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V: 17, 177-190, LNM 191 (1971)

**MEYER, Paul-André**

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (*Proc. Sixth Berkeley Symposium,* **3**, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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VIII: 20, 344-354, LNM 381 (1974)

**WALDENFELS, Wilhelm von**

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

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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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X: 01, 1-18, LNM 511 (1976)

**BRÉMAUD, Pierre**

La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (Martingale theory, Point processes)

This paper discusses martingale methods (as developed by Jacod,*Z. für W-theorie,* **31,** 1975) in the filtering theory of point processes

Comment: The author has greatly developed this topic in his book*Poisson Processes and Queues,* Springer 1981

Keywords: Point processes, Previsible representation, Filtering theory

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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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: 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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Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

Retrieve article from Numdam

VIII: 20, 344-354, LNM 381 (1974)

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

Retrieve article from Numdam

IX: 02, 97-153, LNM 465 (1975)

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 (

Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes

Nature: Original

Retrieve article from Numdam

IX: 06, 226-236, LNM 465 (1975)

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,

Keywords: Point processes, Previsible representation

Nature: Original

Retrieve article from Numdam

X: 01, 1-18, LNM 511 (1976)

La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (Martingale theory, Point processes)

This paper discusses martingale methods (as developed by Jacod,

Comment: The author has greatly developed this topic in his book

Keywords: Point processes, Previsible representation, Filtering theory

Nature: Original

Retrieve article from Numdam

X: 09, 118-124, LNM 511 (1976)

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,

Keywords: Point processes

Nature: Original

Retrieve article from Numdam

XV: 23, 311-319, LNM 850 (1981)

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

Retrieve article from Numdam

XV: 42, 618-626, LNM 850 (1981)

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

Retrieve article from Numdam