IV: 13, 151-161, LNM 124 (1970)
MAISONNEUVE, Bernard;
MORANDO, Philippe
Temps locaux pour les ensembles régénératifs (
Markov processes)
This paper uses the results of the preceding one
412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set
Comment: Same references as the preceding paper
412Keywords: Renewal theory,
Regenerative sets,
Local timesNature: Original
Retrieve article from Numdam
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 theoryNature: Original Retrieve article from Numdam
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 topologyNature: Original Retrieve article from Numdam
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,
IncursionsNature: Original Retrieve article from Numdam
IX: 32, 518-521, LNM 465 (1975)
MAISONNEUVE, Bernard;
MEYER, Paul-André
Ensembles aléatoires markoviens homogènes. Mise au point et compléments (
Markov processes)
This paper corrects or simplifies many details in the long paper
713 by the same authors
Comment: See also the next paper
933Keywords: Regenerative systems,
Last-exit decompositions,
ExcursionsNature: Original Retrieve article from Numdam
IX: 33, 522-529, LNM 465 (1975)
MAISONNEUVE, Bernard
Le comportement de dernière sortie (
Markov processes)
This paper contains improvements to the paper
813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned
Comment: More detailed versions of these results appear in Maisonneuve,
Ann. Prob.,
3, 1975,
Z. für W-theorie, 80, 1989, and in Chapter XX of Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel, Hermann 1992
Keywords: Regenerative systems,
Last-exit decompositions,
ExcursionsNature: Original Retrieve article from Numdam
XI: 16, 303-323, LNM 581 (1977)
BERNARD, Alain;
MAISONNEUVE, Bernard
Décomposition atomique de martingales de la classe $H^1$ (
Martingale theory)
Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (
Studia Math. 51, 1974). An atomic decomposition of a Banach space consists in finding simple elements (called atoms) in its unit ball, such that every element is a linear combination of atoms $\sum_n \lambda_n a_n$ with $\sum_n \|\lambda_n\|<\infty$, the infimum of this sum defining the norm or an equivalent one. Here an atomic decomposition is given for $H^1$ spaces of martingales in continuous time (defined by their maximal function). Atoms are of two kinds: the first kind consists of martingales bounded uniformly by a constant $c$ and supported by an interval $[T,\infty[$ such that $P\{T<\infty\}\le 1/c$. These atoms do not generate the whole space $H^1$ in general, though they do in a few interesting cases (if all martingales are continuous, or in the discrete dyadic case). To generate the whole space it is sufficient to add martingales of integrable variation (those whose total variation has an $L^1$ norm smaller than $1$ constitute the second kind of atoms). This approach leads to a proof of the $H^1$-$BMO$ duality and the Davis inequality
Comment: See also
1117Keywords: Atomic decompositions,
$H^1$ space,
$BMO$Nature: Original Retrieve article from Numdam
XI: 30, 435-445, LNM 581 (1977)
MAISONNEUVE, Bernard
Une mise au point sur les martingales locales continues définies sur un intervalle stochastique (
Martingale theory)
The following definition is given of a continuous local martingale $M$ on an open interval $[0,T[$, for an arbitrary stopping time $T$: two sequences are assumed to exist, one of stopping times $T_n\uparrow T$, one $(M_n)$ of continuous martingales, such that $M=M_n$ on $[0,T_n[$. Stochastic integration is studied, and the change of variable formula is extended. It is proved that the set where the limit $M_{T-}$ exists and is finite is a.s. the same as that where $\langle M,M\rangle_T<\infty$, a result whose proof under the usual definition (i.e., assuming $T$ is previsible) was not clear
Keywords: Martingales on a random set,
Stochastic integralsNature: Original Retrieve article from Numdam
XI: 37, 529-538, LNM 581 (1977)
MAISONNEUVE, Bernard
Changement de temps d'un processus markovien additif (
Markov processes)
A Markov additive process $(X_t,S_t)$ (Cinlar,
Z. für W-theorie, 24, 1972) is a generalisation of a pair $(X,S)$ where $X$ is a Markov process with arbitrary state space, and $S$ is an additive functional of $X$: in the general situation $S$ is positive real valued, $X$ is a Markov process in itself, and the pair $(X,S)$ is a Markov processes, while $S$ is an additive functional
of the pair. For instance, subordinators are Markov additive processes with trivial $X$. A simpler proof of a basic formula of Cinlar is given, and it is shown also that a Markov additive process gives rise to a regenerative system in a slightly extended sense
Comment: See also
1513Keywords: Markov additive processes,
Additive functionals,
Regenerative sets,
Lévy systemsNature: Original Retrieve article from Numdam
XIII: 58, 642-645, LNM 721 (1979)
MAISONNEUVE, Bernard
Martingales de valeur absolue donnée, d'après Protter-Sharpe (
Martingale theory)
The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (
Ann. Prob.,
7, 1979). This proof is further simplified and slightly generalized
Comment: See also
1407Keywords: Gilat's theoremNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 43, 410-417, LNM 784 (1980)
JACOD, Jean;
MAISONNEUVE, Bernard
Remarque sur les fonctionnelles additives non adaptées des processus de Markov (
Markov processes)
It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone
Keywords: Additive functionalsNature: Original Retrieve article from Numdam
XV: 13, 191-205, LNM 850 (1981)
MAISONNEUVE, Bernard
On Lévy's downcrossing theorem and various extensions (
Excursion theory)
Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in
1137. A gap in the related author's paper
Zeit. für W-Theorie, 52, 1980 is repaired at the end of the paper
Keywords: Excursions,
Lévy's downcrossing theorem,
Local times,
Regenerative systemsNature: Original Retrieve article from Numdam
XV: 25, 347-350, LNM 850 (1981)
MAISONNEUVE, Bernard
Surmartingales-mesures (
Martingale theory)
Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu,
Martingales à temps discret Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (
Zeit. für W-theorie, 1978) in continuous time. This topic is now called Kunita decomposition, see
1005 and the corresponding references
Keywords: Supermartingales,
Kunita decompositionNature: Original Retrieve article from Numdam
XX: 10, 95-100, LNM 1204 (1986)
KASPI, Haya;
MAISONNEUVE, Bernard
Predictable local times and exit systems Retrieve article from Numdam
XXVI: 15, 162-166, LNM 1526 (1992)
BOUTABIA, Hacène;
MAISONNEUVE, Bernard
Lois conditionnelles des excursions markoviennes Retrieve article from Numdam
XXXIII: 23, 405-409, LNM 1709 (1999)
CHRÉTIEN, Karl;
KURTZ, David;
MAISONNEUVE, Bernard
Processus gouvernés par des noyaux Retrieve article from Numdam
