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 timesNature: Exposition,
Original additions
Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
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 theoryNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 processesNature: 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 representationNature: Original Retrieve article from Numdam
