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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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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 1517

Comment: 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 inequalities

Nature: Original

Retrieve article from Numdam

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

XVI: 19, 221-233, LNM 920 (1982)

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 1517

Comment: See an earlier paper of the author on this subject,

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

Retrieve article from Numdam