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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: 02, 19-23, LNM 511 (1976)

**CHACON, Rafael V.**; **WALSH, John B.**

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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X: 03, 24-39, LNM 511 (1976)

**JACOD, Jean**; **MÉMIN, Jean**

Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)

The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration

Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)

Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation

Nature: Original

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X: 04, 40-43, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in*Tôhoku Math. J.*, **28**, 1976

Keywords: Angle bracket, Inequalities

Nature: Original

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X: 05, 44-77, LNM 511 (1976)

**KUNITA, Hiroshi**

Absolute continuity for Markov processes (Markov processes)

This paper is devoted to a ``progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil,*Ann. Sci. ENS,* **5**, 1972); the ``Lebesgue decomposition'' of a Markov process with respect to another, and the fact that if absolute continuity holds on the germ field it also holds up to some maximal terminal time; a condition for this terminal time to be equal to the lifetime, under hypothesis (K)

Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the*Kunita decomposition,* and is not restricted to Markov processes. For the general case, see Yoeurp, in LN **1118**, *Grossissements de filtrations,* 1985

Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition

Nature: Original

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X: 06, 78-85, LNM 511 (1976)

**MANDREKAR, Vidyadhar**

Germ-field Markov property for multiparameter processes (Miscellanea)

The paper studies the relations between several Markov properties of a process indexed by an open set of $**R**^n$

Comment: To be completed

Keywords: Several parameter Brownian motions, Several parameter processes

Nature: Original

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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 theory

Nature: Exposition, Original additions

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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 fields

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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X: 10, 125-183, LNM 511 (1976)

**MEYER, Paul-André**

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book*Topics in Harmonic Analysis related to the Littlewood-Paley theory,* Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $**R**^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $**R**^n\times **R**_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $**R**^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (*CRAS Paris,* **278A**, 1974, p.1103) in potential theory and by Kunita (*Nagoya M. J.*, **36**, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false

Comment: This paper was rediscovered by Varopoulos (*J. Funct. Anal.*, **38**, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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X: 11, 184-193, LNM 511 (1976)

**NAGASAWA, Masao**

A probabilistic approach to non-linear Dirichlet problem (Markov processes)

The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (*J. Math. Kyoto Univ.*, **8**, 1968 and **9**, 1969) and Nagasawa (*Kodai Math. Sem. Rep.* **20**, 1968) leads to the probabilistic solution of a non-linear Dirichlet problem

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 926

Keywords: Branching processes, Dirichlet problem

Nature: Original

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X: 12, 194-208, LNM 511 (1976)

**ROST, Hermann**

Skorohod stopping times of minimal variance (Markov processes)

Root's (*Ann. Math. Stat.*, **40**, 1969) solution of the Shorohod imbedding problem for Brownian motion uses the hitting time of a barrier in space-time. Here Root's construction is extended to general Markov processes, an optimality property of Root's stopping times is proved, as well as the uniqueness of such stopping times

Comment: For previous work of the author on Skorohod's imbedding see*Ann. M. Stat.* **40**, 1969 and *Invent. Math.* **14**, 1971, and in this Seminar 523, 613, 806. A general survey on the Skorohod embedding problem is Ob\lój, *Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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X: 13, 209-215, LNM 511 (1976)

**SEKIGUCHI, Takesi**

On the Krickeberg decomposition of continuous martingales (Martingale theory)

The problem investigated is whether the two positive martingales occurring in the Krickeberg decomposition of a $L^1$-bounded continuous martingale of a filtration $({\cal F}_t)$ are themselves continuous. It is shown that the answer is yes only under very stringent conditions: there exists a sub-filtration $({\cal G}_t)$ such that 1) all ${\cal G}$-martingales are continuous 2) the continuous ${\cal F}$-martingales are exactly the ${\cal G}$-martingales

Comment: For related work of the author see*Tôhoku Math. J.* **28**, 1976

Keywords: Continuous martingales, Krickeberg decomposition

Nature: Original

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X: 14, 216-234, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams*Diffusions, Markov Processes and Martingales,* vol. 1 (second edition), Wiley 1994. See also 1024

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

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X: 15, 235-239, LNM 511 (1976)

**WILLIAMS, David**

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see*Ann. Prob.* **3**, 1975. For modern references, we should ask Yor

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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X: 16, 240-244, LNM 511 (1976)

**YAMADA, Toshio**

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,*J. Math. Kyoto Univ.*, **11**, 1971. Many of these theorems have now simpler proofs using local times, in the spirit of Revuz-Yor, *Continuous Martingales and Brownian Motion,* Chapter IX

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

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X: 17, 245-400, LNM 511 (1976)

**MEYER, Paul-André**

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 18, 401-413, LNM 511 (1976)

**PRATELLI, Maurizio**

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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X: 19, 414-421, LNM 511 (1976)

**PRATELLI, Maurizio**

Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)

This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)

Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals

Nature: Original

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X: 20, 422-431, LNM 511 (1976)

**YAN, Jia-An**; **YOEURP, Chantha**

Représentation des martingales comme intégrales stochastiques des processus optionnels (Martingale theory, Stochastic calculus)

An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one

Comment: Apparently this ``optional representation property'' has not been used since

Keywords: Optional stochastic integrals

Nature: Original

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X: 21, 432-480, LNM 511 (1976)

**YOEURP, Chantha**

Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)

It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality

Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN**1118**, *Grossissements de filtrations,* 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer 1023. The general case is due to Azéma (*Z. für W-theorie,* **45,** 1978, presented in 1321) See also 1622

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

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X: 22, 481-500, LNM 511 (1976)

**YOR, Marc**

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

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X: 23, 501-504, LNM 511 (1976)

**MEYER, Paul-André**; **YOEURP, Chantha**

Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)

This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$

Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620

Keywords: Multiplicative decomposition

Nature: Original

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X: 24, 505-520, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

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X: 25, 521-531, LNM 511 (1976)

**BENVENISTE, Albert**

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in*Ann. Inst. Fourier,* **25**, 1975. See also 1105

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

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X: 26, 532-535, LNM 511 (1976)

**NAGASAWA, Masao**

Note on pasting of two Markov processes (Markov processes)

The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word ``roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret,*Publ. Inst. Stat. Univ. Paris,* **14**, 1965. Here it is reduced to a ``revival theorem'' of Ikeda-Nagasawa-Watanabe, *J. Math. Kyoto Univ.*, **8**, 1968

Comment: The piecing out theorem is also reduced to a revival theorem in Meyer,*Ann. Inst. Fourier,* **25**,1975

Keywords: Piecing-out theorem

Nature: Original

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X: 27, 536-539, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN**1579**, 1994

Keywords: $BMO$

Nature: Original

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X: 28, 540-543, LNM 511 (1976)

**MOKOBODZKI, Gabriel**

Démonstration élémentaire d'un théorème de Novikov (Descriptive set theory)

Novikov's theorem asserts that any sequence of analytic subsets of a compact metric space with empty intersection can be enclosed in a sequence of Borel sets with empty intersection. This result has important consequences in descriptive set theory (see Dellacherie 915). A fairly simple proof of this theorem is given, which relates it to the first separation theorem (rather than the second separation theorem as it used to be)

Comment: Dellacherie in this volume (1032) further simplifies the proof. For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **9**

Keywords: Analytic sets

Nature: Original

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X: 29, 544-544, LNM 511 (1976)

**DELLACHERIE, Claude**

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

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X: 30, 545-577, LNM 511 (1976)

**DELLACHERIE, Claude**

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**. The hypothesis that the space is compact is sometimes troublesome for the applications

Keywords: Pseudo-kernels, Regularization

Nature: Original

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X: 31, 578-578, LNM 511 (1976)

**MEYER, Paul-André**

Un point de priorité (Ergodic theory)

An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)

Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem

Nature: Acknowledgment

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X: 32, 579-593, LNM 511 (1976)

**DELLACHERIE, Claude**

Compléments aux exposés sur les ensembles analytiques (Descriptive set theory)

A new proof of Novikov's theorem (see 1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to 919 and 920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo

Keywords: Analytic sets, Section theorems, Capacities

Nature: Original

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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

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X: 02, 19-23, LNM 511 (1976)

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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X: 03, 24-39, LNM 511 (1976)

Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)

The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration

Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)

Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation

Nature: Original

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X: 04, 40-43, LNM 511 (1976)

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in

Keywords: Angle bracket, Inequalities

Nature: Original

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X: 05, 44-77, LNM 511 (1976)

Absolute continuity for Markov processes (Markov processes)

This paper is devoted to a ``progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil,

Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the

Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition

Nature: Original

Retrieve article from Numdam

X: 06, 78-85, LNM 511 (1976)

Germ-field Markov property for multiparameter processes (Miscellanea)

The paper studies the relations between several Markov properties of a process indexed by an open set of $

Comment: To be completed

Keywords: Several parameter Brownian motions, Several parameter processes

Nature: Original

Retrieve article from Numdam

X: 07, 86-103, LNM 511 (1976)

La théorie de la prédiction de F. Knight (General theory of processes)

This paper is devoted to the work of Knight,

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

Keywords: Prediction theory

Nature: Exposition, Original additions

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X: 08, 104-117, LNM 511 (1976)

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,

Keywords: Prediction theory, Germ fields

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

X: 10, 125-183, LNM 511 (1976)

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book

Comment: This paper was rediscovered by Varopoulos (

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

Retrieve article from Numdam

X: 11, 184-193, LNM 511 (1976)

A probabilistic approach to non-linear Dirichlet problem (Markov processes)

The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 926

Keywords: Branching processes, Dirichlet problem

Nature: Original

Retrieve article from Numdam

X: 12, 194-208, LNM 511 (1976)

Skorohod stopping times of minimal variance (Markov processes)

Root's (

Comment: For previous work of the author on Skorohod's imbedding see

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

X: 13, 209-215, LNM 511 (1976)

On the Krickeberg decomposition of continuous martingales (Martingale theory)

The problem investigated is whether the two positive martingales occurring in the Krickeberg decomposition of a $L^1$-bounded continuous martingale of a filtration $({\cal F}_t)$ are themselves continuous. It is shown that the answer is yes only under very stringent conditions: there exists a sub-filtration $({\cal G}_t)$ such that 1) all ${\cal G}$-martingales are continuous 2) the continuous ${\cal F}$-martingales are exactly the ${\cal G}$-martingales

Comment: For related work of the author see

Keywords: Continuous martingales, Krickeberg decomposition

Nature: Original

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X: 14, 216-234, LNM 511 (1976)

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

Retrieve article from Numdam

X: 15, 235-239, LNM 511 (1976)

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

Retrieve article from Numdam

X: 16, 240-244, LNM 511 (1976)

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

Retrieve article from Numdam

X: 17, 245-400, LNM 511 (1976)

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 18, 401-413, LNM 511 (1976)

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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X: 19, 414-421, LNM 511 (1976)

Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)

This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)

Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

X: 20, 422-431, LNM 511 (1976)

Représentation des martingales comme intégrales stochastiques des processus optionnels (Martingale theory, Stochastic calculus)

An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one

Comment: Apparently this ``optional representation property'' has not been used since

Keywords: Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

X: 21, 432-480, LNM 511 (1976)

Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)

It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality

Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

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X: 22, 481-500, LNM 511 (1976)

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

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X: 23, 501-504, LNM 511 (1976)

Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)

This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$

Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620

Keywords: Multiplicative decomposition

Nature: Original

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X: 24, 505-520, LNM 511 (1976)

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

X: 25, 521-531, LNM 511 (1976)

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

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X: 26, 532-535, LNM 511 (1976)

Note on pasting of two Markov processes (Markov processes)

The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word ``roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret,

Comment: The piecing out theorem is also reduced to a revival theorem in Meyer,

Keywords: Piecing-out theorem

Nature: Original

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X: 27, 536-539, LNM 511 (1976)

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN

Keywords: $BMO$

Nature: Original

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X: 28, 540-543, LNM 511 (1976)

Démonstration élémentaire d'un théorème de Novikov (Descriptive set theory)

Novikov's theorem asserts that any sequence of analytic subsets of a compact metric space with empty intersection can be enclosed in a sequence of Borel sets with empty intersection. This result has important consequences in descriptive set theory (see Dellacherie 915). A fairly simple proof of this theorem is given, which relates it to the first separation theorem (rather than the second separation theorem as it used to be)

Comment: Dellacherie in this volume (1032) further simplifies the proof. For a presentation in book form, see Dellacherie-Meyer,

Keywords: Analytic sets

Nature: Original

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X: 29, 544-544, LNM 511 (1976)

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

Retrieve article from Numdam

X: 30, 545-577, LNM 511 (1976)

Sur la construction de noyaux boréliens (Measure theory)

This answers questions of Getoor 923 and Meyer 924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest

Comment: For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

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X: 31, 578-578, LNM 511 (1976)

Un point de priorité (Ergodic theory)

An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)

Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem

Nature: Acknowledgment

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X: 32, 579-593, LNM 511 (1976)

Compléments aux exposés sur les ensembles analytiques (Descriptive set theory)

A new proof of Novikov's theorem (see 1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to 919 and 920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo

Keywords: Analytic sets, Section theorems, Capacities

Nature: Original

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