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XII: 01, 1-19, LNM 649 (1978)

**PRATELLI, Maurizio**

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,*Comm Pure Appl. Math.*, **26**, 1973

Keywords: Interpolation, $BMO$

Nature: Original

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XII: 02, 20-21, LNM 649 (1978)

**STRICKER, Christophe**

Une remarque sur les changements de temps et les martingales locales (Martingale theory)

It is well known (see 606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)

Keywords: Changes of time, Weak martingales

Nature: Original

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XII: 03, 22-34, LNM 649 (1978)

**JACOD, Jean**

Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)

The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed

Comment: This papers improves on 1021 and 1023

Keywords: Semimartingales, Multiplicative decomposition

Nature: Original

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XII: 04, 35-46, LNM 649 (1978)

**MÉMIN, Jean**

Décompositions multiplicatives de semimartingales exponentielles et applications (General theory of processes)

It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions

Comment: The results of this paper are used in Mémin-Shiryaev 1312

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition

Nature: Original

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XII: 05, 47-50, LNM 649 (1978)

**KAZAMAKI, Norihiko**

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XII: 06, 51-52, LNM 649 (1978)

**GARCIA, M.**; **MAILLARD, P.**; **PELTRAUT, Y.**

Une martingale de saut multiplicatif donné (Martingale theory)

Given a totally inaccessible stopping time $T$, it is shown how to construct a strictly positive martingale $M$ with $M_0=1$, such that its only jump occurs at time $T$ and $M_T/M_{T-}=K$, a strictly positive constant

Comment: See also 1308

Keywords: Totally inaccessible stopping times

Nature: Original

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XII: 07, 53-56, LNM 649 (1978)

**LENGLART, Érik**

Sur la localisation des intégrales stochastiques (Stochastic calculus)

A mapping $T$ from processes to processes is*local * if, whenever two processes $X,Y$ are equal on an event $A\subset\Omega$, the same is true for $TX,TY$. Classical results on locality in stochastic calculus are derived here in a simple way from the generalized Girsanov theorem (which concerns a pair of laws $P,Q$ with $Q$ absolutely continuous with respect to $P$, but not necessarily equivalent to it: see Lenglart, *Zeit. für W-theorie,* 39, 1977). A new result is derived: if $X$ and $Y$ are semimartingales and their difference is of finite variation on an event $A$, then their continuous martingale parts are equal on $A$

Keywords: Girsanov's theorem

Nature: Original

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XII: 08, 57-60, LNM 649 (1978)

**MEYER, Paul-André**

Sur un théorème de J. Jacod (General theory of processes)

Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals

Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration

Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales

Nature: Original

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XII: 09, 61-69, LNM 649 (1978)

**YOR, Marc**

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called*progressively enlarged * filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is *honest * for $\cal F$, i.e., is the end of an $\cal F$-optional set

Comment: This result was independently discovered by Barlow,*Zeit. für W-theorie,* 44, 1978, which also has a huge intersection with 1211. Complements are given in 1210, and an explicit decomposition formula for semimartingales in 1211

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 10, 70-77, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 11, 78-97, LNM 649 (1978)

**JEULIN, Thierry**; **YOR, Marc**

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes**833** (T. Jeulin) and **1118**. See also 1350

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**; **YOR, Marc**

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 13, 114-131, LNM 649 (1978)

**YAMADA, Toshio**

Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien (Stochastic calculus)

The results of this paper improve on those of the author's paper (*Zeit. für W-theorie,* **36**, 1976) concerning a one-dimensional stochastic differential equations of the classical Ito type, whose coefficients satisfy a Hölder-like condition instead of the standard Lipschitz condition. The proofs are simplified, and strong convergence of the Cauchy method is shown

Comment: Such equations play an important role in the theory of Bessel processes (see chapter XI of Revuz-Yor,*Continuous Martingales and Brownian Motion,* Springer 1999

Keywords: Stochastic differential equations, Hölder conditions

Nature: Original

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XII: 14, 132-133, LNM 649 (1978)

**CHOU, Ching Sung**

Extension au cas continu d'un théorème de Dubins (Martingale theory)

After a suitable translation, Dubins' theorem can be stated as follows: if $X$ is a positive submartingale with $X_t\in L^2$ for all $t$, then $X^2-[X,X]$ is a submartingale

Keywords: Submartingales

Nature: Original

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XII: 15, 134-137, LNM 649 (1978)

**LÉPINGLE, Dominique**

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

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XII: 16, 138-147, LNM 649 (1978)

**LÉPINGLE, Dominique**

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

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XII: 17, 148-161, LNM 649 (1978)

**LÉPINGLE, Dominique**

Sur le comportement asymptotique des martingales locales (Martingale theory)

This paper is devoted to the extension of well-known statements (the strong law of large numbers, the Borel-Cantelli lemma, and the easier half of the law of the iterated logarithm) to right-continuous local martingales. An interesting technical point is the definition of a family of exponential supermartingales

Keywords: Law of large numbers, Borel-Cantelli lemma, Exponential martingales, Law of the iterated logarithm

Nature: Original

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XII: 18, 162-169, LNM 649 (1978)

**CAIROLI, Renzo**

Une représentation intégrale pour les martingales fortes (Several parameter processes)

This paper uses the results of Cairoli-Walsh,*Ann. Prob.* 5, 1977, to prove a stochastic integral representation of the strong martingales of the Brownian sheet filtration, without assuming they are square integrable

Keywords: Strong martingales, Brownian sheet

Nature: Original

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XII: 19, 170-179, LNM 649 (1978)

**MÉTRAUX, C.**

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

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XII: 20, 180-264, LNM 649 (1978)

**BISMUT, Jean-Michel**

Contrôle des systèmes linéaires quadratiques~: applications de l'intégrale stochastique (Control theory)

(To be completed) This is a set of lectures in control theory, which makes use of refined techniques in stochastic integration. It should be reviewed in detail

Comment: To be completed

Keywords: Optional stochastic integrals

Nature: Original

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XII: 21, 265-309, LNM 649 (1978)

**YOR, Marc**; **SAM LAZARO, José de**

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (*Michigan Math. J.* 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in $H^1$. The main result can be stated as follows: given a law $P\in{\cal P}$, the set ${\cal N}$ has the previsible representation property, i.e., ${\cal F}_0$ is trivial and stochastic integrals with respect to elements of ${\cal N}$ are dense in $H^1$, if and only if $P$ is an extreme point of ${\cal P}$. Many examples and applications are given

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XII: 22, 310-331, LNM 649 (1978)

**WILLIAMS, David**

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

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XII: 23, 332-341, LNM 649 (1978)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Lois empiriques et distance de Prokhorov (Mathematical statistics)

Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones

Keywords: Empirical distribution function, Prohorov distance

Nature: Original

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XII: 24, 342-363, LNM 649 (1978)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Estimation des densités~: risque minimax (Mathematical statistics)

A sequel to the preceding paper 1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed

Comment: For a correction see 1360. An improved version appeared in (*Zeit. für W-theorie,* **47**, 1979)

Keywords: Empirical distribution function

Nature: Original

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XII: 25, 364-377, LNM 649 (1978)

**STRICKER, Christophe**

Les ralentissements en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}t)$ and a stopping time $T$, we may define a new filtration $({\cal G}_t)$ as follows: we introduce an independent random variable $S$, and in intuitive language, we run the picture of $({\cal F}_t)$ up to time $T$, freeze the image between times $T$ and $T+S$, and then start running it again. The main result of this paper is the possibility, by performing this at all the times of discontinuity of $({\cal F}_t)$, to construct a filtration $({\cal G}_t)$ which is quasi-left-continuous. Though the idea is simple, there are considerable technical difficulties

Nature: Original

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XII: 26, 378-397, LNM 649 (1978)

**BROSSARD, Jean**

Comportement non-tangentiel et comportement brownien des fonctions harmoniques dans un demi-espace. Démonstration probabiliste d'un théorème de Calderon et Stein (Potential theory, Real analysis)

Given a harmonic function $u$ in a half space, Stein (*Acta Math.* 106, 1961) shows that the boundary points $x$ such that 1) $u$ has a non-tangential limit at $x$, 2) $u$ is ``non tangentially bounded'' near $x$, 3) $\nabla u$ is locally $L^2$ in the non-tangential cones at $x$, are the sames, except for sets of measure $0$. This result is given here a probabilistic proof using conditional Brownian motion

Keywords: Harmonic functions in a half-space, Non-tangential limits

Nature: Original

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XII: 27, 398-410, LNM 649 (1978)

**GETOOR, Ronald K.**

Homogeneous potentials (General theory of processes)

This is a development in Knight's prediction theory as described in 1007, 1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$

Keywords: Prediction theory

Nature: Original

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XII: 28, 411-423, LNM 649 (1978)

**MEYER, Paul-André**

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (*Zeit. für W-theorie,* 40, 1977) introduced a topology on the sets of ``fuzzy'' times and of fuzzy stopping times which turn these sets into compact metrizable spaces---a fuzzy r.v. $T$ is a right continuous decreasing process $M_t$ with $M_{0-}=1$, $M_t(\omega)$ being interpreted for each $\omega$ as the distribution function $P_{\omega}\{T>t\}$. When this process is adapted the fuzzy r.v. is a fuzzy stopping time. A number of properties of this topology are investigated

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

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XII: 29, 424-424, LNM 649 (1978)

**DELLACHERIE, Claude**

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

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XII: 30, 425-427, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

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XII: 31, 428-445, LNM 649 (1978)

**KNIGHT, Frank B.**

On the sojourn times of killed Brownian motion (Brownian motion)

To be completed

Keywords: Sojourn times, Laplace transforms

Nature: Original

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XII: 32, 446-456, LNM 649 (1978)

**TAYLOR, John C.**

Some remarks on Malliavin's comparison lemma and related topics (Diffusion theory)

The comparison lemma considered here gives estimates for the hitting probabilities of a several dimensional diffusion in terms of the hitting probabilities of a half line for suitably constructed one-dimensional diffusions. A self-contained proof is given

Keywords: Hitting probabilities

Nature: Original

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XII: 33, 457-467, LNM 649 (1978)

**MAINGUENEAU, Marie Anne**

Temps d'arrêt optimaux et théorie générale (General theory of processes)

This is a general discussion of optimal stopping in continuous time. Fairly advanced tools like strong supermartingales, Mertens' decomposition are used

Comment: The subject is taken up in 1332

Keywords: Optimal stopping, Snell's envelope

Nature: Original

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XII: 34, 468-481, LNM 649 (1978)

**SAINT RAYMOND, Jean**

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in*Comm. Pure Appl. Math.* 28, 1975 and 29, 1976, aiming at simplifying the proofs and weakening a few technical hypotheses

Keywords: Large deviations

Nature: Exposition

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XII: 35, 482-488, LNM 649 (1978)

**YOR, Marc**; **MEYER, Paul-André**

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the*weak * topology of $L[\infty$, or with the topology of $L^0$

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XII: 36, 489-490, LNM 649 (1978)

**MOKOBODZKI, Gabriel**

Domination d'une mesure par une capacité (Measure theory)

A bounded measure $\mu$ is said to be dominated by a capacity $C$ (countably subadditive, continuous along increasing sequences; neither strong subadditivity nor decreasing sequences are mentioned) if all sets of capacity $0$ have also measure $0$. The main result then states that the space can be decomposed into a set $A_0$ of capacity $0$, and disjoint sets $A_n$ on each of which $\mu$ is smaller than a multiple of $C$

Keywords: Radon-Nikodym theorem, Capacities

Nature: Original

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XII: 37, 491-508, LNM 649 (1978)

**MOKOBODZKI, Gabriel**

Ensembles à coupes dénombrables et capacités dominées par une mesure (Measure theory, General theory of processes)

Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times**R**_+$. For $A\subset X$ define $C(A)$ as the outer measure of the projection on $X$ of $F\cap(A\times**R**_+)$. Then it is proved that, if there is some measure $\lambda$ such that $\lambda$-null sets are $C$-null (the relation goes the reverse way from the preceding paper 1236!) then $F$ has ($\mu$-a.s.) countable sections, and if the property is strengthened to an $\epsilon-\delta$ ``absolute continuity'' relation, then $F$ has ($\mu$-a.s.) finite sections

Comment: This was a long-standing conjecture of Dellacherie (707), suggested by the theory of semi-polar sets. For further development see 1602

Keywords: Sets with countable sections

Nature: Original

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XII: 38, 509-511, LNM 649 (1978)

**DELLACHERIE, Claude**

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

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XII: 39, 512-514, LNM 649 (1978)

**DELLACHERIE, Claude**

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

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XII: 40, 515-522, LNM 649 (1978)

**DELLACHERIE, Claude**

Supports optionnels et prévisibles d'une P-mesure et applications (General theory of processes)

A $P$-measure is a measure on $\Omega\times**R**_+$ which does not charge $P$-evanescent sets. A $P$-measure has optional and previsible projections which are themselves $P$-measures. As usual, supports are minimal sets carrying a measure, possessing different properties like being optional/previsible, being right/left closed. The purpose of the paper is to find out which kind of supports do exist. Applications are given to honest times

Comment: See 1339 for a complement concerning honest times

Keywords: Projection theorems, Support, Honest times

Nature: Original

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XII: 41, 523-523, LNM 649 (1978)

**DELLACHERIE, Claude**

Erratum et addendum à ``les dérivations en théorie descriptive des ensembles et le théorème de la borne'' (Descriptive set theory)

A few corrections to 1104

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

Nature: Correction

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XII: 42, 524-563, LNM 649 (1978)

**HILLARD, Gérard**

Exemples de normes en théorie descriptive des ensembles (Descriptive set theory)

The situations described in this paper are special cases of 1104, where a coanalytic set $A$ was represented as the union of an increasing family $A_{\alpha}$ of analytic sets indexed by the countable ordinals, such that every analytic subset of $A$ is contained in some $A_{\alpha}$. The hypotheses of 1104 are not easy to check: they are shown here to include the classical Cantor derivation on the coanalytic space of countable compact sets, and a new example on the coanalytic space of all right continuous functions

Comment: The whole subject has been exposed anew in Chapter XXIV of Dellacherie-Meyer,*Probabilités et Potentiel*

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

Nature: Original

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XII: 43, 564-566, LNM 649 (1978)

**DELLACHERIE, Claude**; **MOKOBODZKI, Gabriel**

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

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XII: 44, 567-690, LNM 649 (1978)

**NANOPOULOS, Constantin**; **NOBELIS, Photis**

Régularité et propriétés limites des fonctions aléatoires (Miscellanea, Gaussian processes)

This paper extends to the non-Gaussian case methods to study the regularity of sample paths which have proved useful in the Gaussian case, notably that of majorizing measures (to be completed)

Comment: To be completed

Keywords: Sample path regularity, Majorizing measures

Nature: Original

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XII: 45, 691-706, LNM 649 (1978)

**FERNIQUE, Xavier**

Caractérisation de processus à trajectoires majorées ou continues (Miscellanea, Gaussian processes)

The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes

Keywords: Sample path regularity

Nature: Original

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XII: 46, 707-738, LNM 649 (1978)

**DELLACHERIE, Claude**

Théorie unifiée des capacités et des ensembles analytiques (Descriptive set theory)

A Choquet capacity takes one set as argument and produces a number. Along the years, one has considered multicapacities (which take as arguments finitely many sets) and capacitary operators (which produce sets instead of numbers). The essential result of this paper is that, if one allows functions of infinitely many arguments which produce sets, then the corresponding ``Choquet theorem'' gives all the classical results at a time, without need of an independent theory of analytic sets

Comment: For a more systematic exposition, see Chapter XI of Dellacherie-Meyer*Probabilités et Potentiel*

Keywords: Capacities, Analytic sets

Nature: Original

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XII: 47, 739-739, LNM 649 (1978)

**CHUNG, Kai Lai**

Correction to "Pedagogic Notes on the Barrier Theorem" (Potential theory)

Corrects an error in 1103

Keywords: Classical potential theory, Barrier, Regular points

Nature: Correction

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XII: 48, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 49, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 50, 739-739, LNM 649 (1978)

**LÉPINGLE, Dominique**

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XII: 51, 740-740, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Sur un théorème de C. Stricker'' (Stochastic calculus)

Fills a gap in a proof in 1132

Keywords: Stochastic integrals

Nature: Correction

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XII: 52, 740-740, LNM 649 (1978)

**DELLACHERIE, Claude**

Correction à ``Un crible généralisé'' (Descriptive set theory)

Acknowledgement of priority and references concerning the result in 703

Keywords: Analytic sets

Nature: Correction

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XII: 53, 741-741, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

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XII: 54, 742-745, LNM 649 (1978)

**DELLACHERIE, Claude**

Quelques applications du lemme de Borel-Cantelli à la théorie des semimartingales (Martingale theory, Stochastic calculus)

The general idea is the following: many constructions relative to one single semimartingale---like finding a sequence of stopping times increasing to infinity which reduce a local martingale, finding a change of law which sends a given semimartingale into $H^1$ or $H^2$ (locally)---can be strengthened to handle at the same time countably many given semimartingales

Nature: Original

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XII: 55, 746-756, LNM 649 (1978)

**DELLACHERIE, Claude**

Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (Descriptive set theory, General theory of processes)

There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel

Keywords: Analytic sets

Nature: Original

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XII: 56, 757-762, LNM 649 (1978)

**MEYER, Paul-André**

Inégalités de normes pour les intégrales stochastiques (Stochastic calculus)

Inequalities of the following kind were introduced by Émery: $$\|X.M\|_{H^p}\le c_p \| X\|_{S^p}\,\| M\|$$ where the left hand side is a stochastic integral of the previsible process $X$ w.r.t. the semimartingale $M$, $S^p$ is a supremum norm, and the norm $H^p$ for semimartingales takes into account the Hardy space norm for the martingale part and the $L^p$ norm of the total variation for the finite variation part. On the right hand side, Émery had used a norm called $H^{\infty}$. Here a weaker $BMO$-like norm for semimartingales is suggested

Keywords: Stochastic integrals, Hardy spaces

Nature: Original

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XII: 57, 763-769, LNM 649 (1978)

**MEYER, Paul-André**

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (*Trans. Amer. Math. Soc.* 149, 1970) on the Ito formula $f(B_t)=...$ for $n$-dimensional Brownian motion, under the weakest possible assumptions: namely up to the first exit time from an open set $W$ and assuming only that $f$ is locally in $L^1$ in $W$, and its Laplacian in the sense of distributions is a measure in $W$

Keywords: Ito formula

Nature: Exposition

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XII: 58, 770-774, LNM 649 (1978)

**MEYER, Paul-André**

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

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XII: 59, 775-803, LNM 649 (1978)

**MEYER, Paul-André**

Martingales locales fonctionnelles additives (two talks) (Markov processes)

The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$

Keywords: Hardy spaces, Additive functionals

Nature: Original

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XII: 60, 804-805, LNM 649 (1978)

**LETTA, Giorgio**

Un système de notations pour les processus de Markov (Markov processes)

Instead of notations like $X_T$, $\Theta_T$, etc, the author suggests to use kernel notations---for instance, $X^T$ is the submarkovian kernel $f\mapsto f\circ X_T$ ($0$ on $\{T=\infty\}$). Then the main properties of Markov processes are expressed by simple kernel equalities

Nature: Original

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Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,

Keywords: Interpolation, $BMO$

Nature: Original

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XII: 02, 20-21, LNM 649 (1978)

Une remarque sur les changements de temps et les martingales locales (Martingale theory)

It is well known (see 606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)

Keywords: Changes of time, Weak martingales

Nature: Original

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XII: 03, 22-34, LNM 649 (1978)

Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)

The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed

Comment: This papers improves on 1021 and 1023

Keywords: Semimartingales, Multiplicative decomposition

Nature: Original

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XII: 04, 35-46, LNM 649 (1978)

Décompositions multiplicatives de semimartingales exponentielles et applications (General theory of processes)

It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions

Comment: The results of this paper are used in Mémin-Shiryaev 1312

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition

Nature: Original

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XII: 05, 47-50, LNM 649 (1978)

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XII: 06, 51-52, LNM 649 (1978)

Une martingale de saut multiplicatif donné (Martingale theory)

Given a totally inaccessible stopping time $T$, it is shown how to construct a strictly positive martingale $M$ with $M_0=1$, such that its only jump occurs at time $T$ and $M_T/M_{T-}=K$, a strictly positive constant

Comment: See also 1308

Keywords: Totally inaccessible stopping times

Nature: Original

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XII: 07, 53-56, LNM 649 (1978)

Sur la localisation des intégrales stochastiques (Stochastic calculus)

A mapping $T$ from processes to processes is

Keywords: Girsanov's theorem

Nature: Original

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XII: 08, 57-60, LNM 649 (1978)

Sur un théorème de J. Jacod (General theory of processes)

Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals

Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration

Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales

Nature: Original

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XII: 09, 61-69, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)

Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called

Comment: This result was independently discovered by Barlow,

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 10, 70-77, LNM 649 (1978)

A propos du travail de Yor sur le grossissement des tribus (General theory of processes)

This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 11, 78-97, LNM 649 (1978)

Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)

This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved

Comment: For additional results on enlargements, see the two Lecture Notes volumes

Keywords: Enlargement of filtrations, Honest times

Nature: Original

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XII: 12, 98-113, LNM 649 (1978)

Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)

The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)

Keywords: Hardy spaces, $BMO$

Nature: Original

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XII: 13, 114-131, LNM 649 (1978)

Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien (Stochastic calculus)

The results of this paper improve on those of the author's paper (

Comment: Such equations play an important role in the theory of Bessel processes (see chapter XI of Revuz-Yor,

Keywords: Stochastic differential equations, Hölder conditions

Nature: Original

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XII: 14, 132-133, LNM 649 (1978)

Extension au cas continu d'un théorème de Dubins (Martingale theory)

After a suitable translation, Dubins' theorem can be stated as follows: if $X$ is a positive submartingale with $X_t\in L^2$ for all $t$, then $X^2-[X,X]$ is a submartingale

Keywords: Submartingales

Nature: Original

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XII: 15, 134-137, LNM 649 (1978)

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

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XII: 16, 138-147, LNM 649 (1978)

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

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XII: 17, 148-161, LNM 649 (1978)

Sur le comportement asymptotique des martingales locales (Martingale theory)

This paper is devoted to the extension of well-known statements (the strong law of large numbers, the Borel-Cantelli lemma, and the easier half of the law of the iterated logarithm) to right-continuous local martingales. An interesting technical point is the definition of a family of exponential supermartingales

Keywords: Law of large numbers, Borel-Cantelli lemma, Exponential martingales, Law of the iterated logarithm

Nature: Original

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XII: 18, 162-169, LNM 649 (1978)

Une représentation intégrale pour les martingales fortes (Several parameter processes)

This paper uses the results of Cairoli-Walsh,

Keywords: Strong martingales, Brownian sheet

Nature: Original

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XII: 19, 170-179, LNM 649 (1978)

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

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XII: 20, 180-264, LNM 649 (1978)

Contrôle des systèmes linéaires quadratiques~: applications de l'intégrale stochastique (Control theory)

(To be completed) This is a set of lectures in control theory, which makes use of refined techniques in stochastic integration. It should be reviewed in detail

Comment: To be completed

Keywords: Optional stochastic integrals

Nature: Original

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XII: 21, 265-309, LNM 649 (1978)

Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (Martingale theory)

This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (

Comment: The second named author's contribution concerns only the appendix on homogeneous martingales

Keywords: Previsible representation, Douglas theorem, Extremal laws

Nature: Original

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XII: 22, 310-331, LNM 649 (1978)

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

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XII: 23, 332-341, LNM 649 (1978)

Lois empiriques et distance de Prokhorov (Mathematical statistics)

Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones

Keywords: Empirical distribution function, Prohorov distance

Nature: Original

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XII: 24, 342-363, LNM 649 (1978)

Estimation des densités~: risque minimax (Mathematical statistics)

A sequel to the preceding paper 1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed

Comment: For a correction see 1360. An improved version appeared in (

Keywords: Empirical distribution function

Nature: Original

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XII: 25, 364-377, LNM 649 (1978)

Les ralentissements en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}t)$ and a stopping time $T$, we may define a new filtration $({\cal G}_t)$ as follows: we introduce an independent random variable $S$, and in intuitive language, we run the picture of $({\cal F}_t)$ up to time $T$, freeze the image between times $T$ and $T+S$, and then start running it again. The main result of this paper is the possibility, by performing this at all the times of discontinuity of $({\cal F}_t)$, to construct a filtration $({\cal G}_t)$ which is quasi-left-continuous. Though the idea is simple, there are considerable technical difficulties

Nature: Original

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XII: 26, 378-397, LNM 649 (1978)

Comportement non-tangentiel et comportement brownien des fonctions harmoniques dans un demi-espace. Démonstration probabiliste d'un théorème de Calderon et Stein (Potential theory, Real analysis)

Given a harmonic function $u$ in a half space, Stein (

Keywords: Harmonic functions in a half-space, Non-tangential limits

Nature: Original

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XII: 27, 398-410, LNM 649 (1978)

Homogeneous potentials (General theory of processes)

This is a development in Knight's prediction theory as described in 1007, 1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$

Keywords: Prediction theory

Nature: Original

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XII: 28, 411-423, LNM 649 (1978)

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

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XII: 29, 424-424, LNM 649 (1978)

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

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XII: 30, 425-427, LNM 649 (1978)

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

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XII: 31, 428-445, LNM 649 (1978)

On the sojourn times of killed Brownian motion (Brownian motion)

To be completed

Keywords: Sojourn times, Laplace transforms

Nature: Original

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XII: 32, 446-456, LNM 649 (1978)

Some remarks on Malliavin's comparison lemma and related topics (Diffusion theory)

The comparison lemma considered here gives estimates for the hitting probabilities of a several dimensional diffusion in terms of the hitting probabilities of a half line for suitably constructed one-dimensional diffusions. A self-contained proof is given

Keywords: Hitting probabilities

Nature: Original

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XII: 33, 457-467, LNM 649 (1978)

Temps d'arrêt optimaux et théorie générale (General theory of processes)

This is a general discussion of optimal stopping in continuous time. Fairly advanced tools like strong supermartingales, Mertens' decomposition are used

Comment: The subject is taken up in 1332

Keywords: Optimal stopping, Snell's envelope

Nature: Original

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XII: 34, 468-481, LNM 649 (1978)

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in

Keywords: Large deviations

Nature: Exposition

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XII: 35, 482-488, LNM 649 (1978)

Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)

Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability

Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the

Keywords: Kernels, Radon-Nikodym theorem

Nature: Original

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XII: 36, 489-490, LNM 649 (1978)

Domination d'une mesure par une capacité (Measure theory)

A bounded measure $\mu$ is said to be dominated by a capacity $C$ (countably subadditive, continuous along increasing sequences; neither strong subadditivity nor decreasing sequences are mentioned) if all sets of capacity $0$ have also measure $0$. The main result then states that the space can be decomposed into a set $A_0$ of capacity $0$, and disjoint sets $A_n$ on each of which $\mu$ is smaller than a multiple of $C$

Keywords: Radon-Nikodym theorem, Capacities

Nature: Original

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XII: 37, 491-508, LNM 649 (1978)

Ensembles à coupes dénombrables et capacités dominées par une mesure (Measure theory, General theory of processes)

Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times

Comment: This was a long-standing conjecture of Dellacherie (707), suggested by the theory of semi-polar sets. For further development see 1602

Keywords: Sets with countable sections

Nature: Original

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XII: 38, 509-511, LNM 649 (1978)

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

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XII: 39, 512-514, LNM 649 (1978)

Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (General theory of processes)

The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes

Keywords: Essential suprema, Evanescent sets

Nature: Original

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XII: 40, 515-522, LNM 649 (1978)

Supports optionnels et prévisibles d'une P-mesure et applications (General theory of processes)

A $P$-measure is a measure on $\Omega\times

Comment: See 1339 for a complement concerning honest times

Keywords: Projection theorems, Support, Honest times

Nature: Original

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XII: 41, 523-523, LNM 649 (1978)

Erratum et addendum à ``les dérivations en théorie descriptive des ensembles et le théorème de la borne'' (Descriptive set theory)

A few corrections to 1104

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

Nature: Correction

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XII: 42, 524-563, LNM 649 (1978)

Exemples de normes en théorie descriptive des ensembles (Descriptive set theory)

The situations described in this paper are special cases of 1104, where a coanalytic set $A$ was represented as the union of an increasing family $A_{\alpha}$ of analytic sets indexed by the countable ordinals, such that every analytic subset of $A$ is contained in some $A_{\alpha}$. The hypotheses of 1104 are not easy to check: they are shown here to include the classical Cantor derivation on the coanalytic space of countable compact sets, and a new example on the coanalytic space of all right continuous functions

Comment: The whole subject has been exposed anew in Chapter XXIV of Dellacherie-Meyer,

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

Nature: Original

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XII: 43, 564-566, LNM 649 (1978)

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

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XII: 44, 567-690, LNM 649 (1978)

Régularité et propriétés limites des fonctions aléatoires (Miscellanea, Gaussian processes)

This paper extends to the non-Gaussian case methods to study the regularity of sample paths which have proved useful in the Gaussian case, notably that of majorizing measures (to be completed)

Comment: To be completed

Keywords: Sample path regularity, Majorizing measures

Nature: Original

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XII: 45, 691-706, LNM 649 (1978)

Caractérisation de processus à trajectoires majorées ou continues (Miscellanea, Gaussian processes)

The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes

Keywords: Sample path regularity

Nature: Original

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XII: 46, 707-738, LNM 649 (1978)

Théorie unifiée des capacités et des ensembles analytiques (Descriptive set theory)

A Choquet capacity takes one set as argument and produces a number. Along the years, one has considered multicapacities (which take as arguments finitely many sets) and capacitary operators (which produce sets instead of numbers). The essential result of this paper is that, if one allows functions of infinitely many arguments which produce sets, then the corresponding ``Choquet theorem'' gives all the classical results at a time, without need of an independent theory of analytic sets

Comment: For a more systematic exposition, see Chapter XI of Dellacherie-Meyer

Keywords: Capacities, Analytic sets

Nature: Original

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XII: 47, 739-739, LNM 649 (1978)

Correction to "Pedagogic Notes on the Barrier Theorem" (Potential theory)

Corrects an error in 1103

Keywords: Classical potential theory, Barrier, Regular points

Nature: Correction

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XII: 48, 739-739, LNM 649 (1978)

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 49, 739-739, LNM 649 (1978)

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 50, 739-739, LNM 649 (1978)

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XII: 51, 740-740, LNM 649 (1978)

Correction à ``Sur un théorème de C. Stricker'' (Stochastic calculus)

Fills a gap in a proof in 1132

Keywords: Stochastic integrals

Nature: Correction

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XII: 52, 740-740, LNM 649 (1978)

Correction à ``Un crible généralisé'' (Descriptive set theory)

Acknowledgement of priority and references concerning the result in 703

Keywords: Analytic sets

Nature: Correction

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XII: 53, 741-741, LNM 649 (1978)

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

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XII: 54, 742-745, LNM 649 (1978)

Quelques applications du lemme de Borel-Cantelli à la théorie des semimartingales (Martingale theory, Stochastic calculus)

The general idea is the following: many constructions relative to one single semimartingale---like finding a sequence of stopping times increasing to infinity which reduce a local martingale, finding a change of law which sends a given semimartingale into $H^1$ or $H^2$ (locally)---can be strengthened to handle at the same time countably many given semimartingales

Nature: Original

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XII: 55, 746-756, LNM 649 (1978)

Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (Descriptive set theory, General theory of processes)

There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel

Keywords: Analytic sets

Nature: Original

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XII: 56, 757-762, LNM 649 (1978)

Inégalités de normes pour les intégrales stochastiques (Stochastic calculus)

Inequalities of the following kind were introduced by Émery: $$\|X.M\|_{H^p}\le c_p \| X\|_{S^p}\,\| M\|$$ where the left hand side is a stochastic integral of the previsible process $X$ w.r.t. the semimartingale $M$, $S^p$ is a supremum norm, and the norm $H^p$ for semimartingales takes into account the Hardy space norm for the martingale part and the $L^p$ norm of the total variation for the finite variation part. On the right hand side, Émery had used a norm called $H^{\infty}$. Here a weaker $BMO$-like norm for semimartingales is suggested

Keywords: Stochastic integrals, Hardy spaces

Nature: Original

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XII: 57, 763-769, LNM 649 (1978)

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (

Keywords: Ito formula

Nature: Exposition

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XII: 58, 770-774, LNM 649 (1978)

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

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XII: 59, 775-803, LNM 649 (1978)

Martingales locales fonctionnelles additives (two talks) (Markov processes)

The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$

Keywords: Hardy spaces, Additive functionals

Nature: Original

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XII: 60, 804-805, LNM 649 (1978)

Un système de notations pour les processus de Markov (Markov processes)

Instead of notations like $X_T$, $\Theta_T$, etc, the author suggests to use kernel notations---for instance, $X^T$ is the submarkovian kernel $f\mapsto f\circ X_T$ ($0$ on $\{T=\infty\}$). Then the main properties of Markov processes are expressed by simple kernel equalities

Nature: Original

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