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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

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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