XIII: 52, 611-613, LNM 721 (1979)
MEYER, Paul-André
Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (
Martingale theory)
In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (
Ann. Prob. 8, 1980) to develop the whole theory of stochastic differential equations
Keywords: Doob's inequality,
Stochastic differential equationsNature: Exposition
Retrieve article from Numdam
XIV: 13, 118-124, LNM 784 (1980)
ÉMERY, Michel
Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (
Stochastic calculus)
Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see
1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales
Comment: See
1352. A general reference on the Métivier-Pellaumail method can be found in their book
Stochastic Integration, Academic Press 1980. See also He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC Press 1992
Keywords: Semimartingales,
Spaces of semimartingales,
Stochastic differential equations,
Doob's inequality,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XIV: 14, 125-127, LNM 784 (1980)
LENGLART, Érik
Sur l'inégalité de Métivier-Pellaumail (
Stochastic calculus)
A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality
Keywords: Doob's inequality,
Métivier-Pellaumail inequalityNature: New proof of known results Retrieve article from Numdam
XV: 17, 251-258, LNM 850 (1981)
PITMAN, James W.
A note on $L_2$ maximal inequalities (
Martingale theory)
This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$
Keywords: Maximal inequality,
Doob's inequalityNature: Original Retrieve article from Numdam
