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XI: 01, 1-20, LNM 581 (1977)

**AVANISSIAN, Vazgain**

Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $**R**^n$, $n\ge1$: 1) Harmonic functions of infinite order (see Avanissian and Fernique, *Ann. Inst. Fourier,* **18-2**, 1968), which are $C^\infty$ functions satisfying a growth condition on their iterated laplacians, and are shown to be real analytic. 2) Infinitely differentiable functions (or distributions) similar to completely monotonic functions on the line, i.e., whose iterated laplacians are alternatively positive and negative (they were introduced by Lelong). Among the results is the fact that the second class is included in the first

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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XI: 02, 21-26, LNM 581 (1977)

**BENVENISTE, Albert**

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of*medial limits * due to Mokobodzki (see 719), which in fact was developed precisely at the author's request to solve this problem

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

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XI: 03, 27-33, LNM 581 (1977)

**CHUNG, Kai Lai**

Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $**R**^n$, and let $z$ be a boundary point. Then a barrier at $z$ is a superharmonic function in $D$, strictly positive and with a limit equal to $0$ at $z$. The barrier theorem asserts that if there is a barrier at $z$, then $z$ is regular. An elegant proof of this is given using Brownian motion. Then it is shown that the expectation of $S$, the hitting time of $D^c$, is bounded, upper semi-continuous in $R^n$ and continuous in $D$, and is a barrier at every regular point

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

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XI: 04, 34-46, LNM 581 (1977)

**DELLACHERIE, Claude**

Les dérivations en théorie descriptive des ensembles et le théorème de la borne (Descriptive set theory)

At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given

Comment: See a correction in 1241, and several examples in Hillard 1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,*Probabilités et Potentiel*

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 05, 47-50, LNM 581 (1977)

**DELLACHERIE, Claude**

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,*Ann. Inst. Fourier,* **25**, 1975. See also Benveniste, 1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed

Keywords: Optional processes, Separability, Changes of time

Nature: Original

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XI: 06, 51-58, LNM 581 (1977)

**DUDLEY, Richard M.**; **GUTMANN, Sam**

Stopping times with given laws (General theory of processes)

Given a filtration ${\cal A}_t$ such that for all $t>0$ the law $P$ restricted to ${\cal A}_t$ is non-atomic, then for every law $\mu$ on the half-line there exists a stopping time with law $\mu$. The proof uses an interesting measure theoretic lemma on decreasing sequences of non-atomic $\sigma$-fields

Comment: It follows that the Brownian filtration contains a process whose law is that of a Poisson process (though of course it is not a Poisson process*of the Brownian filtration *)

Keywords: Stopping

Nature: Original

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XI: 07, 59-64, LNM 581 (1977)

**HOROWITZ, Joseph**

Une remarque sur les bimesures (Measure theory)

A bimeasure is a function $\beta(A,B)$ of two set variables, which is a measure in each variable when the other is kept fixed. It is important to have conditions under which a bimeasure ``is'' a measure, i.e., is of the form $\mu(A\times B)$ for some measure $\mu$ on the product space. This is known to be true for positive bimeasures ( Kingman,*Pacific J. of Math.,* **21**, 1967, see also 315). Here a condition of bounded variation is given, which implies that a bimeasure is a difference of two positive bimeasures, and therefore is a measure

Comment: Signed bimeasures which are not measures occur naturally, see for instance Bakry 1742, and Émery-Stricker on Gaussian semimartingales

Keywords: Bimeasures

Nature: Original

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XI: 08, 65-78, LNM 581 (1977)

**EL KAROUI, Nicole**; **MEYER, Paul-André**

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

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XI: 09, 79-108, LNM 581 (1977)

**EL KAROUI, Nicole**; **WEIDENFELD, Gérard**

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

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XI: 10, 109-119, LNM 581 (1977)

**MEYER, Paul-André**

Convergence faible de processus, d'après Mokobodzki (General theory of processes)

The following simple question of Benveniste was answered positively by Mokobodzki (*Séminaire de Théorie du Potentiel,* Lect. Notes in M. 563, 1976): given a sequence $(U^n)$ of optional processes such that $U^n_T$ converges weakly in $L^1$ for every stopping time $T$, does there exist an optional process $U$ such that $U^n_T$ converges to $U_T$? The proof is rather elaborate

Keywords: Weak convergence in $L^1$

Nature: Exposition

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XI: 11, 120-131, LNM 581 (1977)

**MEYER, Paul-André**

Résultats récents de A. Benveniste en théorie des flots (Ergodic theory)

A filtered flow is said to be diffuse if there exists a r.v. ${\cal F}_0$-measurable $J$ such that given any ${\cal F}_0$-measurable r.v.'s $T$ and $H$, $P\{J\circ\theta_T=H, 0<T<\infty\}=0$. The main result of the paper is the fact that a diffuse flow contains all Lévy flows (flows of increments of Lévy processes, no invariant measure is involved). In particular, the Brownian flow contains a Poisson counter

Comment: This result on the whole line is similar to 1106, which concerns a half-line. The original paper of Benveniste appeared in*Z. für W-theorie,* **41**, 1977/78

Keywords: Filtered flows, Poisson flow

Nature: Exposition

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XI: 12, 132-195, LNM 581 (1977)

**MEYER, Paul-André**

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(**R**^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XI: 13, 196-256, LNM 581 (1977)

**WEBER, Michel**

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in*Math. Reviews,* **56**, **13343**

Nature: Exposition, Original additions

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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 theory

Nature: Original

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XI: 15, 298-302, LNM 581 (1977)

**ZANZOTTO, Pio Andrea**

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

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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 1117

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 17, 324-326, LNM 581 (1977)

**BERNARD, Alain**

Complément à l'exposé précédent (Martingale theory)

This paper is a sequel to 1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 18, 327-339, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in*Acta Math.* **4**, 1975. These processes have stochastic integral representations along (increasing) paths, with a common kernel called their derivative. Under an integrability restriction, a process holomorphic in a region of the plane is shown to be extendable as a holomorphic process to a larger region of a canonical shape (intersection of a rectangle and a disk centered at the origin)

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 19, 340-348, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in*Ann. Prob.* **5**, 1971 for additional results

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 20, 349-355, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

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XI: 21, 356-361, LNM 581 (1977)

**CHOU, Ching Sung**

Le processus des sauts d'une martingale locale (Martingale theory)

Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$

Comment: The same result is proved independently by D. Lépingle in this volume, see 1129. For an application see 1308, and for another approach 1335

Keywords: Local martingales, Jumps

Nature: Original

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XI: 22, 362-364, LNM 581 (1977)

**DELLACHERIE, Claude**

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

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XI: 23, 365-375, LNM 581 (1977)

**DELLACHERIE, Claude**; **STRICKER, Christophe**

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

Nature: Original

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XI: 24, 376-382, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in*Zeit. für W-theorie,* **36**, 1976 and by Protter in *Ann. Prob.* **5**, 1977. The theory has become now so classical that the paper has only historical interest

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XI: 25, 383-389, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

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XI: 26, 390-410, LNM 581 (1977)

**JACOD, Jean**

Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (Martingale theory)

This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (1121, 1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined

Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,*Calcul stochastique et problèmes de martingales,* Springer 1979. Other, equivalent, definitions are given in 1415, 1417, 1424 and 1530

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

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XI: 27, 411-414, LNM 581 (1977)

**KOSKAS, Maurice**

Images d'équations différentielles stochastiques (Stochastic calculus)

This paper answers a natural question: can one take computations performed on ``canonical'' versions of processes back to their original spaces? It is related to Stricker's work (*Zeit. für W-theorie,* **39**, 1977) on the restriction of filtrations

Keywords: Stochastic differential equations

Nature: Original

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XI: 28, 415-417, LNM 581 (1977)

**LENGLART, Érik**

Une caractérisation des processus prévisibles (General theory of processes)

One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale

Keywords: Previsible processes

Nature: Original

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XI: 29, 418-434, LNM 581 (1977)

**LÉPINGLE, Dominique**

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

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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 integrals

Nature: Original

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XI: 31, 446-481, LNM 581 (1977)

**MEYER, Paul-André**

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

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XI: 32, 482-489, LNM 581 (1977)

**MEYER, Paul-André**

Sur un théorème de C. Stricker (Martingale theory)

Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex

Comment: A gap in a proof is filled in 1251

Keywords: Hardy spaces, Changes of measure

Nature: Original

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XI: 33, 490-492, LNM 581 (1977)

**WALSH, John B.**

A property of conformal martingales (Martingale theory)

Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense

Comment: See also 1408

Keywords: Conformal martingales

Nature: Original

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XI: 34, 493-501, LNM 581 (1977)

**YOR, Marc**

A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)

Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales

Keywords: Yoeurp's lemma, Square bracket

Nature: Original

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XI: 35, 502-517, LNM 581 (1977)

**YOR, Marc**

Remarques sur la représentation des martingales comme intégrales stochastiques (Martingale theory)

The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor,*Z. für W-theorie,* **38**, 1977. Several concrete applications are given here, in particular a complete proof of a ``folklore'' result on the representation of local martingales of a Lévy process, and a discussion of the commutation problem of 1123

Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see 1221

Keywords: Previsible representation, Extreme points, Independent increments, Lévy processes

Nature: Original

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XI: 36, 518-528, LNM 581 (1977)

**YOR, Marc**

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe*Publ. RIMS, Kyoto Univ.* **15**, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer 1505

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

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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 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

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XI: 38, 539-565, LNM 581 (1977)

**TORTRAT, Albert**

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

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XI: 39, 566-573, LNM 581 (1977)

**ÉMERY, Michel**

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (*Comm. Pure and Appl. Math.*, 1975)

Nature: Exposition

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Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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XI: 02, 21-26, LNM 581 (1977)

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

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XI: 03, 27-33, LNM 581 (1977)

Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

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XI: 04, 34-46, LNM 581 (1977)

Les dérivations en théorie descriptive des ensembles et le théorème de la borne (Descriptive set theory)

At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given

Comment: See a correction in 1241, and several examples in Hillard 1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 05, 47-50, LNM 581 (1977)

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,

Keywords: Optional processes, Separability, Changes of time

Nature: Original

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XI: 06, 51-58, LNM 581 (1977)

Stopping times with given laws (General theory of processes)

Given a filtration ${\cal A}_t$ such that for all $t>0$ the law $P$ restricted to ${\cal A}_t$ is non-atomic, then for every law $\mu$ on the half-line there exists a stopping time with law $\mu$. The proof uses an interesting measure theoretic lemma on decreasing sequences of non-atomic $\sigma$-fields

Comment: It follows that the Brownian filtration contains a process whose law is that of a Poisson process (though of course it is not a Poisson process

Keywords: Stopping

Nature: Original

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XI: 07, 59-64, LNM 581 (1977)

Une remarque sur les bimesures (Measure theory)

A bimeasure is a function $\beta(A,B)$ of two set variables, which is a measure in each variable when the other is kept fixed. It is important to have conditions under which a bimeasure ``is'' a measure, i.e., is of the form $\mu(A\times B)$ for some measure $\mu$ on the product space. This is known to be true for positive bimeasures ( Kingman,

Comment: Signed bimeasures which are not measures occur naturally, see for instance Bakry 1742, and Émery-Stricker on Gaussian semimartingales

Keywords: Bimeasures

Nature: Original

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XI: 08, 65-78, LNM 581 (1977)

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

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XI: 09, 79-108, LNM 581 (1977)

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

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XI: 10, 109-119, LNM 581 (1977)

Convergence faible de processus, d'après Mokobodzki (General theory of processes)

The following simple question of Benveniste was answered positively by Mokobodzki (

Keywords: Weak convergence in $L^1$

Nature: Exposition

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XI: 11, 120-131, LNM 581 (1977)

Résultats récents de A. Benveniste en théorie des flots (Ergodic theory)

A filtered flow is said to be diffuse if there exists a r.v. ${\cal F}_0$-measurable $J$ such that given any ${\cal F}_0$-measurable r.v.'s $T$ and $H$, $P\{J\circ\theta_T=H, 0<T<\infty\}=0$. The main result of the paper is the fact that a diffuse flow contains all Lévy flows (flows of increments of Lévy processes, no invariant measure is involved). In particular, the Brownian flow contains a Poisson counter

Comment: This result on the whole line is similar to 1106, which concerns a half-line. The original paper of Benveniste appeared in

Keywords: Filtered flows, Poisson flow

Nature: Exposition

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XI: 12, 132-195, LNM 581 (1977)

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XI: 13, 196-256, LNM 581 (1977)

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in

Nature: Exposition, Original additions

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XI: 14, 257-297, LNM 581 (1977)

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 theory

Nature: Original

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XI: 15, 298-302, LNM 581 (1977)

Sur l'existence d'un noyau induisant un opérateur sous markovien donné (Measure theory)

The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$

Comment: See 923, 924, 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

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XI: 16, 303-323, LNM 581 (1977)

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 (

Comment: See also 1117

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 17, 324-326, LNM 581 (1977)

Complément à l'exposé précédent (Martingale theory)

This paper is a sequel to 1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

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XI: 18, 327-339, LNM 581 (1977)

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 19, 340-348, LNM 581 (1977)

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 20, 349-355, LNM 581 (1977)

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

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XI: 21, 356-361, LNM 581 (1977)

Le processus des sauts d'une martingale locale (Martingale theory)

Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$

Comment: The same result is proved independently by D. Lépingle in this volume, see 1129. For an application see 1308, and for another approach 1335

Keywords: Local martingales, Jumps

Nature: Original

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XI: 22, 362-364, LNM 581 (1977)

Sur la régularisation des surmartingales (Martingale theory)

It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)

Comment: See 1524

Keywords: General filtrations, Strong supermartingales

Nature: Original

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XI: 23, 365-375, LNM 581 (1977)

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

Nature: Original

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XI: 24, 376-382, LNM 581 (1977)

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XI: 25, 383-389, LNM 581 (1977)

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

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XI: 26, 390-410, LNM 581 (1977)

Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (Martingale theory)

This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (1121, 1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined

Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

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XI: 27, 411-414, LNM 581 (1977)

Images d'équations différentielles stochastiques (Stochastic calculus)

This paper answers a natural question: can one take computations performed on ``canonical'' versions of processes back to their original spaces? It is related to Stricker's work (

Keywords: Stochastic differential equations

Nature: Original

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XI: 28, 415-417, LNM 581 (1977)

Une caractérisation des processus prévisibles (General theory of processes)

One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale

Keywords: Previsible processes

Nature: Original

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XI: 29, 418-434, LNM 581 (1977)

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

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XI: 30, 435-445, LNM 581 (1977)

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 integrals

Nature: Original

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XI: 31, 446-481, LNM 581 (1977)

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

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XI: 32, 482-489, LNM 581 (1977)

Sur un théorème de C. Stricker (Martingale theory)

Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex

Comment: A gap in a proof is filled in 1251

Keywords: Hardy spaces, Changes of measure

Nature: Original

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XI: 33, 490-492, LNM 581 (1977)

A property of conformal martingales (Martingale theory)

Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense

Comment: See also 1408

Keywords: Conformal martingales

Nature: Original

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XI: 34, 493-501, LNM 581 (1977)

A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)

Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales

Keywords: Yoeurp's lemma, Square bracket

Nature: Original

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XI: 35, 502-517, LNM 581 (1977)

Remarques sur la représentation des martingales comme intégrales stochastiques (Martingale theory)

The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor,

Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see 1221

Keywords: Previsible representation, Extreme points, Independent increments, Lévy processes

Nature: Original

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XI: 36, 518-528, LNM 581 (1977)

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

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XI: 37, 529-538, LNM 581 (1977)

Changement de temps d'un processus markovien additif (Markov processes)

A Markov additive process $(X_t,S_t)$ (Cinlar,

Comment: See also 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

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XI: 38, 539-565, LNM 581 (1977)

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

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XI: 39, 566-573, LNM 581 (1977)

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (

Nature: Exposition

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