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VII: 14, 136-145, LNM 321 (1973)

**MEYER, Paul-André**

Le dual de $H^1$ est $BMO$ (cas continu) (Martingale theory)

The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $**R**$ and $**R**^n$ and the duality between $BMO$ and $H^1$ were almost immediately translated into discrete martingale theory by Herz and Garsia. The next step, due to Getoor-Sharpe ({\sl Invent. Math.} **16**, 1972), delt with continuous martingales. The extension to right continuous martingales, a good exercise in martingale theory, is given here

Comment: See 907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer,*Probabilités et Potentiel,* Vol. B, Chapter VII

Keywords: $BMO$, Hardy spaces, Fefferman inequality

Nature: Original

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IX: 07, 237-238, LNM 465 (1975)

**MEYER, Paul-André**

Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)

Fills a gap in the proof of the duality theorem in 714

Keywords: $BMO$

Nature: Correction

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X: 17, 245-400, LNM 511 (1976)

**MEYER, Paul-André**

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

X: 18, 401-413, LNM 511 (1976)

**PRATELLI, Maurizio**

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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X: 27, 536-539, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN**1579**, 1994

Keywords: $BMO$

Nature: Original

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XI: 12, 132-195, LNM 581 (1977)

**MEYER, Paul-André**

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(**R**^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XI: 16, 303-323, LNM 581 (1977)

**BERNARD, Alain**; **MAISONNEUVE, Bernard**

Décomposition atomique de martingales de la classe $H^1$ (Martingale theory)

Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (*Studia Math.* **51**, 1974). An atomic decomposition of a Banach space consists in finding simple elements (called atoms) in its unit ball, such that every element is a linear combination of atoms $\sum_n \lambda_n a_n$ with $\sum_n \|\lambda_n\|<\infty$, the infimum of this sum defining the norm or an equivalent one. Here an atomic decomposition is given for $H^1$ spaces of martingales in continuous time (defined by their maximal function). Atoms are of two kinds: the first kind consists of martingales bounded uniformly by a constant $c$ and supported by an interval $[T,\infty[$ such that $P\{T<\infty\}\le 1/c$. These atoms do not generate the whole space $H^1$ in general, though they do in a few interesting cases (if all martingales are continuous, or in the discrete dyadic case). To generate the whole space it is sufficient to add martingales of integrable variation (those whose total variation has an $L^1$ norm smaller than $1$ constitute the second kind of atoms). This approach leads to a proof of the $H^1$-$BMO$ duality and the Davis inequality

Comment: See also 1117

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 17, 324-326, LNM 581 (1977)

**BERNARD, Alain**

Complément à l'exposé précédent (Martingale theory)

This paper is a sequel to 1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 25, 383-389, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

Retrieve article from Numdam

XI: 31, 446-481, LNM 581 (1977)

**MEYER, Paul-André**

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

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XII: 01, 1-19, LNM 649 (1978)

**PRATELLI, Maurizio**

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,*Comm Pure Appl. Math.*, **26**, 1973

Keywords: Interpolation, $BMO$

Nature: Original

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XII: 05, 47-50, LNM 649 (1978)

**KAZAMAKI, Norihiko**

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**; **YOR, Marc**

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 16, 138-147, LNM 649 (1978)

**LÉPINGLE, Dominique**

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

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XII: 48, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 49, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 50, 739-739, LNM 649 (1978)

**LÉPINGLE, Dominique**

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XIII: 30, 360-370, LNM 721 (1979)

**JEULIN, Thierry**; **YOR, Marc**

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XIII: 34, 400-406, LNM 721 (1979)

**YOR, Marc**

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

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XV: 18, 259-277, LNM 850 (1981)

**BRU, Bernard**; **HEINICH, Henri**; **LOOTGIETER, Jean-Claude**

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XV: 19, 278-284, LNM 850 (1981)

**ÉMERY, Michel**

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

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XVIII: 32, 500-500, LNM 1059 (1984)

**ÉMERY, Michel**

Sur l'exponentielle d'une martingale de $BMO$ (Martingale theory)

This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XXXIII: 16, 342-348, LNM 1709 (1999)

**GRANDITS, Peter**

Some remarks on L$^\infty $, H$^\infty $, and $BMO$ (Martingale theory)

It is known from 1212 that neither $L^\infty$ nor $H^\infty$ is dense in $BMO$. This article answers a question raised by Durrett (*Brownian Motion and Martingales in Analysis,* Wadworth 1984): Does there exist a $BMO$-martingale which has a best approximation in $L^\infty$? The answer is negative, but becomes positive if $L^\infty$ is replaced with $H^\infty$

Keywords: $BMO$, Hardy spaces

Nature: Original

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Le dual de $H^1$ est $BMO$ (cas continu) (Martingale theory)

The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $

Comment: See 907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer,

Keywords: $BMO$, Hardy spaces, Fefferman inequality

Nature: Original

Retrieve article from Numdam

IX: 07, 237-238, LNM 465 (1975)

Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)

Fills a gap in the proof of the duality theorem in 714

Keywords: $BMO$

Nature: Correction

Retrieve article from Numdam

X: 17, 245-400, LNM 511 (1976)

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

X: 18, 401-413, LNM 511 (1976)

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

Retrieve article from Numdam

X: 27, 536-539, LNM 511 (1976)

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN

Keywords: $BMO$

Nature: Original

Retrieve article from Numdam

XI: 12, 132-195, LNM 581 (1977)

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 16, 303-323, LNM 581 (1977)

Décomposition atomique de martingales de la classe $H^1$ (Martingale theory)

Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (

Comment: See also 1117

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

Retrieve article from Numdam

XI: 17, 324-326, LNM 581 (1977)

Complément à l'exposé précédent (Martingale theory)

This paper is a sequel to 1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

Retrieve article from Numdam

XI: 25, 383-389, LNM 581 (1977)

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

Retrieve article from Numdam

XI: 31, 446-481, LNM 581 (1977)

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

Retrieve article from Numdam

XII: 01, 1-19, LNM 649 (1978)

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,

Keywords: Interpolation, $BMO$

Nature: Original

Retrieve article from Numdam

XII: 05, 47-50, LNM 649 (1978)

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

Retrieve article from Numdam

XII: 12, 98-113, LNM 649 (1978)

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XII: 16, 138-147, LNM 649 (1978)

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

Retrieve article from Numdam

XII: 48, 739-739, LNM 649 (1978)

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

Retrieve article from Numdam

XII: 49, 739-739, LNM 649 (1978)

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

Retrieve article from Numdam

XII: 50, 739-739, LNM 649 (1978)

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

Retrieve article from Numdam

XIII: 30, 360-370, LNM 721 (1979)

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XV: 18, 259-277, LNM 850 (1981)

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XV: 19, 278-284, LNM 850 (1981)

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

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XVIII: 32, 500-500, LNM 1059 (1984)

Sur l'exponentielle d'une martingale de $BMO$ (Martingale theory)

This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XXXIII: 16, 342-348, LNM 1709 (1999)

Some remarks on L$^\infty $, H$^\infty $, and $BMO$ (Martingale theory)

It is known from 1212 that neither $L^\infty$ nor $H^\infty$ is dense in $BMO$. This article answers a question raised by Durrett (

Keywords: $BMO$, Hardy spaces

Nature: Original

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