a counable dense subse of R d. We se: n 1, ψ n x) = sup h aj x). 1 j n Then: x R d, lim ψ n x) = ψx). n The funcions ψ n are convex and Lipschiz coninuous. Le φ be a nonnegaive funcion, of C class, wih compac suppor and such ha φx) dx = 1. We se, for n, p 1, x R d, ψ n,p x) = ψ n x 1 p y ) φy) dy. Clearly, ψ n,p is convex, of C class and Lipschiz coninuous. Consequenly, is derivaive is bounded on R d. Moreover, lim p ψ n,p = ψ n uniformly on R d. The desired resul now follows direcly. The nex resul will be useful in he sequel. Proposiion 2.2 Le X, ) be an R d -valued peacock. Then: 1. he map: E[X ] is consan; 2. he map: E[ X ] is increasing, and herefore, for every T, sup E[ X ] = E[ X T ] < ; T 5

6 3. for every T, he random variables X ; T ) are uniformly inegrable. Proof Properie and 2 are obvious. If c, x 1 { x c} 2 x c) +. As he funcion x 2 x c) + Now, by dominaed convergence, is convex, sup E [ X 1 { X c}] E[2 XT c) + ]. [,T ] lim E[2 X T c) + ] =. c + Hence, propery 3 holds. 2.3 Examples The following examples are given in [HPRY] for d = 1. The proofs given below are essenially he same as in [HPRY]. Proposiion 2.3 Le X be a cenered R d -valued random variable. Then X, ) is a peacock. Proof Le ψ : R d R be a convex funcion, and s <. Then, ψs X) 1 s ) ψ) + s ψ X). Since X is cenered, by Jensen s inequaliy: ψ) = ψ E[ X]) E[ψ X)]. Hence, E[ψs X)] 1 s ) E[ψ X)] + s E[ψ X)] = E[ψ X)]. 6

7 Proposiion 2.4 Le X, ) be a family of cenered, R d -valued, Gaussian variables. We denoe by C) = c i,j )) 1 i,j d he covariance marix of X. Then, X, ) is a peacock if and only if he map: C) is increasing in he sense of quadraic forms, i.e: a = a 1,, a d ) R d, c i,j ) a i a j is increasing. 1 i,j d Proof 1) For every a R d, he funcion: x R d 1 i,j d d ) 2 a i a j x i x j = a i x i is convex. This enails ha, if X, ) is a peacock, hen he map: C) is increasing in he sense of quadraic forms. 2) Conversely, suppose ha he map: C) is increasing in he sense of quadraic forms. By he proof of [HPRY, Theorem 2.16, p.132], here exiss a cenered R d -valued Gaussian process: Γ = Γ 1,,, Γ d, ), ), such ha: i=1 s,, 1 i, j d, E[Γ i,s Γ j, ] = c i,j s ). Therefrom we deduce ha Γ, ) is a maringale which is associaed o X, ), and consequenly, X, ) is a peacock. Corollary 2.1 Le A be a d d marix. We consider he R d -valued Ornsein-Uhlenbeck process U, ), defined as he unique) soluion, sared from, of he SDE: du = db + A U d where B, ) denoes a d-dimensional Brownian moion. Then, U, ) is a peacock. 7

8 Proof One has: U = Hence, for every, U covariance marix is: C) = exp s) A) db s. is a cenered, R d -valued Gaussian variable whose exps A) exps A ) ds where A denoes he adjoin marix of A. Therefrom i is clear ha he map: C) is increasing in he sense of quadraic forms, and Proposiion 2.4 applies. Proposiion 2.5 Le M, ) be an R d -valued, righ-coninuous maringale such ha: [ ] T >, E sup T M <. Then, 1. X := 1 2. X := ) M s ds ; ) M s M ) ds ; is a peacock, is a peacock. Proof Using Proposiion 2.1, we may use he proof of [HPRY, Theorem 1.4, p.26]. For he convenience of he reader, we reproduce his proof below. 1) Le ψ : R d R be a convex funcion, of C class and such ha he derivaive ψ is bounded on R d. Seing: M = one has, by inegraion by pars: s dm s, X = M 1 M and dx = 2 M d. Denoing by F s he σ-algebra generaed by {M u ; u s}, one ges, for s, E[X F s ] = X s + s 1 1 ) M s. 8

9 Consequenly, by Jensen s inequaliy, E[ψX )] E[ψX s + s 1 1 ) M s )]. Using again he fac ha ψ is convex, one obains: Now, E[ψX )] E[ψX s )] + s 1 1 ) E[ψ X s ) M s ]. ψ X s ) M s = and herefore s u 2 ψ X u ) M u, M u ) du + s u ψ X u ) dm u E[ψX )] E[ψX s )] s 1 1 ) E[ψ X s ) M s ], which, by Proposiion 2.1, yields he desired resul. 2) Le ψ be as above. One may suppose ha M =. One has, for s, E[ X F s ] = X s + s) M s. Consequenly, by Jensen s inequaliy, E[ψ X )] E[ψ X s + s) M s )]. Using again he fac ha ψ is convex, one obains: Now, and herefore E[ψ X )] E[ψ X s )] + s) E[ψ X s ) M s ]. ψ X s ) M s = s ψ X u )M u, M u ) du + s ψ X u ) dm u E[ψ X )] E[ψ X s )] s) E[ψ X s ) M s ], which, by Proposiion 2.1, yields he desired resul. 9

10 3 Righ-coninuous peacoks In his secion, we shall show ha any righ coninuous peacock admis an associaed righ-coninuous maringale. For his, we sar from Srassen s heorem, which we now recall. Theorem 3.1 Srassen [S], Theorem 8) Le µ n, n N) be a sequence in M. Then µ n, n N) is a peacock if and only if here exiss a maringale M n, n N) which is associaed o µ n, n N). We shall exend his heorem o righ-coninuous peacocks indexed by R +. In he case d = 1, he following heorem is proven in [HR], by a quie differen mehod. In paricular, in [HR], we do no use Srassen s heorem, nor he Hahn-Banach heorem, bu an explici approximaion by soluions of SDE s. Theorem 3.2 Le µ, ) be a family in M. Then he following properies are equivalen: i) There exiss a righ-coninuous maringale associaed o µ, ). ii) µ, ) is a peacock and he map: µ M Proof is righ-coninuous. 1) We firs assume ha propery i) is saisfied. Then, he fac ha µ, ) is a peacock follows classically from Jensen s inequaliy. Le M, ) be a righ-coninuous maringale associaed o µ, ). Then, if f C b R d ), dominaed convergence yields ha, for any, lim s,s> fx) µ s dx) = Therefore, he map: lim E[fM s)] = E[fM )] = s,s> µ M is righ-coninuous, and propery ii) is saisfied. fx) µ dx). 2) Conversely, we now assume ha propery ii) is saisfied. For every n N, we se: µ n) k = µ k2 n, k N. 1

11 By Srassen s heorem Theorem 3.1), here exiss a maringale M n) k, k N) which is associaed o µ n) k, k N). We se: X n) = M n) k if = k 2 n and X n) = oherwise. Consequenly, he law of X n) is µ if {k 2 n ; k N}, and is δ he Dirac measure a ) if {k 2 n ; k N}. Noe ha, due o he lack of uniqueness in Srassen s heorem, he law of X n) k2, k N) may be no he same as he law of X n+1) n k2, k N). n Only he one-dimensional marginals are idenical. 3) Le D = {k 2 n ; k, n N} he se of dyadic numbers. For every n N, for every r 1 and τ r = 1, 2,, r ) D r, we denoe by Π r,n) τ r he law of X n) 1,, X n) r ), a probabiliy on R d ) r. Lemma 3.1 For every τ r D r, he se of probabiliy measures: {Π r,n) τ r N} is igh. Proof We se, for x = x 1,, x r ) R d ) r, x r = r j=1 xj. Then, for p >, ; n Π r,n) τ r x r p) 1 p Πr,n) τ r x r ) = 1 p r j=1 E[ X n) j ] 1 p r µ j x ) j=1 since, by poin 2), he law of X n) j is eiher µ j or δ. Hence, lim sup Π r,n) τ p r x r p) =. n 4) As a consequence of he previous lemma, and wih he help of he diagonal procedure, here exiss a subsequence n l ) l such ha, for every τ r D r, he sequence of probabiliies on R d ) r : Π r,n l) τ r, l ), weakly converges o a probabiliy which we denoe by Π τ r) r. We remark ha, for l large enough, he law of X n l) j is µ j. Then, here exiss an R d -valued process X, D) such ha, for every r N and every τ r = 1,, r ) D r, he law of X 1,, X r ) is Π r) τ r, and Π 1) = µ for every D. 11

12 Lemma 3.2 The process X, D) is a maringale associaed o µ, D). Proof As we have already seen, he process X, D) is associaed o µ, D). We now prove ha i is a maringale. We se: Then, p >, x R d, ϕ p x) = 1 x ) 1 x. p ϕ p C b R d ; R d ) and ϕ p x) = x for x p. Le < s 2 < < s r s be elemens of D, and le f C b R d ) r ). We se: f = sup{ fx) ; x R d ) r }. Then, for l large enough, E[fX n l),, X n l) s r ) X n l) ] = E[fX n l),, X n l) s r ) X n l) s ]. On he oher hand, E[fX s1,, X sr ) ϕ p X )] E[fX s1,, X sr ) X ] f µ x 1{ x p} ), for every p >, E[fX n l),, X n l) s r ) ϕ p X n l) )] E[fX n l),, X n l) s r ) X n l) ] f µ x 1{ x p} ), for every l and every p >, and likewise, replacing by s. Moreover, lim l E[fXn l),, X n l) s r ) ϕ p X n l) )] = E[fX s1,, X sr ) ϕ p X )], and likewise, replacing by s. Finally, we obain, for p >, E[fX s1,, X sr ) X ] E[fX s1,, X sr ) X s ] 2 f [ µ x 1{ x p} ) + µs x 1{ x p} )], and he desired resul follows, leing p go o. 12

13 5) By he classical heory of maringales see, for example, [DM]), almos surely, for every, M = lim s,s D,s> X s is well defined, and M, ) is a righ-coninuous maringale. Besides, since, by hypohesis, he map: µ M is righconinuous, we deduce from Lemma 3.2 ha his maringale M, ) is associaed o µ, ). 4 The general case Theorem 3.2 shall now be exended, by approximaion, o he general case. Theorem 4.1 Le µ, ) be a family in M. Then he following properies are equivalen: i) There exiss a maringale associaed o µ, ). ii) µ, ) is a peacock. Proof Le µ, ) be a peacock. Lemma 4.1 There exiss a counable se R + such ha he map: µ M is coninuous a any s. Proof Le χ : R d R + be defined by: χx) = 1 x ) + = 1 x ) x. Then χ C c + R d ) and χ is he difference of wo convex funcions. We se: χ m x) = m d χm x), and we define he counable se H by: { r } H = a j χ m x q j ) ; r N, m N, a j Q +, q j Q d. j= 13

14 For h H, he funcion: µ h) is he difference of wo increasing funcions, and hence admis a counable se h of disconinuiies. We se = h H h. Then is a counable subse of R +, and µ h) is coninuous a any s, for every h H. Now, i is easy o see ha H is dense in C c + R d ) in he following sense: for every ϕ C c + R d ), here exis a compac se K R d and a sequence h n ) n H such ha: n, Supp h n K and lim n h n = ϕ uniformly. Consequenly, µ is vaguely coninuous a any s, and, since measures µ are probabiliies, µ is also weakly coninuous a any s. We may wrie = {d j ; j N}. For n N, we denoe by k n) l, l ) he increasing rearrangemen of he se: We define µ n) {k 2 n ; k N} {d j ; j n}., ) by: µ n) = µ k n) l if here exiss l such ha = k n) l, and by: µ n) l+1 k n) l+1 kn) l = kn) µ k n) l + kn) l k n) l+1 kn) l µ k n) l+1 if [k n) l, k n) l+1 ]. Lemma 4.2 The following properies hold: 1. For every n, µ n), ) is a peacock and he map: M is coninuous. µ n) 2. For any, sup{µ n) x ) ; n N} <. 3. For any, he se {µ n) 4. For, lim n µ n) = µ in M. ; n N} is uniformly inegrable. Proof Properie and 4 are clear by consrucion. Propery 2 resp. propery 3) follows direcly from propery 2 resp. propery 3) in Proposiion 2.2. By Theorem 3.2, here exiss, for each n, a righ-coninuous maringale 14

15 M n), ) which is associaed o µ n) and τ r = 1,, r ) R r +, we denoe by Π r,n) τ r a probabiliy measure on R d ) r., ). For any r N he law of M n) 1,, M n) r ), Lemma 4.3 For every τ r R r +, he se of probabiliy measures: {Π r,n) τ r N} is igh. Proof As in Lemma 3.1, for p >, Π r,n) τ r x r p) 1 p and by propery 2 in Lemma 4.2, r j=1 µ n) j x ), ; n lim sup Π r,n) τ p r x r p) =. n Le now U be an ulrafiler on N, which refines Fréche s filer. As a consequence of he previous lemma, for every r N and every τ r R r +, lim Π r,n) τ U r exiss for he weak convergence and we denoe his limi by Π τ r) r. By propery 4 in Lemma 4.2, Π 1) = µ. There exiss a process M, ) such ha, for every r N and every τ r = 1,, r ) R r +, he law of M 1,, M r ) is Π r) τ r. In paricular, his process M, ) is associaed o µ, ). Lemma 4.4 The process M, ) is a maringale. Proof The proof is quie similar o ha of Lemma 3.2, bu we give he deails for he sake of compleeness. We recall he noaion: p >, x R d, ϕ p x) = 1 x ) 1 x. p Le < s 2 < < s r s be elemens of R +, and le f C b R d ) r ). We se: f = sup{ fx) ; x R d ) r }. Then, for every n, E[fM n) On he oher hand,,, M n) ) M n) s r ] = E[fM n),, M s n) r ) M s n) ]. E[fM s1,, M sr ) ϕ p M )] E[fM s1,, M sr ) M ] f µ x 1{ x p} ), for every p >, 15

16 E[fM n) f µ n),, M s n) r )] E[fM s n) 1,, M s n) r ) x 1{ x p}, for every n and every p >, ) ϕ p M n) and likewise, replacing by s. Moreover, lim U E[fM n) ) M n) ],, M s n) r ) ϕ p M n) )] = E[fM s1,, M sr ) ϕ p M )], and likewise, replacing by s. Finally, we obain, for p >, E[fX s1,, X sr ) X ] E[fX s1,, X sr ) X s ] [ 2 f sup µ n) ) ) ] x 1{ x p} + µ n) s x 1{ x p}, n and, by propery 3 in Lemma 4.2, he desired resul follows, leing p go o. This lemma complees he proof of Theorem 4.1. Acknowledgmen We are graeful o Marc Yor for his help during he preparaion of his paper. References [CFM] P. Carier; J.M.G. Fell; P.-A. Meyer. Comparaison des mesures porées par un convexe compac. Bull. Soc. Mah. France, ), p [DM] C. Dellacherie; P.-A. Meyer. Probabiliés e poeniel, Chapires V à VIII, Théorie des maringales. Hermann 198). [D] J.L. Doob. Generalized sweeping-ou and probabiliy. J. Func. Anal., ), p [HPRY] F. Hirsch; C. Profea; B. Roynee; M. Yor. Peacocks and associaed maringales, wih explici consrucions. Bocconi & Springer Series, vol. 3, Springer 211). [HR] F. Hirsch; B. Roynee. A new proof of Kellerer s heorem. Prépublicaion Universié d Evry, n o 323, 9/ ). 16

17 [K] H.G. Kellerer. Markov-Komposiion und eine Anwendung auf Maringale. Mah. Ann., ), p [L] G. Lowher. Fiing maringales o given marginals. hp://arxiv.org/abs/ v1 28). [S] V. Srassen. The exisence of probabiliy measures wih given marginals. Ann. Mah. Sa ), p