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12 matches found
II: 08, 140-165, LNM 51 (1968)
MEYER, Paul-André
Guide détaillé de la théorie ``générale'' des processus (General theory of processes)
This paper states and comments the essential results of a theory which was considered difficult in those times. New terminology was introduced (for instance, the accessible and previsible $\sigma$-fields) though not quite the definitive one (the word ``optional'' only timidly appears instead of the awkward ``well-measurable''). A few new results on the $\sigma$-fields ${\cal F}_{T-}$ and increasing processes are given at the end, the only ones to be proved
Comment: This paper had pedagogical importance in its time, but is now obsolete
Keywords: Previsible processes, Section theorems
Nature: Exposition
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III: 11, 155-159, LNM 88 (1969)
MEYER, Paul-André
Une nouvelle démonstration des théorèmes de section (General theory of processes)
The proof of the section theorems has improved over the years, from complicated-false to complicated-true, and finally to easy-true. This was a step on the way, due to Dellacherie (inspired by Cornea-Licea, Z. für W-theorie, 10, 1968)
Comment: This is essentially the definitive proof, using a general section theorem instead of capacity theory
Keywords: Section theorems, Optional processes, Previsible processes
Nature: Original
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V: 10, 87-102, LNM 191 (1971)
Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)
Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections
Comment: See Dellacherie-Meyer, Probabilités et Potentiel, Chap. XI
Keywords: Analytic sets, Random sets, Section theorems
Nature: New exposition of known results
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VI: 02, 35-50, LNM 258 (1972)
AZÉMA, Jacques
Une remarque sur les temps de retour. Trois applications (Markov processes, General theory of processes)
This paper is the first step in the investigations of Azéma on the ``dual'' form of the general theory of processes (for which see Azéma (Ann. Sci. ENS, 6, 1973, and 814). Here the $\sigma$-fields of cooptional and coprevisible sets are introduced in a Markovian set-up, and without their definitive names. A section theorem by return times is proved, and applications to the theory of Markov processes are given
Keywords: Homogeneous processes, Coprevisible processes, Cooptional processes, Section theorems, Projection theorems, Time reversal
Nature: Original
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VII: 05, 38-47, LNM 321 (1973)
Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)
This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved
Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan, Z. für W-Theorie 44, 1978
Keywords: Stopping times, Section theorems
Nature: Original
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VIII: 04, 22-24, LNM 381 (1974)
Un ensemble progressivement mesurable... (General theory of processes)
The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections
Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails
Keywords: Progressive sets, Section theorems
Nature: Original
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IX: 08, 239-245, LNM 465 (1975)
DELLACHERIE, Claude; MEYER, Paul-André
Un nouveau théorème de projection et de section (General theory of processes)
Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration
Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer, Probabilités et Potentiel, theorems IV.84 of vol. A and App.1, \no~6
Keywords: Section theorems, Optional processes, Projection theorems
Nature: Original
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X: 25, 521-531, LNM 511 (1976)
Séparabilité optionnelle, d'après Doob (General theory of processes)
A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given
Comment: Doob's original paper appeared in Ann. Inst. Fourier, 25, 1975. See also 1105
Keywords: Optional processes, Separability, Section theorems
Nature: Exposition, Original additions
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X: 32, 579-593, LNM 511 (1976)
Compléments aux exposés sur les ensembles analytiques (Descriptive set theory)
A new proof of Novikov's theorem (see 1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to 919 and 920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo
Keywords: Analytic sets, Section theorems, Capacities
Nature: Original
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XIV: 49, 500-546, LNM 784 (1980)
Tribus de Meyer et théorie des processus (General theory of processes, Stochastic calculus)
The subject of this paper is the study of the $\sigma$-field on $R_+\times\Omega$ generated by a family of cadlag processes including the deterministic ones, and stable under stopping at non-random times. Of course the optional and previsible $\sigma$-fields are Meyer $\sigma$-fields in this very general sense. It is a matter of wonder to see how far one can go with such simple hypotheses, which were suggested by Dellacherie 705
Comment: This beautiful paper was generally ignored. If a suggestive name had been used instead of the terminology ``Meyer $\sigma$-field'', its fate might have been different. See 1524 for an interesting application. The work of Fourati (partly unpublished) follows along the same lines, but including time reversal: see 2119
Keywords: Projection theorems, Section theorems
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: 26, 351-370, LNM 850 (1981)
Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (General theory of processes)
One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was 511, in which Dellacherie introduced ``rabotages'' (306) to develop the theory without analytic sets; see also 1246, 1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications
Keywords: Section theorems, Capacities, Sierpinski's ``rabotages''
Nature: Original
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