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III: 13, 163-174, LNM 88 (1969)

**MEYER, Paul-André**

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,*Ann. Math. Stat.*, **39**, 1968

Keywords: Burkholder inequalities

Nature: Exposition

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VI: 13, 151-158, LNM 258 (1972)

**MEYER, Paul-André**

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (*Acta Math.,* **124**, 1970), Burkholder-Davis-Gundy (*Proc. 6-th Berkeley Symposium,* **3**, 1972). The technique of proof is now obsolete

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

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VI: 19, 198-201, LNM 258 (1972)

**RAO, Murali**

Doob's decomposition and Burkholder's inequalities (Martingale theory)

The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (*Ann. Math. Stat.,* **37**, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale

Comment: This little-known paper would probably deserve a modern translation in continuous time

Keywords: Burkholder inequalities, Decomposition of supermartingales

Nature: Original

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IX: 04, 206-212, LNM 465 (1975)

**CHOU, Ching Sung**

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

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IX: 05, 213-225, LNM 465 (1975)

**CHOU, Ching Sung**

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia*Martingale Inequalities: Seminar Notes on Recent Progress,* Benjamin, 1973, are extended to continuous time

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

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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

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XII: 19, 170-179, LNM 649 (1978)

**MÉTRAUX, C.**

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

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XIII: 26, 294-306, LNM 721 (1979)

**BONAMI, Aline**; **LÉPINGLE, Dominique**

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $**P**$ two functionals of a $**P**$-martingale---remains true, possibly with new constants, when $**P**$ is replaced by an equivalent law $Z.**P**$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 27, 307-312, LNM 721 (1979)

**IZUMISAWA, Masataka**; **SEKIGUCHI, Takesi**

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in*Tôhoku Math. J.* **29**, 1977. For a modern reference see also Kazamaki, *Continuous Exponential Martingales and $\,BMO$,* LNM. 1579, 1994

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIV: 04, 26-48, LNM 784 (1980)

**LENGLART, Érik**; **LÉPINGLE, Dominique**; **PRATELLI, Maurizio**

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,*Probabilités et Potentiel E *; see also 1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.} **49**,1982

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

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XV: 20, 285-289, LNM 850 (1981)

**CHOU, Ching Sung**

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

Retrieve article from Numdam

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,

Keywords: Burkholder inequalities

Nature: Exposition

Retrieve article from Numdam

VI: 13, 151-158, LNM 258 (1972)

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

Retrieve article from Numdam

VI: 19, 198-201, LNM 258 (1972)

Doob's decomposition and Burkholder's inequalities (Martingale theory)

The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (

Comment: This little-known paper would probably deserve a modern translation in continuous time

Keywords: Burkholder inequalities, Decomposition of supermartingales

Nature: Original

Retrieve article from Numdam

IX: 04, 206-212, LNM 465 (1975)

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

Retrieve article from Numdam

IX: 05, 213-225, LNM 465 (1975)

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

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

XII: 19, 170-179, LNM 649 (1978)

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 26, 294-306, LNM 721 (1979)

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIII: 27, 307-312, LNM 721 (1979)

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIV: 04, 26-48, LNM 784 (1980)

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

Retrieve article from Numdam

XV: 20, 285-289, LNM 850 (1981)

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

Retrieve article from Numdam