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 inequalitiesNature: Original
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
XVI: 10, 151-152, LNM 920 (1982)
MEYER, Paul-André
Sur une inégalité de Stein (
Applications of martingale theory)
In his book
Topics in harmonic analysis related to the Littlewood-Paley theory (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation
Keywords: Littlewood-Paley theory,
Martingale inequalitiesNature: Exposition,
Original additions Retrieve article from Numdam
XVI: 19, 221-233, LNM 920 (1982)
YOR, Marc
Application de la relation de domination à certains renforcements des inégalités de martingales (
Martingale theory)
The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper
1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman
1517Comment: See an earlier paper of the author on this subject,
Stochastics, 3, 1979. The author mentions that part of the results were discovered slightly earlier by R.~Gundy
Keywords: Martingale inequalities,
Domination inequalitiesNature: Original Retrieve article from Numdam
