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VIII: 01, 1-10, LNM 381 (1974)
AZÉMA, Jacques; MEYER, Paul-André
Une nouvelle représentation du type de Skorohod (Markov processes)
A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized left'' terminal time. A uniqueness result is proved
Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (Invent. Math. 18, 1973 and this volume, 814). A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding, Multiplicative functionals
Nature: Original
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VIII: 02, 11-19, LNM 381 (1974)
BRETAGNOLLE, Jean
Une remarque sur le problème de Skorohod (Brownian motion)
The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded
Comment: A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding
Nature: Original
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VIII: 03, 20-21, LNM 381 (1974)
CHUNG, Kai Lai
Note on last exit decomposition (Markov processes)
This is a useful complement to the monograph of Chung Lectures on Boundary Theory for Markov Chains, Annals of Math. Studies 65, Princeton 1970
Keywords: Markov chains
Nature: Original
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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: 06, 27-36, LNM 381 (1974)
DINGES, Hermann
Stopping sequences (Markov processes, Potential theory)
Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's filling scheme'', and several operations on stopping sequences, aiming at the construction of short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process
Comment: This is a development of the research of H.~Rost on the filling scheme'', for which see 523, 524, 612. This article contains announcements of further results
Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme
Nature: Original
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VIII: 07, 37-77, LNM 381 (1974)
DUPUIS, Claire
Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (Independent increments)
The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (J. Math. Mech., 16, 1967) whose methods are generalized here
Keywords: Hausdorff measures, Lévy processes
Nature: Original
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VIII: 08, 78-79, LNM 381 (1974)
FERNIQUE, Xavier
Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (Miscellanea)
The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper
Keywords: Stable measures
Nature: Original
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VIII: 09, 80-133, LNM 381 (1974)
GEBUHRER, Marc Olivier
Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)
The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (Arkiv för Math., 6, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups
Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces
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VIII: 10, 134-149, LNM 381 (1974)
KNIGHT, Frank B.
Existence of small oscillations at zeros of brownian motion (Brownian motion)
The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros
Comment: This result may be compared to Kahane, C.R. Acad. Sci. 248, 1974
Keywords: Law of the iterated logarithm, Local times
Nature: Original
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VIII: 11, 150-154, LNM 381 (1974)
HEATH, David C.
Skorohod stopping via Potential Theory (Potential theory, Markov processes)
The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $R^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool
Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding
Nature: Original
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VIII: 12, 155-171, LNM 381 (1974)
HEINKEL, Bernard
Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (Gaussian processes)
To be completed
Comment: To be completed
Keywords: Continuity of paths of Gaussian processes
Nature: Original
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VIII: 13, 172-261, LNM 381 (1974)
MAISONNEUVE, Bernard; MEYER, Paul-André
Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)
This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (Ann. Prob. 1, 1973; Indiana Math. J. 23, 1973). The basic and earlier paper of Dynkin on the same subject ( Teor. Ver. Prim. 16, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in 932-933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains
Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as Systèmes Régénératifs, Astérisque, 15, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer, Probabilités et Potentiel, Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard, Compactification et balayage de processus droits, Astérisque 21, 1975. A recent book on excursion theory is Blumenthal, Excursions of Markov Processes, Birkhäuser 1992
Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions
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
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VIII: 15, 289-289, LNM 381 (1974)
MEYER, Paul-André
Une note sur la compactification de Ray (Markov processes)
This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification
Comment: This is now considered a standard fact
Keywords: Ray compactification, Right processes
Nature: Original
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VIII: 16, 290-309, LNM 381 (1974)
MEYER, Paul-André
Noyaux multiplicatifs (Markov processes)
This paper presents results due to Jacod (Mém. Soc. Math. France, 35, 1973): given a pair $(X,Y)$ which jointly is a Markov process, and whose first component $X$ is a Markov process by itself, describe the conditional distribution of the joint path of $(X,Y)$ over a given path of $X$. These distributions constitute a multiplicative kernel, and attempts are made to regularize it
Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results
Keywords: Multiplicative kernels, Semimarkovian processes
Nature: Exposition
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VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma, Z. für W-theorie, 45, 1978 and its exposition 1321
Keywords: Supermartingales, Multiplicative decomposition
Nature: Original
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VIII: 18, 316-328, LNM 381 (1974)
PRIOURET, Pierre
Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)
The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic
Comment: To be completed
Keywords: Processes with jumps
Nature: Original
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VIII: 19, 329-343, LNM 381 (1974)
SMYTHE, Robert T.
Remarks on the hypotheses of duality (Markov processes)
To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh , Invent. Math., 19, 1973)
Comment: The independent paper Garsia Alvarez-Meyer Ann. Prob. 1, 1973, has some results in common with this one
Keywords: Dual semigroups
Nature: Original
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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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