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 inequalitiesNature: Exposition
Retrieve article from Numdam
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 functionsNature: Exposition Retrieve article from Numdam
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 supermartingalesNature: Original Retrieve article from Numdam
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 inequalitiesNature: Exposition Retrieve article from Numdam
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 inequalitiesNature: Original Retrieve article from Numdam
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 theoremNature: Exposition,
Original additions Retrieve article from Numdam
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 inequalitiesNature: Original Retrieve article from Numdam
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 inequalitiesNature: Original Retrieve article from Numdam
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 inequalitiesNature: Original Retrieve article from Numdam
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 inequalitiesNature: Original Retrieve article from Numdam
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 functionsNature: Original Retrieve article from Numdam
