VI: 09, 109-112, LNM 258 (1972)
SAM LAZARO, José de;
MEYER, Paul-André
Un gros processus de Markov. Application à certains flots (
Markov processes)
In a vague but useful sense, a ``big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (
Trans. Amer. Math. Soc.,
131, 1968)
Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer,
Probabilités et Potentiel, \no XIV.45
Keywords: Prediction theory,
Filtered flowsNature: Original
Retrieve article from Numdam
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
1007Keywords: Previsible projections,
Optional projections,
Prediction theoryNature: Exposition,
Original additions Retrieve article from Numdam
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 theoryNature: Exposition,
Original additions Retrieve article from Numdam
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 fieldsNature: Original Retrieve article from Numdam
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 theoryNature: 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 theoryNature: 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 theoryNature: Original Retrieve article from Numdam
