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
XII: 58, 770-774, LNM 649 (1978)
MEYER, Paul-André
Sur le lemme de La Vallée Poussin et un théorème de Bismut (
Measure theory,
General theory of processes)
Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces
Keywords: Uniform integrability,
Class (D) processes,
Moderate convex functionsNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 31, 371-377, LNM 721 (1979)
DELLACHERIE, Claude
Inégalités de convexité pour les processus croissants et les sousmartingales (
Martingale theory)
Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,
Probabilités et Potentiels B, Chapter VI
Keywords: Martingale inequalities,
Convex functionsNature: Exposition,
Original proofs 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
XIV: 25, 220-222, LNM 784 (1980)
YAN, Jia-An
Caractérisation d'ensembles convexes de $L^1$ ou $H^1$ (
Stochastic calculus,
Functional analysis)
This is a new and simpler approach to the crucial functional analytic lemma in
1354 (the proof that semimartingales are the stochastic integrators in $L^0$). That is, given a convex set $K\subset L^1$ containing $0$, find a condition for the existence of $Z>0$ in $L^\infty$ such that $\sup_{X\in K}E[ZX]<\infty$. A similar result is discussed for $H^1$ instead of $L^1$
Comment: This lemma, instead of the original one, has proved very useful in mathematical finance
Keywords: Semimartingales,
Stochastic integrals,
Convex functionsNature: 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
XVI: 11, 153-158, LNM 920 (1982)
MEYER, Paul-André
Interpolation entre espaces d'Orlicz (
Functional analysis)
This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in
1609Keywords: Interpolation,
Orlicz spaces,
Moderate convex functionsNature: Exposition Retrieve article from Numdam
XVI-S: 59, 217-236, LNM 921 (1982)
DARLING, Richard W.R.
Martingales in manifolds - Definition, examples and behaviour under maps (
Stochastic differential geometry)
Martingales in manifolds have been introduced independently by Meyer
1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions
Comment: More details are given in
Bull. L.M.S. 15 (1983),
Publ R.I.M.S. Kyoto~
19 (1983) and
Zeit. für W-theorie 65 (1984). Characterizating of manifold-valued martingales by convex functions has become a powerful tool: see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989) and his St-Flour lectures (Springer LNM 1738)
Keywords: Martingales in manifolds,
Semimartingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XVIII: 33, 501-518, LNM 1059 (1984)
ÉMERY, Michel;
ZHENG, Wei-An
Fonctions convexes et semimartingales dans une variété (
Stochastic differential geometry)
On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling
1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XXV: 19, 220-233, LNM 1485 (1991)
ÉMERY, Michel;
MOKOBODZKI, Gabriel
Sur le barycentre d'une probabilité dans une variété (
Stochastic differential geometry)
In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$
Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (
J. London Math. Soc. 46, 1992), as pointed out in
2650Keywords: Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
