Browse by: Author name - Classification - Keywords - Nature
105 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: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (Descriptive set theory, Markov processes, General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie, Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sections
Nature: Original
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III: 10, 152-154, LNM 88 (1969)
MEYER, Paul-André
Un résultat élémentaire sur les temps d'arrêt (General theory of processes)
This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times
Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront
Keywords: Stopping times, Accessible times, Previsible times
Nature: Original
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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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IV: 05, 60-70, LNM 124 (1970)
DELLACHERIE, Claude
Un exemple de la théorie générale des processus (General theory of processes)
In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly
Comment: This example has become classical. See for example Dellacherie-Meyer, Probabilités et Potentiel, Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer 906
Keywords: Stopping times, Accessible times, Previsible times
Nature: Original
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V: 15, 147-169, LNM 191 (1971)
MAISONNEUVE, Bernard
Ensembles régénératifs, temps locaux et subordinateurs (General theory of processes, Renewal theory)
New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen, Math. Scand., 24, 1969), including a general definition of local time of a random set
Comment: See Meyer 412, Morando-Maisonneuve 413, later work of Maisonneuve in 813 and later
Keywords: Local times, Subordinators, Renewal theory
Nature: Original
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V: 29, 290-310, LNM 191 (1971)
WALSH, John B.
Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)
It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called essential topology, used in the paper of Chung and Walsh 522 in the same volume
Comment: See Doob Bull. Amer. Math. Soc., 72, 1966. An important application in given by Walsh 623 in the next volume. See the paper 1025 of Benveniste. For the use of a different topology see Ito J. Math. Soc. Japan, 20, 1968
Keywords: Essential topology
Nature: Original
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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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VI: 05, 90-97, LNM 258 (1972)
CHUNG, Kai Lai
On universal field equations (General theory of processes)
There is a pun in the title, since ``field'' here is a $\sigma$-field and not a quantum field. The author proves useful results on the $\sigma$-fields ${\cal F}_{T-}$ and ${\cal F}_{T+}$ associated with an arbitrary random variable $T$ in the paper of Chung-Doob, Amer. J. Math., 87, 1965. As a corollary, he can prove easily that for a Hunt process, accessible = previsible
Keywords: Filtrations
Nature: Original
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VI: 10, 113-117, LNM 258 (1972)
MAISONNEUVE, Bernard
Topologies du type de Skorohod (General theory of processes)
This paper presents an adaptation of the well known Skorohod topology, to the case of an arbitrary (i.e., non-compact) interval of the line
Keywords: Skorohod topology
Nature: Original
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VI: 14, 159-163, LNM 258 (1972)
MEYER, Paul-André
Temps d'arrêt algébriquement prévisibles (General theory of processes)
The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$
Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers
Keywords: Stopping times, Previsible processes
Nature: Original
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VII: 04, 36-37, LNM 321 (1973)
DELLACHERIE, Claude
Temps d'arrêt totalement inaccessibles (General theory of processes)
Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left
Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times
Nature: Original
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VII: 05, 38-47, LNM 321 (1973)
DELLACHERIE, Claude
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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VII: 18, 180-197, LNM 321 (1973)
MEYER, Paul-André
Résultats d'Azéma en théorie générale des processus (General theory of processes)
This paper presents several results from a paper of Azéma (Invent. Math., 18, 1972) which have become (in a slightly extended version) standard tools in the general theory of processes. The problem is that of ``localizing'' a time $L$ which is not a stopping time. With $L$ are associated the supermartingale $c^L_t=P\{L>t|{\cal F}_t\}$ and the previsible increasing processes $p^L$ which generates it (and is the dual previsible projection of the unit mass on the graph of $L$). Then the left support of $dp^L$ is the smallest left-closed previsible set containing the graph of $L$, while the set $\{c^L_-=1\}$ is the greatest previsible set to the left of $L$. Other useful results are the following: given a progressive process $X$, the process $\limsup_{s\rightarrow t} X_s$ is optional, previsible if $s<t$ is added, and a few similar results
Comment: These results have been included (with their optional counterpart, whose interest was discovered later) in Dellacherie-Meyer, Probabilités et Potentiel, Vol. E, Chapter XX 12--17
Keywords: Optimal stopping, Previsible processes
Nature: Exposition
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VII: 22, 217-222, LNM 321 (1973)
MEYER, Paul-André
Sur les désintégrations régulières de L. Schwartz (General theory of processes)
This paper presents a small part of an important article of L.~Schwartz (J. Anal. Math., 26, 1973). The main result is an extension of the classical theorem on the existence of regular conditional probability distributions: on a good filtered probability space, the previsible and optional projections of a process can be computed by means of true kernels
Comment: The contents of this paper have been considerably developed by F.~Knight in his theory of prediction. See 1007
Keywords: Previsible projections, Optional projections, Prediction theory
Nature: Exposition, Original additions
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VII: 23, 223-247, LNM 321 (1973)
MEYER, Paul-André
Sur un problème de filtration (General theory of processes)
This is an attempt to present, in the usual language of the seminar, a a simple case from filtering theory (additive white noise in one dimension). The results proved are the Kallianpur-Striebel formula (Ann. Math. Stat., 39, 1968) and a theorem of Clark
Keywords: Filtering theory, Innovation
Nature: Exposition
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VIII: 04, 22-24, LNM 381 (1974)
DELLACHERIE, Claude
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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VIII: 05, 25-26, LNM 381 (1974)
DELLACHERIE, Claude
Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)
This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems
Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor, Z. für W-theorie, 38, 1977 and Yor 1221. For another approach to the restricted case considered here, see Ruiz de Chavez 1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property
Keywords: Brownian motion, Poisson processes, Previsible representation
Nature: Original
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VIII: 14, 262-288, LNM 381 (1974)
MEYER, Paul-André
Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)
This paper is an exposition of a paper by Azéma (Ann. Sci. ENS, 6, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional
Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer, Probabilités et Potentiel, Chapter XVIII, 1992
Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals
Nature: Exposition, Original additions
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IX: 02, 97-153, LNM 465 (1975)
BENVENISTE, Albert
Processus stationnaires et mesures de Palm du flot spécial sous une fonction (Ergodic theory, General theory of processes)
This paper takes over several topics of 901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (Ann. Inst. H. Poincaré, 9, 1973). Then a deeper study of flows under a function is performed, leading to section theorems of optional or previsible homogeneous sets by optional or previsible counters. The last section (written in collaboration with J.~Jacod) concerns a stationary counter (discrete point process) in its natural filtration, and its stochastic intensity: here it is shown (contrary to the case of processes indexed by a half-line) that the stochastic intensity does not determine the law of the counter
Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes
Nature: Original
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IX: 06, 226-236, LNM 465 (1975)
CHOU, Ching Sung; MEYER, Paul-André
Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (General theory of processes)
Dellacherie has studied in 405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property
Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan, Semimartingale Theory and Stochastic Calculus, CRC~Press 1992
Keywords: Point processes, Previsible representation
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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IX: 25, 466-470, LNM 465 (1975)
MEYER, Paul-André; YAN, Jia-An
Génération d'une famille de tribus par un processus croissant (General theory of processes)
The previsible and optional $\sigma$-fields of a filtration $({\cal F}_t)$ on $\Omega$ are studied without the usual hypotheses: no measure is involved, and the filtration is not right continuous. It is proved that if the $\sigma$-fields ${\cal F}_{t-}$ are separable, then so is the previsible $\sigma$-field, and the filtration is the natural one for a continuous strictly increasing process. A similar result is proved for the optional $\sigma$-field assuming $\Omega$ is a Blackwell space, and then every measurable adapted process is optional
Comment: Making the filtration right continuous generally destroys the separability of the optional $\sigma$-field
Keywords: Previsible processes, Optional processes
Nature: Original
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IX: 28, 494-494, LNM 465 (1975)
DELLACHERIE, Claude
Correction à ``Intégrales stochastiques par rapport...'' (General theory of processes)
This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805
Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002
Keywords: Previsible representation
Nature: Original
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IX: 37, 556-564, LNM 465 (1975)
MEYER, Paul-André
Retour aux retournements (Markov processes, General theory of processes)
The first part of the talk is devoted to an important correction to the theorem on p.285 of 814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way
Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer Probabilités et potentiel
Keywords: Time reversal, Cooptional processes, Coprevisible processes, Homogeneous processes
Nature: Exposition, Original additions
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X: 07, 86-103, LNM 511 (1976)
MEYER, Paul-André
La théorie de la prédiction de F. Knight (General theory of processes)
This paper is devoted to the work of Knight, Ann. Prob. 3, 1975, the main idea of which is to associate with every reasonable process $(X_t)$ another process, taking values in a space of probability measures, and whose value at time $t$ is a conditional distribution of the future of $X$ after $t$ given its past before $t$. It is shown that the prediction process contains essentially the same information as the original process (which can be recovered from it), and that it is a time-homogeneous Markov process
Comment: The results are related to those of Schwartz (presented in 722), the main difference being that the future is predicted instead of the whole path. Knight has devoted to this subject the Essays on the Prediction Process, Hayward Inst. of Math. Stat., 1981, and a book, Foundations of the Prediction Process, Oxford Science Publ. 1992
Keywords: Prediction theory
Nature: Exposition, Original additions
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X: 08, 104-117, LNM 511 (1976)
MEYER, Paul-André; YOR, Marc
Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (General theory of processes)
This paper contains another version of Knight's theory (preceding paper 1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$
Comment: On the pathology of germ fields, see H. von Weizsäcker, Ann. Inst. Henri Poincaré, 19, 1983
Keywords: Prediction theory, Germ fields
Nature: Original
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X: 09, 118-124, LNM 511 (1976)
MEYER, Paul-André
Generation of $\sigma$-fields by step processes (General theory of processes)
On a Blackwell measurable space, let ${\cal F}_t$ be a right continuous filtration, such that for any stopping time $T$ the $\sigma$-field ${\cal F}_T$ is countably generated. Then (discarding possibly one single null set), this filtration is the natural filtration of a right-continuous step process
Comment: This answers a question of Knight, Ann. Math. Stat., 43, 1972
Keywords: Point processes
Nature: Original
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X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books
Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem
Nature: Exposition, Original additions
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X: 25, 521-531, LNM 511 (1976)
BENVENISTE, Albert
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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XI: 02, 21-26, LNM 581 (1977)
BENVENISTE, Albert
Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)
The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of medial limits due to Mokobodzki (see 719), which in fact was developed precisely at the author's request to solve this problem
Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits
Nature: Original
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XI: 05, 47-50, LNM 581 (1977)
DELLACHERIE, Claude
Deux remarques sur la séparabilité optionnelle (General theory of processes)
Optional separability was defined by Doob, Ann. Inst. Fourier, 25, 1975. See also Benveniste, 1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed
Keywords: Optional processes, Separability, Changes of time
Nature: Original
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XI: 06, 51-58, LNM 581 (1977)
DUDLEY, Richard M.; GUTMANN, Sam
Stopping times with given laws (General theory of processes)
Given a filtration ${\cal A}_t$ such that for all $t>0$ the law $P$ restricted to ${\cal A}_t$ is non-atomic, then for every law $\mu$ on the half-line there exists a stopping time with law $\mu$. The proof uses an interesting measure theoretic lemma on decreasing sequences of non-atomic $\sigma$-fields
Comment: It follows that the Brownian filtration contains a process whose law is that of a Poisson process (though of course it is not a Poisson process of the Brownian filtration )
Keywords: Stopping
Nature: Original
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XI: 08, 65-78, LNM 581 (1977)
EL KAROUI, Nicole; MEYER, Paul-André
Les changements de temps en théorie générale des processus (General theory of processes)
Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care
Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case
Keywords: Changes of time
Nature: Original
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XI: 09, 79-108, LNM 581 (1977)
EL KAROUI, Nicole; WEIDENFELD, Gérard
Théorie générale et changement de temps (General theory of processes)
The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care
Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333
Keywords: Changes of time
Nature: Original
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XI: 10, 109-119, LNM 581 (1977)
MEYER, Paul-André
Convergence faible de processus, d'après Mokobodzki (General theory of processes)
The following simple question of Benveniste was answered positively by Mokobodzki (Séminaire de Théorie du Potentiel, Lect. Notes in M. 563, 1976): given a sequence $(U^n)$ of optional processes such that $U^n_T$ converges weakly in $L^1$ for every stopping time $T$, does there exist an optional process $U$ such that $U^n_T$ converges to $U_T$? The proof is rather elaborate
Keywords: Weak convergence in $L^1$
Nature: Exposition
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XI: 14, 257-297, LNM 581 (1977)
YOR, Marc
Sur les théories du filtrage et de la prédiction (General theory of processes, Markov processes)
To be completed
Comment: MR 57, 10801
Keywords: Filtering theory, Prediction theory
Nature: Original
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XI: 28, 415-417, LNM 581 (1977)
LENGLART, Érik
Une caractérisation des processus prévisibles (General theory of processes)
One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale
Keywords: Previsible processes
Nature: Original
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XI: 34, 493-501, LNM 581 (1977)
YOR, Marc
A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)
Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales
Keywords: Yoeurp's lemma, Square bracket
Nature: Original
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XII: 03, 22-34, LNM 649 (1978)
JACOD, Jean
Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)
The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed
Comment: This papers improves on 1021 and 1023
Keywords: Semimartingales, Multiplicative decomposition
Nature: Original
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XII: 04, 35-46, LNM 649 (1978)
MÉMIN, Jean
Décompositions multiplicatives de semimartingales exponentielles et applications (General theory of processes)
It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions
Comment: The results of this paper are used in Mémin-Shiryaev 1312
Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition
Nature: Original
Retrieve article from Numdam
XII: 08, 57-60, LNM 649 (1978)
MEYER, Paul-André
Sur un théorème de J. Jacod (General theory of processes)
Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals
Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration
Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales
Nature: Original
Retrieve article from Numdam
XII: 09, 61-69, LNM 649 (1978)
YOR, Marc
Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)
Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called progressively enlarged filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is honest for $\cal F$, i.e., is the end of an $\cal F$-optional set
Comment: This result was independently discovered by Barlow, Zeit. für W-theorie, 44, 1978, which also has a huge intersection with 1211. Complements are given in 1210, and an explicit decomposition formula for semimartingales in 1211
Keywords: Enlargement of filtrations, Honest times
Nature: Original
Retrieve article from Numdam
XII: 10, 70-77, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André
A propos du travail de Yor sur le grossissement des tribus (General theory of processes)
This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$
Keywords: Enlargement of filtrations, Honest times
Nature: Original
Retrieve article from Numdam
XII: 11, 78-97, LNM 649 (1978)
JEULIN, Thierry; YOR, Marc
Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)
This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved
Comment: For additional results on enlargements, see the two Lecture Notes volumes 833 (T. Jeulin) and 1118. See also 1350
Keywords: Enlargement of filtrations, Honest times
Nature: Original
Retrieve article from Numdam
XII: 25, 364-377, LNM 649 (1978)
STRICKER, Christophe
Les ralentissements en théorie générale des processus (General theory of processes)
Given a filtration $({\cal F}t)$ and a stopping time $T$, we may define a new filtration $({\cal G}_t)$ as follows: we introduce an independent random variable $S$, and in intuitive language, we run the picture of $({\cal F}_t)$ up to time $T$, freeze the image between times $T$ and $T+S$, and then start running it again. The main result of this paper is the possibility, by performing this at all the times of discontinuity of $({\cal F}_t)$, to construct a filtration $({\cal G}_t)$ which is quasi-left-continuous. Though the idea is simple, there are considerable technical difficulties
Nature: Original
Retrieve article from Numdam
XII: 27, 398-410, LNM 649 (1978)
GETOOR, Ronald K.
Homogeneous potentials (General theory of processes)
This is a development in Knight's prediction theory as described in 1007, 1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$
Keywords: Prediction theory
Nature: Original
Retrieve article from Numdam
XII: 28, 411-423, LNM 649 (1978)
MEYER, Paul-André
Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)
Baxter and Chacón (Zeit. für W-theorie, 40, 1977) introduced a topology on the sets of ``fuzzy'' times and of fuzzy stopping times which turn these sets into compact metrizable spaces---a fuzzy r.v. $T$ is a right continuous decreasing process $M_t$ with $M_{0-}=1$, $M_t(\omega)$ being interpreted for each $\omega$ as the distribution function $P_{\omega}\{T>t\}$. When this process is adapted the fuzzy r.v. is a fuzzy stopping time. A number of properties of this topology are investigated
Comment: See 1536 for an extension to Polish spaces
Keywords: Stopping times, Fuzzy stopping times
Nature: Exposition, Original additions
Retrieve article from Numdam
XII: 29, 424-424, LNM 649 (1978)
DELLACHERIE, Claude
Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)
It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability
Keywords: Stopping times
Nature: Original
Retrieve article from Numdam
XII: 30, 425-427, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André
Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)
Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$
Keywords: Stopping times
Nature: Original
Retrieve article from Numdam
XII: 33, 457-467, LNM 649 (1978)
MAINGUENEAU, Marie Anne
Temps d'arrêt optimaux et théorie générale (General theory of processes)
This is a general discussion of optimal stopping in continuous time. Fairly advanced tools like strong supermartingales, Mertens' decomposition are used
Comment: The subject is taken up in 1332
Keywords: Optimal stopping, Snell's envelope
Nature: Original
Retrieve article from Numdam
XII: 37, 491-508, LNM 649 (1978)
MOKOBODZKI, Gabriel
Ensembles à coupes dénombrables et capacités dominées par une mesure (Measure theory, General theory of processes)
Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times R_+$. For $A\subset X$ define $C(A)$ as the outer measure of the projection on $X$ of $F\cap(A\timesR_+)$. Then it is proved that, if there is some measure $\lambda$ such that $\lambda$-null sets are $C$-null (the relation goes the reverse way from the preceding paper 1236!) then $F$ has ($\mu$-a.s.) countable sections, and if the property is strengthened to an $\epsilon-\delta$ ``absolute continuity'' relation, then $F$ has ($\mu$-a.s.) finite sections
Comment: This was a long-standing conjecture of Dellacherie (707), suggested by the theory of semi-polar sets. For further development see 1602
Keywords: Sets with countable sections
Nature: Original
Retrieve article from Numdam
XII: 38, 509-511, LNM 649 (1978)
DELLACHERIE, Claude
Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)
Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707
Keywords: Sets with countable sections, Semi-polar sets
Nature: Original
Retrieve article from Numdam
XII: 39, 512-514, LNM 649 (1978)
DELLACHERIE, Claude
Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)
The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes
Keywords: Essential suprema, Evanescent sets
Nature: Original
Retrieve article from Numdam
XII: 40, 515-522, LNM 649 (1978)
DELLACHERIE, Claude
Supports optionnels et prévisibles d'une P-mesure et applications (General theory of processes)
A $P$-measure is a measure on $\Omega\timesR_+$ which does not charge $P$-evanescent sets. A $P$-measure has optional and previsible projections which are themselves $P$-measures. As usual, supports are minimal sets carrying a measure, possessing different properties like being optional/previsible, being right/left closed. The purpose of the paper is to find out which kind of supports do exist. Applications are given to honest times
Comment: See 1339 for a complement concerning honest times
Keywords: Projection theorems, Support, Honest times
Nature: Original
Retrieve article from Numdam
XII: 55, 746-756, LNM 649 (1978)
DELLACHERIE, Claude
Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (Descriptive set theory, General theory of processes)
There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel
Keywords: Analytic sets
Nature: Original
Retrieve article from Numdam
XII: 58, 770-774, LNM 649 (1978)
MEYER, Paul-André
Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)
Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces
Keywords: Uniform integrability, Class (D) processes, Moderate convex functions
Nature: Exposition, Original additions
Retrieve article from Numdam
XIII: 08, 118-125, LNM 721 (1979)
YOEURP, Chantha
Sauts additifs et sauts multiplicatifs des semi-martingales (Martingale theory, General theory of processes)
First of all, the jump processes of special semimartingales are characterized (using a result of 1121, 1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al. 1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes
Keywords: Jump processes
Nature: Original
Retrieve article from Numdam
XIII: 15, 199-203, LNM 721 (1979)
MEYER, Paul-André
Une remarque sur le calcul stochastique dépendant d'un paramètre (General theory of processes)
Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod, Zeit. für W-Theorie, 31, 1975. The main feature is the corresponding use of random measures, previsible random measures, and previsible dual projections
Keywords: Processes depending on a parameter, Previsible processes, Previsible projections, Random measures
Nature: Exposition, Original additions
Retrieve article from Numdam
XIII: 17, 216-226, LNM 721 (1979)
LETTA, Giorgio
Quasimartingales et formes linéaires associées (General theory of processes, Martingale theory)
This paper gives a definition of a quasimartingale $X$ different from the usual one using the stochastic variation: the linear functional $E[\int_0^{\infty} H_s dX_s]$ should be relatively bounded on the vector lattice (Riesz space) of elementary previsible processes. Many classical theorems have simple proofs or elegant interpretations in this language
Keywords: Quasimartingales, Riesz spaces
Nature: Original
Retrieve article from Numdam
XIII: 24, 260-280, LNM 721 (1979)
ÉMERY, Michel
Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)
The stability theory for stochastic differential equations was developed independently by Émery (Zeit. für W-Theorie, 41, 1978) and Protter (same journal, 44, 1978). However, these results were stated in the language of convergent subsequences instead of true topological results. Here a linear topology (like convergence in probability: metrizable, complete, not locally convex) is defined on the space of semimartingales. Side results concern the Banach spaces $H^p$ and $S^p$ of semimartingales. Several useful continuity properties are proved
Comment: This topology has become a standard tool. For its main application, see the next paper 1325
Keywords: Semimartingales, Spaces of semimartingales
Nature: Original
Retrieve article from Numdam
XIII: 29, 332-359, LNM 721 (1979)
JEULIN, Thierry; YOR, Marc
Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)
The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears
Keywords: Hardy's inequality, Previsible representation
Nature: Original
Retrieve article from Numdam
XIII: 32, 378-384, LNM 721 (1979)
SZPIRGLAS, Jacques; MAZZIOTTO, Gérald
Théorème de séparation dans le problème d'arrêt optimal (General theory of processes)
Let $({\cal G}_t)$ be an enlargement of a filtration $({\cal F}_t)$ with the property that for every $t$, if $X$ is ${\cal G}_t$-measurable, then $E[X\,|\,{\cal F}_t]=E[X\,|\,{\cal F}_\infty]$. Then if $(X_t)$ is a ${\cal F}$-optional process, its Snell envelope is the same in both filtrations. Applications are given to filtering theory
Keywords: Optimal stopping, Snell's envelope, Filtering theory
Nature: Original
Retrieve article from Numdam
XIII: 34, 400-406, LNM 721 (1979)
YOR, Marc
Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)
This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$
Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$
Nature: Original
Retrieve article from Numdam
XIII: 38, 443-452, LNM 721 (1979)
EL KAROUI, Nicole
Temps local et balayage des semimartingales (General theory of processes)
This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the balayage formula (see Azéma-Yor, introduction to Temps Locaux , Astérisque , 52-53): if $Z$ is a locally bounded previsible process, then $$Z_{g_t}X_t=\int_0^t Z_{g_s}dX_s$$ and therefore $Y_t=Z_{g_t}X_t$ is a semimartingale. The main problem of the series of reports is: what can be said if $Z$ is not previsible, but optional, or even progressive?\par This particular paper is devoted to the study of the non-adapted process $$K_t=\sum_{g\in G,g\le t } (M_{D_g}-M_g)$$ which turns out to have finite variation
Comment: This paper is completed by 1357
Keywords: Local times, Balayage, Balayage formula
Nature: Original
Retrieve article from Numdam
XIII: 39, 453-471, LNM 721 (1979)
YOR, Marc
Sur le balayage des semi-martingales continues (General theory of processes)
For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$
Comment: See 1357
Keywords: Local times, Balayage, Balayage formula
Nature: Original
Retrieve article from Numdam
XIII: 40, 472-477, LNM 721 (1979)
STRICKER, Christophe
Semimartingales et valeur absolue (General theory of processes)
For the general notation, see 1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale
Comment: A complement is given in the next paper 1341. See also 1351
Keywords: Balayage, Quasimartingales
Nature: Original
Retrieve article from Numdam
XIII: 41, 478-487, LNM 721 (1979)
MEYER, Paul-André; STRICKER, Christophe; YOR, Marc
Sur une formule de la théorie du balayage (General theory of processes)
For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional
Comment: See 1351, 1357
Keywords: Balayage, Balayage formula
Nature: Original
Retrieve article from Numdam
XIII: 42, 488-489, LNM 721 (1979)
MEYER, Paul-André
Construction de semimartingales s'annulant sur un ensemble donné (General theory of processes)
The title states exactly the subject of this short report, whose conclusion is: in the Brownian filtration, every closed optional set is the set of zeros of a continuous semimartingale
Keywords: Semimartingales
Nature: Original
Retrieve article from Numdam
XIII: 50, 574-609, LNM 721 (1979)
JEULIN, Thierry
Grossissement d'une filtration et applications (General theory of processes, Markov processes)
This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths
Keywords: Enlargement of filtrations, Williams decomposition
Nature: Original
Retrieve article from Numdam
XIII: 51, 610-610, LNM 721 (1979)
STRICKER, Christophe
Encore une remarque sur la ``formule de balayage'' (General theory of processes)
A slight extension of 1341
Keywords: Balayage
Nature: Original
Retrieve article from Numdam
XIII: 55, 624-624, LNM 721 (1979)
YOR, Marc
Un exemple de J. Pitman (General theory of processes)
The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form
Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622
Keywords: Balayage, Balayage formula
Nature: Exposition
Retrieve article from Numdam
XIII: 57, 634-641, LNM 721 (1979)
EL KAROUI, Nicole
A propos de la formule d'Azéma-Yor (General theory of processes)
For the problem and notation, see the review of 1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)
Comment: This paper ends happily the whole series of papers on balayage in this volume
Keywords: Balayage, Balayage formula
Nature: Original
Retrieve article from Numdam
XIV: 16, 140-147, LNM 784 (1980)
ÉMERY, Michel
Métrisabilité de quelques espaces de processus aléatoires (General theory of processes, Stochastic calculus)
As a sequel to the main work of 1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete
Keywords: Spaces of semimartingales
Nature: Original
Retrieve article from Numdam
XIV: 18, 152-160, LNM 784 (1980)
ÉMERY, Michel
Compensation de processus à variation finie non localement intégrables (General theory of processes, Stochastic calculus)
First an example is given of a local martingale $M$ and an unbounded previsible process $H$ such that $H.M$ exists in the sense of 1126 and 1415, but is not a local martingale. This leads to a natural enlargement of the class of local martingales, which turns out to be the same suggested by Chou in 1322 under the name of class $(\Sigma_m)$. Once the class has been so extended, the operation of previsible compensation can be extended to a class of processes with finite variation, but not locally integrable variation, and the class of special semimartingales can be also enlarged
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997). Using the language of L. Schwartz 1530, it is the intersection of the set of (usual) semimartingales with the set of formal martingales
Keywords: Local martingales, Stochastic integrals, Compensators
Nature: Original
Retrieve article from Numdam
XIV: 19, 161-172, LNM 784 (1980)
JACOD, Jean
Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration (Stochastic calculus, General theory of processes)
Given a square integrable vector martingale $M$ and a previsible vector process $H$, the conditions implying the existence of the (scalar valued) stochastic integral $H.M$ are less restrictive than the existence of the ``componentwise'' stochastic integral, unless the components of $M$ are orthogonal (this result was due to Galtchouk, 1975). The theory of vector stochastic integrals, though parallel to the scalar theory, requires a careful theory given in this paper
Comment: Another approach, yielding an equivalent definition, is followed by L. Schwartz in his article 1530 on formal semimartingales
Keywords: Semimartingales, Stochastic integrals
Nature: Original
Retrieve article from Numdam
XIV: 20, 173-188, LNM 784 (1980)
MEYER, Paul-André
Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)
This is an introduction to beautiful results of Jeulin on enlargements, for which see Zeit. für W-Theorie, 52, 1980, and above all the Lecture Notes vol. 833, Semimartingales et grossissement d'une filtration
Comment: See also 1329, 1350
Keywords: Enlargement of filtrations, Semimartingales
Nature: Exposition
Retrieve article from Numdam
XIV: 21, 189-199, LNM 784 (1980)
YOR, Marc
Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)
The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment
Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')
Keywords: Enlargement of filtrations
Nature: Original
Retrieve article from Numdam
XIV: 22, 200-204, LNM 784 (1980)
AUERHAN, J.; LÉPINGLE, Dominique; YOR, Marc
Construction d'une martingale réelle continue de filtration naturelle donnée (General theory of processes)
It is proved in 1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed
Nature: Original
Retrieve article from Numdam
XIV: 23, 205-208, LNM 784 (1980)
SEYNOU, Aboubakary
Sur la compatibilité temporelle d'une tribu et d'une filtration discrète (General theory of processes)
Let us say that a $\sigma$-field ${\cal G}$ is embedded into a filtration $({\cal H}_n)$ if ${\cal G}= {\cal H}_T$ for some ${\cal H}$-stopping time $T$, that a filtration $({\cal F}_n)$ is embedded into $({\cal H}_n)$ if ${\cal F}_S$ can be embedded into $({\cal H}_n)$ for every ${\cal F}$-stopping time $S$, and finally that ${\cal G}$ and ${\cal F}$ or $({\cal F}_n)$ are compatible if they can be embedded into one single filtration $({\cal H}_n)$. Then the question investigated is whether the compatibility of ${\cal G}$ and the individual $\sigma$-fields ${\cal F}_n$ implies that of ${\cal G}$ and the whole filtration $({\cal F}_n)$
Comment: This problem arose from the spectral point of view on stochastic integration as in 1123
Keywords: Filtrations
Nature: Original
Retrieve article from Numdam
XIV: 27, 227-248, LNM 784 (1980)
JACOD, Jean; MÉMIN, Jean
Sur la convergence des semimartingales vers un processus à accroissements indépendants (General theory of processes, Stochastic calculus, Martingale theory)
A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)
Keywords: Convergence in law, Tightness
Nature: Original
Retrieve article from Numdam
XIV: 30, 255-255, LNM 784 (1980)
REBOLLEDO, Rolando
Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (General theory of processes, Martingale theory)
Concerns 1311. For the definitive version, see Mém. Soc. Math. France, 62, 1979
Keywords: Central limit theorem, Skorohod topology, Local martingales, Jumps
Nature: Correction
Retrieve article from Numdam
XIV: 31, 256-281, LNM 784 (1980)
FUJISAKI, Masatoshi
Contrôle stochastique continu et martingales (General theory of processes)
To be completed
Keywords: Control theory
Nature: Original
Retrieve article from Numdam
XIV: 34, 316-317, LNM 784 (1980)
ÉMERY, Michel
Une propriété des temps prévisibles (General theory of processes)
The idea is to prove that the general theory of processes on a random interval $[0,T[$, where $T$ is previsible, is essentially the same as on $[0,\infty[$. To this order, a continuous, strictly increasing adapted process $(A_t)$ is constructed, such that $A_0=0$, $A_T=1$
Keywords: Previsible times
Nature: Original
Retrieve article from Numdam
XIV: 35, 318-323, LNM 784 (1980)
ÉMERY, Michel
Annonçabilité des temps prévisibles. Deux contre-exemples (General theory of processes)
It is shown that two standard results on previsible stopping times on a probability space, namely that every previsible time $T$ can be ``foretold'' by a strictly increasing sequence $T_n\uparrow T$, and that the $T_n$ can themselves be taken previsible, become false if exceptional sets of measure zero are not allowed
Keywords: Previsible times
Nature: Original
Retrieve article from Numdam
XIV: 49, 500-546, LNM 784 (1980)
LENGLART, Érik
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
Retrieve article from Numdam
XV: 21, 290-306, LNM 850 (1981)
CHACON, Rafael V.; LE JAN, Yves; WALSH, John B.
Spatial trajectories (Markov processes, General theory of processes)
It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated
Comment: See Chacon-Jamison, Israel J. of M., 33, 1979
Keywords: Spatial trajectories
Nature: Original
Retrieve article from Numdam
XV: 22, 307-310, LNM 850 (1981)
LE JAN, Yves
Tribus markoviennes et prédiction (Markov processes, General theory of processes)
The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (1007) is used
Keywords: Prediction theory
Nature: Original
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XV: 23, 311-319, LNM 850 (1981)
ALDOUS, David J.; BARLOW, Martin T.
On countable dense random sets (General theory of processes, Point processes)
This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection
Keywords: Poisson point processes
Nature: Original
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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 theorems
Nature: Original
Retrieve article from Numdam
XV: 26, 351-370, LNM 850 (1981)
DELLACHERIE, Claude
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
Retrieve article from Numdam
XV: 30, 413-489, LNM 850 (1981)
SCHWARTZ, Laurent
Les semi-martingales formelles (Stochastic calculus, General theory of processes)
This is a natural development of the 1--1correspondence between semimartingales and $\sigma$-additive $L^0$-valued vector measures on the previsible $\sigma$-field, which satisfy a suitable boundedness property. What if boundedness is replaced by a $\sigma$-finiteness property? It turns out that these measures can be represented as formal stochastic integrals $H{\cdot} X$ where $X$ is a standard semimartingale, and $H$ is a (finitely valued, but possibly non-integrable) previsible process. The basic definition is quite elementary: $H{\cdot}X$ is an equivalence class of pairs $(H,X)$, where two pairs $(H,X)$ and $(K,Y)$ belong to the same class iff for some (hence for all) bounded previsible process $U>0$ such that $LH$ and $LK$ are bounded, the (usual) stochastic integrals $(UH){\cdot}X$ and $(UK){\cdot}Y$ are equal. (One may take for instance $U=1/(1{+}|H|{+}|K|)$.)\par As a consequence, the author gives an elegant and pedagogical characterization of the space $L(X)$ of all previsible processes integrable with respect to $X$ (introduced by Jacod, 1126; see also 1415, 1417 and 1424). This works just as well in the case when $X$ is vector-valued, and gives a new definition of vector stochastic integrals (see Galtchouk, Proc. School-Seminar Vilnius, 1975, and Jacod 1419). \par Some topological considerations (that can be skipped if the reader is not interested in convergences of processes) are delicate to follow, specially since the theory of unbounded vector measures (in non-locally convex spaces!) requires much care and is difficult to locate in the literature
Keywords: Semimartingales, Formal semimartingales, Stochastic integrals
Nature: Original
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XV: 34, 523-525, LNM 850 (1981)
STRICKER, Christophe
Sur la caractérisation des semi-martingales (General theory of processes, Stochastic calculus)
This is a sequel to the preceding paper 1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability
Keywords: Semimartingales, Spaces of semimartingales
Nature: Original
Retrieve article from Numdam
XV: 35, 526-528, LNM 850 (1981)
YOR, Marc
Sur certains commutateurs d'une filtration (General theory of processes)
Let $({\cal F}_t)$ be a filtration satisfying the usual conditions and ${\cal G}$ be a $\sigma$-field. Then the conditional expectation $E[.|{\cal G}]$ commutes with $E[.|{\cal F}_T]$ for all stopping times $T$ if and only if for some stopping time $S$ ${\cal G}$ lies between ${\cal F}_{S-}]$ and ${\cal F}_S]$
Keywords: Conditional expectations
Nature: Original
Retrieve article from Numdam
XV: 37, 547-560, LNM 850 (1981)
JACOD, Jean
Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)
The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets
Keywords: Semimartingales, Skorohod topology, Convergence in law
Nature: Original
Retrieve article from Numdam
XV: 38, 561-586, LNM 850 (1981)
PELLAUMAIL, Jean
Solutions faibles et semi-martingales (Stochastic calculus, General theory of processes)
From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of 1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions
Keywords: Stochastic differential equations, Weak solutions, Fuzzy random variables
Nature: Original
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XV: 41, 604-617, LNM 850 (1981)
LÉPINGLE, Dominique; MEYER, Paul-André; YOR, Marc
Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)
This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case
Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps
Nature: Original
Retrieve article from Numdam
XV: 42, 618-626, LNM 850 (1981)
ITMI, Mhamed
Processus ponctuels marqués stochastiques. Représentation des martingales et filtration naturelle quasicontinue à gauche (General theory of processes)
This paper contains a study of the filtration generated by a point process (multivariate: it takes values in a Polish space), and in particular of its quasi-left continuity, and previsible representation
Keywords: Point processes, Previsible representation
Nature: Original
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XV: 45, 643-668, LNM 850 (1981)
AUERHAN, J.; LÉPINGLE, Dominique
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)
This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given
Comment: Further results are given by Malric Ann. Inst. H. Poincaré 26 (1990)
Nature: Original
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XVI: 17, 213-218, LNM 920 (1982)
FALKNER, Neil; STRICKER, Christophe; YOR, Marc
Temps d'arrêt riches et applications (General theory of processes)
This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$
Keywords: Stopping times, Local times, Semimartingales, Previsible processes
Nature: Original
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XVI: 22, 248-256, LNM 920 (1982)
JEULIN, Thierry
Sur la convergence absolue de certaines intégrales (General theory of processes)
This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too
Comment: This paper greatly simplifies a proof in the author's Semimartingales et Grossissement de Filtrations, LNM 833, p.44
Keywords: Enlargement of filtrations
Nature: Original
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XVI: 26, 298-313, LNM 920 (1982)
DELLACHERIE, Claude; LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)
This paper is a sequel to 1524. Let $\Theta$ be a chronology, i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in 1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right
Keywords: Stopping times
Nature: Original
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XVI: 27, 314-318, LNM 920 (1982)
LENGLART, Érik
Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)
Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110
Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals
Nature: Original
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XVI: 30, 348-354, LNM 920 (1982)
HE, Sheng-Wu; WANG, Jia-Gang
The total continuity of natural filtrations (General theory of processes)
Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity
Keywords: Filtrations, Independent increments, Previsible representation, Total continuity, Lévy processes
Nature: Original
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XLIII: 14, 341-349, LNM 2006 (2011)
BOULEAU, Nicolas
The Lent Particle Method for Marked Point Processes (General theory of processes, Point processes)
Nature: Original