XI: 22, 362-364, LNM 581 (1977)
DELLACHERIE, Claude
Sur la régularisation des surmartingales (
Martingale theory)
It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)
Comment: See
1524Keywords: General filtrations,
Strong supermartingalesNature: Original
Retrieve article from Numdam
XV: 24, 320-346, LNM 850 (1981)
DELLACHERIE, Claude;
LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (
General theory of processes)
The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in
1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)
Keywords: General filtrations,
Strong supermartingales,
Snell's envelope,
Section theoremsNature: Original Retrieve article from Numdam
XVI: 25, 285-297, LNM 920 (1982)
UPPMAN, Are
Un théorème de Helly pour les surmartingales fortes (
Martingale theory)
Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences
Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory
Keywords: Supermartingales,
Strong supermartingalesNature: Original Retrieve article from Numdam
