I: 04, 52-53, LNM 39 (1967)
DELLACHERIE, Claude
Un complément au théorème de Weierstrass-Stone (
Functional analysis)
An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore
Keywords: Stone-Weierstrass theoremNature: Original
Retrieve article from Numdam
III: 04, 93-96, LNM 88 (1969)
DELLACHERIE, Claude
Une application aux fonctionnelles additives d'un théorème de Mokobodzki (
Markov processes)
Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,
Probabilité et potentiels, Chap. II,
27). They are used here to prove that every continuous additive functional of a Markov process has a ``perfect'' version
Comment: See also
203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
III: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (
Descriptive set theory,
Markov processes,
General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
III: 06, 115-136, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires II (
Descriptive set theory,
Markov processes)
Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (
Fund. Math.,
5, 1924) can be extended to an abstract set-up to show that a non-semi-polar Borel set contains a non-semi-polar compact set
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972. More recent proofs no longer depend on ``rabotages'': Dellacherie-Meyer,
Probabilités et potentiel, Appendix to Chapter IV
Keywords: Sierpinski's ``rabotages'',
Semi-polar setsNature: Original Retrieve article from Numdam
IV: 05, 60-70, LNM 124 (1970)
DELLACHERIE, Claude
Un exemple de la théorie générale des processus (
General theory of processes)
In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly
Comment: This example has become classical. See for example Dellacherie-Meyer,
Probabilités et Potentiel, Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer
906Keywords: Stopping times,
Accessible times,
Previsible timesNature: Original Retrieve article from Numdam
IV: 06, 71-72, LNM 124 (1970)
DELLACHERIE, Claude
Au sujet des sauts d'un processus de Hunt (
Markov processes)
Two a.s. results on jumps: the process cannot jump
from a semi-polar set; at the first hitting time of any finely closed set $F$, either the process does not jump, or it jumps from outside $F$
Comment: Both results are improvements of previous results of Meyer and Weil
Keywords: Hunt processes,
Semi-polar setsNature: Original Retrieve article from Numdam
IV: 07, 73-75, LNM 124 (1970)
DELLACHERIE, Claude
Potentiels de Green et fonctionnelles additives (
Markov processes,
Potential theory)
Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$
Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces
Keywords: Green potentials,
Additive functionalsNature: Original Retrieve article from Numdam
IV: 08, 76-76, LNM 124 (1970)
DELLACHERIE, Claude
Un lemme de théorie de la mesure (
Measure theory)
A lemma used by Erdös, Kesterman and Rogers (
Coll. Math., XI, 1963) is reduced to the fact that a sequence of bounded r.v.'s contains a weakly convergent subsequence
Keywords: Convergence in norm,
SubsequencesNature: Original proofs Retrieve article from Numdam
IV: 19, 240-282, LNM 124 (1970)
DELLACHERIE, Claude;
DOLÉANS-DADE, Catherine;
LETTA, Giorgio;
MEYER, Paul-André
Diffusions à coefficients continus, d'après Stroock et Varadhan (
Markov processes,
Diffusion theory)
This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (
Comm. Pure Appl. Math.,
22, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof
Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,
Multidimensional Diffusion Processes, Springer 1979
Keywords: Elliptic differential operators,
Uniqueness in lawNature: Exposition Retrieve article from Numdam
V: 07, 77-81, LNM 191 (1971)
DELLACHERIE, Claude
Quelques commentaires sur les prolongements de capacités (
Descriptive set theory)
Remarks on the extension of capacities from sets to functions. Probably superseded by the work of Mokobodzki on functional capacities
Comment: See Dellacherie-Meyer,
Probabilités et Potentiel, Chap. XI: capacités fonctionnelles
Keywords: CapacitiesNature: Original Retrieve article from Numdam
V: 08, 82-85, LNM 191 (1971)
DELLACHERIE, Claude
Une démonstration du théorème de séparation des ensembles analytiques (
Descriptive set theory)
The first separation theorem can be deduced from Choquet's capacity theorem
Comment: Starting point in Sion,
Ann. Inst. Fourier, 13, 1963. This proof has become standard, see Dellacherie-Meyer,
Probabilités et Potentiel, Chap. III
Keywords: Analytic sets,
Capacities,
Separation theoremNature: Original Retrieve article from Numdam
V: 09, 86-86, LNM 191 (1971)
DELLACHERIE, Claude
Correction à ``Ensembles Aléatoires II'' (
Descriptive set theory)
Correction to Dellacherie
306Comment: See Dellacherie
511Keywords: Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
V: 10, 87-102, LNM 191 (1971)
DELLACHERIE, Claude
Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (
Descriptive set theory)
Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections
Comment: See Dellacherie-Meyer,
Probabilités et Potentiel, Chap. XI
Keywords: Analytic sets,
Random sets,
Section theoremsNature: New exposition of known results Retrieve article from Numdam
V: 11, 103-126, LNM 191 (1971)
DELLACHERIE, Claude
Ensembles pavés et rabotages (
Descriptive set theory)
A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie
306 to solve several problems in probabilistic potential theory. The main paper on this subject
Comment: See Dellacherie,
Capacités et Processus Stochastiques, 1970. Author should be consulted on recent developments (see
1526)
Keywords: Analytic sets,
Capacities,
Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
V: 12, 127-137, LNM 191 (1971)
DELLACHERIE, Claude;
DOLÉANS-DADE, Catherine
Un contre-exemple au problème des laplaciens approchés (
Martingale theory)
The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years
Comment: Problem originated in Meyer,
Ill. J. Math.,
7, 1963
Keywords: Submartingales,
SupermartingalesNature: Original Retrieve article from Numdam
VII: 03, 33-35, LNM 321 (1973)
DELLACHERIE, Claude
Un crible généralisé (
Descriptive set theory)
Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic
Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in
1252Keywords: Analytic setsNature: Original Retrieve article from Numdam
VII: 04, 36-37, LNM 321 (1973)
DELLACHERIE, Claude
Temps d'arrêt totalement inaccessibles (
General theory of processes)
Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left
Keywords: Stopping times,
Accessible sets,
Totally inaccessible stopping timesNature: Original Retrieve article from Numdam
VII: 05, 38-47, LNM 321 (1973)
DELLACHERIE, Claude
Sur les théorèmes fondamentaux de la théorie générale des processus (
General theory of processes)
This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved
Comment: The idea of this paper has proved fruitful. See for instance Lenglart,
1449; Le Jan,
Z. für W-Theorie 44, 1978
Keywords: Stopping times,
Section theoremsNature: Original Retrieve article from Numdam
VII: 06, 48-50, LNM 321 (1973)
DELLACHERIE, Claude
Une démonstration du théorème de Souslin-Lusin (
Descriptive set theory)
The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets
Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See
Un cours sur les ensembles analytiques, in
Analytic Sets, C.A. Rogers ed., Academic Press 1980
Keywords: Borel sets,
Analytic sets,
Separation theoremNature: New exposition of known results Retrieve article from Numdam
VII: 07, 51-57, LNM 321 (1973)
DELLACHERIE, Claude
Une conjecture sur les ensembles semi-polaires (
Markov processes)
For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets
Comment: The conjecture can be proved, using a general result of Mokobodzki, see
1238Keywords: Polar sets,
Semi-polar setsNature: Original Retrieve article from Numdam
VII: 08, 58-60, LNM 321 (1973)
DELLACHERIE, Claude
Potentiels de fonctionnelles additives. Un contre-exemple de Knight (
Markov processes)
An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point
Keywords: Additive functionalsNature: Exposition,
Original additions Retrieve article from Numdam
VIII: 04, 22-24, LNM 381 (1974)
DELLACHERIE, Claude
Un ensemble progressivement mesurable... (
General theory of processes)
The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections
Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails
Keywords: Progressive sets,
Section theoremsNature: Original Retrieve article from Numdam
VIII: 05, 25-26, LNM 381 (1974)
DELLACHERIE, Claude
Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (
General theory of processes)
This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems
Comment: A gap in the proof is filled in
928 and
2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,
Z. für W-theorie, 38, 1977 and Yor
1221. For another approach to the restricted case considered here, see Ruiz de Chavez
1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property
Keywords: Brownian motion,
Poisson processes,
Previsible representationNature: Original Retrieve article from Numdam
IX: 08, 239-245, LNM 465 (1975)
DELLACHERIE, Claude;
MEYER, Paul-André
Un nouveau théorème de projection et de section (
General theory of processes)
Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration
Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie
705. The result takes its definitive form in Dellacherie-Meyer,
Probabilités et Potentiel, theorems IV.84 of vol. A and App.1, \no~6
Keywords: Section theorems,
Optional processes,
Projection theoremsNature: Original Retrieve article from Numdam
IX: 15, 336-372, LNM 465 (1975)
DELLACHERIE, Claude
Ensembles analytiques, théorèmes de séparation et applications (
Descriptive set theory)
According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition
Comment: See also the next paper
916, the set of lectures by Dellacherie in C.A. Rogers,
Analytic Sets, Academic Press 1981, and chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Second separation theoremNature: Exposition Retrieve article from Numdam
IX: 16, 373-389, LNM 465 (1975)
DELLACHERIE, Claude;
MEYER, Paul-André
Ensembles analytiques et temps d'arrêt (
Descriptive set theory)
This is a sequel to the preceding paper
915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $
N^
N$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language
Comment: See also the next paper
917, the set of lectures by Dellacherie in C.A. Rogers,
Analytic Sets, Academic Press 1981, and chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Second separation theorem,
Stopping timesNature: Original Retrieve article from Numdam
IX: 17, 390-405, LNM 465 (1975)
DELLACHERIE, Claude
Jeux infinis avec information complète et temps d'arrêt (
Descriptive set theory)
This is a sequel to the preceding paper
916. It shows how well the language of stopping times applies, not only to the second separation theorem, but to the Gale-Stewart theorem on the determinacy of open games
Comment: The original remark on the relation between game determinacy and separation theorems, due to Blackwell (1967), led to a huge literature. More details can be found in chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Determinacy of games,
Gale and Stewart theoremNature: Original Retrieve article from Numdam
IX: 18, 406-407, LNM 465 (1975)
DELLACHERIE, Claude
Une remarque sur les espaces sousliniens de Bourbaki (
Descriptive set theory)
The paper claims to prove that given any space with a Souslin topology i.e., which is Hausdorff and a continuous image of a Polish space, it is possible to strengthen the topology so that it becomes metrizable without losing the Souslin property. The author discovered later a serious mistake in his proof
Comment: The problem is still open, and interesting
Keywords: Analytic setsNature: False Retrieve article from Numdam
IX: 28, 494-494, LNM 465 (1975)
DELLACHERIE, Claude
Correction à ``Intégrales stochastiques par rapport...'' (
General theory of processes)
This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie
805Comment: Another way of filling this gap is given by Ruiz de Chavez
1821. The same gap for the Poisson process is corrected in
2002Keywords: Previsible representationNature: Original Retrieve article from Numdam
IX: 29, 495-495, LNM 465 (1975)
DELLACHERIE, Claude
Une propriété des ensembles semi-polaires (
Markov processes)
It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)
Keywords: Semi-polar setsNature: Original Retrieve article from Numdam
X: 29, 544-544, LNM 511 (1976)
DELLACHERIE, Claude
Correction à des exposés de 1973/74 (
Descriptive set theory)
Corrections to
915 and
918Keywords: Analytic sets,
Semi-polar sets,
Suslin spacesNature: Original Retrieve article from Numdam
X: 30, 545-577, LNM 511 (1976)
DELLACHERIE, Claude
Sur la construction de noyaux boréliens (
Measure theory)
This answers questions of Getoor
923 and Meyer
924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest
Comment: For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
41. The hypothesis that the space is compact is sometimes troublesome for the applications
Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
X: 32, 579-593, LNM 511 (1976)
DELLACHERIE, Claude
Compléments aux exposés sur les ensembles analytiques (
Descriptive set theory)
A new proof of Novikov's theorem (see
1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to
919 and
920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo
Keywords: Analytic sets,
Section theorems,
CapacitiesNature: Original Retrieve article from Numdam
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 PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Exposition,
Original additions Retrieve article from Numdam
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 timeNature: Original Retrieve article from Numdam
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
1524Keywords: General filtrations,
Strong supermartingalesNature: Original Retrieve article from Numdam
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
1135Comment: The last section states an interesting open problem
Keywords: Changes of time,
Spectral representationNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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
707Keywords: Sets with countable sections,
Semi-polar setsNature: Original Retrieve article from Numdam
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 setsNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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
1104Keywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Correction Retrieve article from Numdam
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 supremaNature: Original Retrieve article from Numdam
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 PotentielKeywords: Capacities,
Analytic setsNature: Original Retrieve article from Numdam
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
703Keywords: Analytic setsNature: Correction Retrieve article from Numdam
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 Retrieve article from Numdam
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 setsNature: Original Retrieve article from Numdam
XIII: 31, 371-377, LNM 721 (1979)
DELLACHERIE, Claude
Inégalités de convexité pour les processus croissants et les sousmartingales (
Martingale theory)
Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,
Probabilités et Potentiels B, Chapter VI
Keywords: Martingale inequalities,
Convex functionsNature: Exposition,
Original proofs Retrieve article from Numdam
XV: 24, 320-346, LNM 850 (1981)
DELLACHERIE, Claude;
LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (
General theory of processes)
The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in
1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)
Keywords: General filtrations,
Strong supermartingales,
Snell's envelope,
Section theoremsNature: Original Retrieve article from Numdam
XV: 26, 351-370, LNM 850 (1981)
DELLACHERIE, Claude
Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (
General theory of processes)
One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was
511, in which Dellacherie introduced ``rabotages'' (
306) to develop the theory without analytic sets; see also
1246,
1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications
Keywords: Section theorems,
Capacities,
Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
XV: 27, 371-387, LNM 850 (1981)
DELLACHERIE, Claude
Sur les noyaux $\sigma$-finis (
Measure theory)
This paper is an improvement of
1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called
measurability of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $
R$ is all one needs). The case of ``basic'' kernels, considered in
1235, is thoroughly discussed
Keywords: Kernels,
Radon-Nikodym theoremNature: Original Retrieve article from Numdam
XVI: 02, 8-28, LNM 920 (1982)
DELLACHERIE, Claude;
FEYEL, Denis;
MOKOBODZKI, Gabriel
Intégrales de capacités fortement sous-additives Retrieve article from Numdam
XVI: 03, 29-40, LNM 920 (1982)
DELLACHERIE, Claude
Appendice à l'exposé précédent Retrieve article from Numdam
XVI: 26, 298-313, LNM 920 (1982)
DELLACHERIE, Claude;
LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (
General theory of processes)
This paper is a sequel to
1524. Let $\Theta$ be a
chronology, i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in
1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right
Keywords: Stopping timesNature: Original Retrieve article from Numdam
XIX: 16, 222-229, LNM 1123 (1985)
DELLACHERIE, Claude
Quelques résultats sur les maisons de jeu analytiques Retrieve article from Numdam
XXIV: 05, 52-104, LNM 1426 (1990)
DELLACHERIE, Claude
Théorie des processus de production Retrieve article from Numdam
XXV: 01, 1-9, LNM 1485 (1991)
DELLACHERIE, Claude
Théorie non-linéaire du potentiel : Un principe unifié de domination et du maximum et quelques applications Retrieve article from Numdam
XXXII: 01, 1-5, LNM 1686 (1998)
DELLACHERIE, Claude;
IWANIK, Anzelm
Sous-mesures symétriques sur un ensemble fini Retrieve article from Numdam
XLIII: 06, 187-189, LNM 2006 (2011)
DELLACHERIE, Claude
On isomorphic probability spacesNature: Original
XLV: 04, 141-157, LNM 2078 (2013)
DELLACHERIE, Claude;
ÉMERY, Michel
Filtrations Indexed by Ordinals; Application to a Conjecture of S. LaurentKeywords: FiltrationNature: Original
