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8 matches found
V: 12, 127-137, LNM 191 (1971)
Un contre-exemple au problème des laplaciens approchés (Martingale theory)
The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years
Comment: Problem originated in Meyer, Ill. J. Math., 7, 1963
Keywords: Submartingales, Supermartingales
Nature: Original
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VI: 11, 118-129, LNM 258 (1972)
MEYER, Paul-André
La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)
The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer, Zeit für X-theorie, 21, 1970
Keywords: Supermartingales, Föllmer measures
Nature: Exposition, Original additions
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VI: 19, 198-201, LNM 258 (1972)
RAO, Murali
Doob's decomposition and Burkholder's inequalities (Martingale theory)
The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (Ann. Math. Stat., 37, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale
Comment: This little-known paper would probably deserve a modern translation in continuous time
Keywords: Burkholder inequalities, Decomposition of supermartingales
Nature: Original
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VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma, Z. für W-theorie, 45, 1978 and its exposition 1321
Keywords: Supermartingales, Multiplicative decomposition
Nature: Original
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XI: 22, 362-364, LNM 581 (1977)
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 1524
Keywords: General filtrations, Strong supermartingales
Nature: Original
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XV: 24, 320-346, LNM 850 (1981)
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 theorems
Nature: Original
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XV: 25, 347-350, LNM 850 (1981)
Surmartingales-mesures (Martingale theory)
Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu, Martingales à temps discret
Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (Zeit. für W-theorie, 1978) in continuous time. This topic is now called Kunita decomposition, see 1005 and the corresponding references
Keywords: Supermartingales, Kunita decomposition
Nature: Original
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XVI: 25, 285-297, LNM 920 (1982)
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 supermartingales
Nature: Original
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