I: 06, 72-162, LNM 39 (1967)
MEYER, Paul-André
Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)
This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (Nagoya Math. J. 30, 1967) on square integrable martingales. The filtration is assumed to be free from fixed times of discontinuity, a restriction lifted in the modern theory. A new feature is the definition of the second increasing process associated with a square integrable martingale (a ``square bracket'' in the modern terminology). In the second talk, stochastic integrals are defined with respect to local martingales (introduced from Ito-Watanabe, Ann. Inst. Fourier, 15, 1965), and the general integration by parts formula is proved. Also a restricted class of semimartingales is defined and an ``Ito formula'' for change of variables is given, different from that of Kunita-Watanabe. The third talk contains the famous Kunita-Watanabe theorem giving the structure of martingale additive functionals of a Hunt process, and a new proof of Lévy's description of the structure of processes with independent increments (in the time homogeneous case). The fourth talk deals mostly with Lévy systems (Motoo-Watanabe, J. Math. Kyoto Univ., 4, 1965; Watanabe, Japanese J. Math., 36, 1964)
Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312
Keywords: Square integrable martingales, Angle bracket, Stochastic integrals
Nature: Exposition, Original additions
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II: 09, 166-170, LNM 51 (1968)
MEYER, Paul-André
Une majoration du processus croissant associé à une surmartingale (Martingale theory)
Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$
Comment: This paper became obsolete after the $H^1$-$BMO$ theory
Keywords: Inequalities, Potential of an increasing process
Nature: Original
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III: 08, 143-143, LNM 88 (1969)
MEYER, Paul-André
Un lemme de théorie des martingales (Martingale theory)
The author apparently believed that this classical and useful remark was new (it is often called ``Hunt's lemma'', see Hunt, Martingales et Processus de Markov, Masson 1966, p.47)
Keywords: Almost sure convergence
Nature: Well-known
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III: 12, 160-162, LNM 88 (1969)
MEYER, Paul-André
Rectification à des exposés antérieurs (Markov processes, Martingale theory)
Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer
Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov
Keywords: Time reversal, Stochastic integrals
Nature: Correction
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III: 13, 163-174, LNM 88 (1969)
MEYER, Paul-André
Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)
A proof of the famous Burkholder inequalities in discrete time, from Gundy, Ann. Math. Stat., 39, 1968
Keywords: Burkholder inequalities
Nature: Exposition
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IV: 03, 37-46, LNM 124 (1970)
CHERSI, Franco
Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)
Gundy's result (Studia Math., 33, 1968) is a converse to Doob's inequality: for a positive martingale such that $X_n\leq cX_{n-1}$, the integrability of $\sup_n X_n$ implies boundedness in $L\log^+L$. All martingales satisfy this condition on regular filtrations
Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales
Keywords: Inequalities, Regular martingales
Nature: Exposition, Original additions
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IV: 09, 77-107, LNM 124 (1970)
DOLÉANS-DADE, Catherine; MEYER, Paul-André
Intégrales stochastiques par rapport aux martingales locales (Martingale theory, Stochastic calculus)
This is a continuation of Meyer 106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in 106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in 106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality
Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer 1017
Keywords: Local martingales, Stochastic integrals, Change of variable formula
Nature: Original
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IV: 14, 162-169, LNM 124 (1970)
MEYER, Paul-André
Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)
The original paper appeared in Ann. Math. Stat., 36, 1965, and the inequalities are extensions to martingales of the Borel-Cantelli lemma and the strong law of large numbers. For martingales with bounded jumps, exponential bounds are given (Neveu, Martingales à temps discret, gives a better one)
Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete
Keywords: Inequalities
Nature: Exposition
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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, Supermartingales
Nature: Original
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V: 13, 138-140, LNM 191 (1971)
DOLÉANS-DADE, Catherine
Une martingale uniformément intégrable, non localement de carré intégrable (Martingale theory)
Now well known! This paper helped to set the basic notions of the theory
Keywords: Square integrable martingales
Nature: Original
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V: 16, 170-176, LNM 191 (1971)
MEYER, Paul-André
Sur un article de Dubins (Martingale theory)
Description of a Skorohod imbedding procedure for real valued r.v.'s due to Dubins (Ann. Math. Stat., 39, 1968), using a remarkable discrete approximation of measures. It does not use randomization
Comment: This beautiful method to realize Skorohod's imbedding is related to that of Chacon and Walsh in 1002. For a deeper study see Bretagnolle 802. A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding
Nature: Exposition
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V: 18, 191-195, LNM 191 (1971)
MEYER, Paul-André
Démonstration simplifiée d'un théorème de Knight (Martingale theory)
A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M 190) asserts that this operation performed on $n$ orthogonal martingales yields $n$ independent Brownian motions. The result is extended to Poisson processes
Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor Continuous Martingales and Brownian Motion, Chapter V)
Keywords: Continuous martingales, Changes of time
Nature: Exposition, Original additions
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VI: 06, 98-100, LNM 258 (1972)
KAZAMAKI, Norihiko
Examples on local martingales (Martingale theory)
Two simple examples are given, the first one concerning the filtration generated by an exponential stopping time, the second one showing that local martingales are not preserved under time changes (Kazamaki, Zeit. für W-theorie, 22, 1972)
Keywords: Changes of time, Local martingales, Weak martingales
Nature: Original
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VI: 07, 101-104, LNM 258 (1972)
KAZAMAKI, Norihiko
Krickeberg's decomposition for local martingales (Martingale theory)
It is shown that a local martingale bounded in $L^1$ is a difference of two (minimal) positive local martingales
Keywords: Local martingales, Krickeberg decomposition
Nature: Original
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VI: 11, 118-129, LNM 258 (1972)
MEYER, Paul-André
La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)
The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer, Zeit für X-theorie, 21, 1970
Keywords: Supermartingales, Föllmer measures
Nature: Exposition, Original additions
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VI: 13, 151-158, LNM 258 (1972)
MEYER, Paul-André
Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)
The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (Acta Math., 124, 1970), Burkholder-Davis-Gundy (Proc. 6-th Berkeley Symposium, 3, 1972). The technique of proof is now obsolete
Keywords: Burkholder inequalities, Moderate convex functions
Nature: Exposition
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VI: 19, 198-201, LNM 258 (1972)
RAO, Murali
Doob's decomposition and Burkholder's inequalities (Martingale theory)
The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (Ann. Math. Stat., 37, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale
Comment: This little-known paper would probably deserve a modern translation in continuous time
Keywords: Burkholder inequalities, Decomposition of supermartingales
Nature: Original
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VII: 12, 118-121, LNM 321 (1973)
KAZAMAKI, Norihiko
Une note sur les martingales faibles (Martingale theory)
Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or weak martingales. It is shown that in the filtration generated by one single stopping time, processes which are prelocally martingales (square integrable martingales) are so globally. It follows that prelocal martingales may not be prelocally square integrable
Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki, Zeit. für W-theorie, 22, 1972
Keywords: Weak martingales
Nature: Original
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VII: 14, 136-145, LNM 321 (1973)
MEYER, Paul-André
Le dual de $H^1$ est $BMO$ (cas continu) (Martingale theory)
The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $R$ and $R^n$ and the duality between $BMO$ and $H^1$ were almost immediately translated into discrete martingale theory by Herz and Garsia. The next step, due to Getoor-Sharpe ({\sl Invent. Math.} 16, 1972), delt with continuous martingales. The extension to right continuous martingales, a good exercise in martingale theory, is given here
Comment: See 907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer, Probabilités et Potentiel, Vol. B, Chapter VII
Keywords: $BMO$, Hardy spaces, Fefferman inequality
Nature: Original
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VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma, Z. für W-theorie, 45, 1978 and its exposition 1321
Keywords: Supermartingales, Multiplicative decomposition
Nature: Original
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IX: 04, 206-212, LNM 465 (1975)
CHOU, Ching Sung
Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)
A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities
Keywords: Inequalities, Burkholder inequalities
Nature: Exposition
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IX: 05, 213-225, LNM 465 (1975)
CHOU, Ching Sung
Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)
The methods developed in discrete time by Garsia Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, 1973, are extended to continuous time
Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books
Keywords: Inequalities, Burkholder inequalities
Nature: Original
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IX: 07, 237-238, LNM 465 (1975)
MEYER, Paul-André
Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)
Fills a gap in the proof of the duality theorem in 714
Keywords: $BMO$
Nature: Correction
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IX: 19, 408-419, LNM 465 (1975)
STRICKER, Christophe
Mesure de Föllmer en théorie des quasimartingales (Martingale theory)
The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer, Z. für W-theorie, 21, 1972; Ann. Prob. 1, 1973) is constructed using a weak limit procedure instead of a projective limit
Comment: On Föllmer measures see 611. This paper corresponds to an early stage in the theory of quasimartingales, for which the main reference was Orey, Proc. Fifth Berkeley Symp., 2
Keywords: Quasimartingales, Föllmer measures
Nature: Original
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IX: 20, 420-424, LNM 465 (1975)
STRICKER, Christophe
Une caractérisation des quasimartingales (Martingale theory)
An integral criterion is shown to be equivalent to the usual definition of a quasimartingale using the stochastic variation
Keywords: Quasimartingales
Nature: Original
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X: 01, 1-18, LNM 511 (1976)
BRÉMAUD, Pierre
La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (Martingale theory, Point processes)
This paper discusses martingale methods (as developed by Jacod, Z. für W-theorie, 31, 1975) in the filtering theory of point processes
Comment: The author has greatly developed this topic in his book Poisson Processes and Queues, Springer 1981
Keywords: Point processes, Previsible representation, Filtering theory
Nature: Original
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X: 03, 24-39, LNM 511 (1976)
JACOD, Jean; MÉMIN, Jean
Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)
The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration
Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation
Nature: Original
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X: 04, 40-43, LNM 511 (1976)
KAZAMAKI, Norihiko
A simple remark on the conditioned square functions for martingale transforms (Martingale theory)
This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time
Comment: The author has published a paper on a related subject in Tôhoku Math. J., 28, 1976
Keywords: Angle bracket, Inequalities
Nature: Original
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X: 10, 125-183, LNM 511 (1976)
MEYER, Paul-André
Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)
This long paper consists of four talks, suggested by E.M.~Stein's book Topics in Harmonic Analysis related to the Littlewood-Paley theory, Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $R^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $R^n\times R_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $R^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (CRAS Paris, 278A, 1974, p.1103) in potential theory and by Kunita (Nagoya M. J., 36, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false
Comment: This paper was rediscovered by Varopoulos (J. Funct. Anal., 38, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253
Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory
Nature: Original
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X: 13, 209-215, LNM 511 (1976)
SEKIGUCHI, Takesi
On the Krickeberg decomposition of continuous martingales (Martingale theory)
The problem investigated is whether the two positive martingales occurring in the Krickeberg decomposition of a $L^1$-bounded continuous martingale of a filtration $({\cal F}_t)$ are themselves continuous. It is shown that the answer is yes only under very stringent conditions: there exists a sub-filtration $({\cal G}_t)$ such that 1) all ${\cal G}$-martingales are continuous 2) the continuous ${\cal F}$-martingales are exactly the ${\cal G}$-martingales
Comment: For related work of the author see Tôhoku Math. J. 28, 1976
Keywords: Continuous martingales, Krickeberg decomposition
Nature: Original
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X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books
Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem
Nature: Exposition, Original additions
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X: 18, 401-413, LNM 511 (1976)
PRATELLI, Maurizio
Sur certains espaces de martingales de carré intégrable (Martingale theory)
The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$
Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250
Keywords: Inequalities, Angle bracket, $BMO$
Nature: Original
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X: 19, 414-421, LNM 511 (1976)
PRATELLI, Maurizio
Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)
This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)
Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals
Nature: Original
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X: 20, 422-431, LNM 511 (1976)
YAN, Jia-An; YOEURP, Chantha
Représentation des martingales comme intégrales stochastiques des processus optionnels (Martingale theory, Stochastic calculus)
An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one
Comment: Apparently this ``optional representation property'' has not been used since
Keywords: Optional stochastic integrals
Nature: Original
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X: 21, 432-480, LNM 511 (1976)
YOEURP, Chantha
Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)
It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $&#279;lta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality
Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN 1118, Grossissements de filtrations, 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer 1023. The general case is due to Azéma (Z. für W-theorie, 45, 1978, presented in 1321) See also 1622
Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures
Nature: Original
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X: 22, 481-500, LNM 511 (1976)
YOR, Marc
Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)
This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up
Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems
Nature: Original
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X: 23, 501-504, LNM 511 (1976)
MEYER, Paul-André; YOEURP, Chantha
Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)
This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$
Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620
Keywords: Multiplicative decomposition
Nature: Original
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X: 27, 536-539, LNM 511 (1976)
KAZAMAKI, Norihiko
A characterization of $BMO$ martingales (Martingale theory)
A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$
Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN 1579, 1994
Keywords: $BMO$
Nature: Original
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XI: 12, 132-195, LNM 581 (1977)
MEYER, Paul-André
Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)
This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(R^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right
Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected
Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms
Nature: Exposition, Original additions
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XI: 16, 303-323, LNM 581 (1977)
BERNARD, Alain; MAISONNEUVE, Bernard
Décomposition atomique de martingales de la classe $H^1$ (Martingale theory)
Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (Studia Math. 51, 1974). An atomic decomposition of a Banach space consists in finding simple elements (called atoms) in its unit ball, such that every element is a linear combination of atoms $\sum_n \lambda_n a_n$ with $\sum_n \|\lambda_n\|<\infty$, the infimum of this sum defining the norm or an equivalent one. Here an atomic decomposition is given for $H^1$ spaces of martingales in continuous time (defined by their maximal function). Atoms are of two kinds: the first kind consists of martingales bounded uniformly by a constant $c$ and supported by an interval $[T,\infty[$ such that $P\{T<\infty\}\le 1/c$. These atoms do not generate the whole space $H^1$ in general, though they do in a few interesting cases (if all martingales are continuous, or in the discrete dyadic case). To generate the whole space it is sufficient to add martingales of integrable variation (those whose total variation has an $L^1$ norm smaller than $1$ constitute the second kind of atoms). This approach leads to a proof of the $H^1$-$BMO$ duality and the Davis inequality
Comment: See also 1117
Keywords: Atomic decompositions, $H^1$ space, $BMO$
Nature: Original
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XI: 17, 324-326, LNM 581 (1977)
BERNARD, Alain
Complément à l'exposé précédent (Martingale theory)
This paper is a sequel to 1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$
Keywords: Atomic decompositions, $H^1$ space, $BMO$
Nature: Original
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XI: 21, 356-361, LNM 581 (1977)
CHOU, Ching Sung
Le processus des sauts d'une martingale locale (Martingale theory)
Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $&#279;lta M_t$ of some local martingale $M$
Comment: The same result is proved independently by D. Lépingle in this volume, see 1129. For an application see 1308, and for another approach 1335
Keywords: Local martingales, Jumps
Nature: Original
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XI: 22, 362-364, LNM 581 (1977)
DELLACHERIE, Claude
Sur la régularisation des surmartingales (Martingale theory)
It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)
Comment: See 1524
Keywords: General filtrations, Strong supermartingales
Nature: Original
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XI: 23, 365-375, LNM 581 (1977)
DELLACHERIE, Claude; STRICKER, Christophe
Changements de temps et intégrales stochastiques (Martingale theory)
A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135
Comment: The last section states an interesting open problem
Keywords: Changes of time, Spectral representation
Nature: Original
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XI: 25, 383-389, LNM 581 (1977)
DOLÉANS-DADE, Catherine; MEYER, Paul-André
Une caractérisation de $BMO$ (Martingale theory)
Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention
Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible
Keywords: $BMO$, Stochastic exponentials, Martingale inequalities
Nature: Original
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XI: 26, 390-410, LNM 581 (1977)
JACOD, Jean
Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (Martingale theory)
This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (1121, 1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined
Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod, Calcul stochastique et problèmes de martingales, Springer 1979. Other, equivalent, definitions are given in 1415, 1417, 1424 and 1530
Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales
Nature: Original
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XI: 29, 418-434, LNM 581 (1977)
LÉPINGLE, Dominique
Sur la représentation des sauts des martingales (Martingale theory)
The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool
Comment: See also 1335
Keywords: Local martingales, Jumps, Optional stochastic integrals
Nature: Original
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XI: 30, 435-445, LNM 581 (1977)
MAISONNEUVE, Bernard
Une mise au point sur les martingales locales continues définies sur un intervalle stochastique (Martingale theory)
The following definition is given of a continuous local martingale $M$ on an open interval $[0,T[$, for an arbitrary stopping time $T$: two sequences are assumed to exist, one of stopping times $T_n\uparrow T$, one $(M_n)$ of continuous martingales, such that $M=M_n$ on $[0,T_n[$. Stochastic integration is studied, and the change of variable formula is extended. It is proved that the set where the limit $M_{T-}$ exists and is finite is a.s. the same as that where $\langle M,M\rangle_T<\infty$, a result whose proof under the usual definition (i.e., assuming $T$ is previsible) was not clear
Keywords: Martingales on a random set, Stochastic integrals
Nature: Original
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XI: 31, 446-481, LNM 581 (1977)
MEYER, Paul-André
Notes sur les intégrales stochastiques (Martingale theory)
This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times
Comment: Three errors are corrected in 1248 and 1249
Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$
Nature: Original
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XI: 32, 482-489, LNM 581 (1977)
MEYER, Paul-André
Sur un théorème de C. Stricker (Martingale theory)
Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex
Comment: A gap in a proof is filled in 1251
Keywords: Hardy spaces, Changes of measure
Nature: Original
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XI: 33, 490-492, LNM 581 (1977)
WALSH, John B.
A property of conformal martingales (Martingale theory)
Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense
Comment: See also 1408
Keywords: Conformal martingales
Nature: Original
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XI: 34, 493-501, LNM 581 (1977)
YOR, Marc
A propos d'un lemme de Ch. Yoeurp (General theory of processes, Martingale theory)
Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales
Keywords: Yoeurp's lemma, Square bracket
Nature: Original
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XI: 35, 502-517, LNM 581 (1977)
YOR, Marc
Remarques sur la représentation des martingales comme intégrales stochastiques (Martingale theory)
The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor, Z. für W-theorie, 38, 1977. Several concrete applications are given here, in particular a complete proof of a ``folklore'' result on the representation of local martingales of a Lévy process, and a discussion of the commutation problem of 1123
Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see 1221
Keywords: Previsible representation, Extreme points, Independent increments, Lévy processes
Nature: Original
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XI: 36, 518-528, LNM 581 (1977)
YOR, Marc
Sur quelques approximations d'intégrales stochastiques (Martingale theory)
The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form
Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe Publ. RIMS, Kyoto Univ. 15, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer 1505
Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals
Nature: Original
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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: 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: 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: 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: 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: 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: 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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XIII: 07, 116-117, LNM 721 (1979)
ÉMERY, Michel; STRICKER, Christophe
Démonstration élémentaire d'un résultat d'Azéma et Jeulin (Martingale theory)
A short proof is given for the following result: given a positive supermartingale $X$ and $h>0$, the supermartingale $E[X_{t+h}\,|\,{\cal F}_t]$ belongs to the class (D). The original proof (Ann. Inst. Henri Poincaré, 12, 1976) used Föllmer's measures
Keywords: Class (D) processes
Nature: Original
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XIII: 08, 118-125, LNM 721 (1979)
YOEURP, Chantha
Sauts additifs et sauts multiplicatifs des semi-martingales (Martingale theory, General theory of processes)
First of all, the jump processes of special semimartingales are characterized (using a result of 1121, 1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al. 1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes
Keywords: Jump processes
Nature: Original
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XIII: 09, 126-131, LNM 721 (1979)
PRATELLI, Maurizio
Le support exact du temps local d'une martingale continue (Martingale theory)
It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$
Keywords: Local times
Nature: Original
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XIII: 10, 132-137, LNM 721 (1979)
SIDIBÉ, Ramatoulaye
Martingales locales à accroissements indépendants (Martingale theory, Independent increments)
It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale
Comment: The case of non-stationary increments is considered in 1544. See also the errata sheet of vol. XV
Keywords: Local martingales, Lévy processes
Nature: Original
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XIII: 11, 138-141, LNM 721 (1979)
REBOLLEDO, Rolando
Décomposition des martingales locales et raréfaction des sauts (Martingale theory)
The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property
Comment: A correction is given as 1430
Keywords: Convergence in law, Tightness
Nature: Original
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XIII: 12, 142-161, LNM 721 (1979)
MÉMIN, Jean; SHIRYAEV, Albert N.
Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (Martingale theory)
A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales
Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition, Local characteristics
Nature: Original
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XIII: 17, 216-226, LNM 721 (1979)
LETTA, Giorgio
Quasimartingales et formes linéaires associées (General theory of processes, Martingale theory)
This paper gives a definition of a quasimartingale $X$ different from the usual one using the stochastic variation: the linear functional $E[\int_0^{\infty} H_s dX_s]$ should be relatively bounded on the vector lattice (Riesz space) of elementary previsible processes. Many classical theorems have simple proofs or elegant interpretations in this language
Keywords: Quasimartingales, Riesz spaces
Nature: Original
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XIII: 18, 227-232, LNM 721 (1979)
BRUNEAU, Michel
Sur la $p$-variation d'une surmartingale continue (Martingale theory)
The $p$-variation of a deterministic function being defined in the obvious way as a supremum over all partitions, the sample functions of a continuous martingale (and therefore semimartingale) are known to be of finite $p$-variation for $p>2$ (not for $p=2$ in general: non-anticipating partitions are not sufficient to compute the $p$-variation). If $X$ is a continuous supermartingale, a universal bound is given on the expected $p$-variation of $X$ on the interval $[0,T_\lambda]$, where $T_\lambda=\inf\{t:|X_t-X_0|\ge\lambda\}$. The main tool is Doob's classical upcrossing inequality
Comment: For an extension see 1319. These properties are used in T.~Lyons' pathwise theory of stochastic differential equations; see his long article in Rev. Math. Iberoamericana 14, 1998
Keywords: $p$-variation, Upcrossings
Nature: Original
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XIII: 19, 233-237, LNM 721 (1979)
STRICKER, Christophe
Sur la $p$-variation des surmartingales (Martingale theory)
The method of the preceding paper of Bruneau 1318 is extended to all right-continuous semimartingales
Keywords: $p$-variation, Upcrossings
Nature: Original
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XIII: 20, 238-239, LNM 721 (1979)
STRICKER, Christophe
Une remarque sur l'exposé précédent (Martingale theory)
A few comments are added to the preceding paper 1319, concerning in particular its relationship with results of Lépingle, Zeit. für W-Theorie, 36, 1976
Keywords: $p$-variation, Upcrossings
Nature: Original
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XIII: 21, 240-249, LNM 721 (1979)
MEYER, Paul-André
Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)
The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (Zeit. für W-Theorie, 45, 1978) is the introduction of a multiplicative system as a two-parameter process $C_{st}$ taking values in $[0,1]$, defined for $s\le t$, such that $C_{tt}=1$, $C_{st}C_{tu}=C_{su}$ for $s\le t\le u$, decreasing and previsible in $t$ for fixed $s$, such that $E[C_{st}Y_t\,|\,{\cal F}_s]=Y_s$ for $s<t$. Then for fixed $s$ the process $(C_{st}Y_t)$ turns out to be a right-continuous martingale on $[s,\infty[$, and what we have done amounts to pasting together all the multiplicative decompositions on zero-free intervals. Existence (and uniqueness of multiplicative systems are proved, though the uniqueness result is slightly different from Azéma's
Keywords: Multiplicative decomposition
Nature: Exposition, Original additions
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XIII: 22, 250-252, LNM 721 (1979)
CHOU, Ching Sung
Caractérisation d'une classe de semimartingales (Martingale theory, Stochastic calculus)
The class of semimartingales $X$ such that the stochastic integral $J\,.\,X$ is a martingale for some nowhere vanishing previsible process $J$ is a natural class of martingale-like processes. Local martingales are exactly the members of this class which are special semimartingales
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)
Keywords: Local martingales, Stochastic integrals
Nature: Original
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XIII: 26, 294-306, LNM 721 (1979)
BONAMI, Aline; LÉPINGLE, Dominique
Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)
Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $P$ two functionals of a $P$-martingale---remains true, possibly with new constants, when $P$ is replaced by an equivalent law $Z.P$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities
Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance
Keywords: Weighted norm inequalities, Burkholder inequalities
Nature: Original
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XIII: 27, 307-312, LNM 721 (1979)
IZUMISAWA, Masataka; SEKIGUCHI, Takesi
Weighted norm inequalities for martingales (Martingale theory)
See the review of 1326. The topic is the same, though the proof is different
Comment: See the paper by Kazamaki-Izumisawa in Tôhoku Math. J. 29, 1977. For a modern reference see also Kazamaki, Continuous Exponential Martingales and $\,BMO$, LNM. 1579, 1994
Keywords: Weighted norm inequalities, Burkholder inequalities
Nature: Original
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XIII: 28, 313-331, LNM 721 (1979)
DOLÉANS-DADE, Catherine; MEYER, Paul-André
Inégalités de normes avec poids (Martingale theory)
See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory
Comment: An exponent $1/\lambda$ is missing in formula (4), p.315
Keywords: Weighted norm inequalities
Nature: Exposition, Original additions
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XIII: 29, 332-359, LNM 721 (1979)
JEULIN, Thierry; YOR, Marc
Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)
The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears
Keywords: Hardy's inequality, Previsible representation
Nature: Original
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XIII: 30, 360-370, LNM 721 (1979)
JEULIN, Thierry; YOR, Marc
Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)
The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$
Comment: On the same topic see 1518
Keywords: $BMO$, $H^1$ space, Hardy spaces
Nature: Original
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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 functions
Nature: Exposition, Original proofs
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XIII: 33, 385-399, LNM 721 (1979)
LE JAN, Yves
Martingales et changement de temps (Martingale theory, Markov processes)
The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary
Keywords: Changes of time, Energy, Douglas formula
Nature: Original
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XIII: 34, 400-406, LNM 721 (1979)
YOR, Marc
Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)
This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$
Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$
Nature: Original
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XIII: 52, 611-613, LNM 721 (1979)
MEYER, Paul-André
Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)
In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (Ann. Prob. 8, 1980) to develop the whole theory of stochastic differential equations
Keywords: Doob's inequality, Stochastic differential equations
Nature: Exposition
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XIII: 58, 642-645, LNM 721 (1979)
MAISONNEUVE, Bernard
Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)
The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (Ann. Prob., 7, 1979). This proof is further simplified and slightly generalized
Comment: See also 1407
Keywords: Gilat's theorem
Nature: Exposition, Original additions
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XIV: 04, 26-48, LNM 784 (1980)
LENGLART, Érik; LÉPINGLE, Dominique; PRATELLI, Maurizio
Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)
This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes
Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer, Probabilités et Potentiel E ; see also 1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.} 49,1982
Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities
Nature: Original
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XIV: 05, 49-52, LNM 784 (1980)
LENGLART, Érik
Appendice à l'exposé précédent~: inégalités de semimartingales (Martingale theory, Stochastic calculus)
This paper contains several applications of the methods of 1404 to the case of semimartingales instead of martingales
Keywords: Inequalities, Semimartingales
Nature: Original
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XIV: 06, 53-61, LNM 784 (1980)
AZÉMA, Jacques; GUNDY, Richard F.; YOR, Marc
Sur l'intégrabilité uniforme des martingales exponentielles (Martingale theory)
The main result of this paper is the following: Let $X$ be a martingale which is continuous and bounded in $L^1$ (both conditions are essential). Then $X$ is uniformly integrable if and only if $tP\{X^{*}>t\}$ or equivalently $tP\{S(X)>t\}$ tend to $0$ as $t\rightarrow\infty$, where $S(X)$ is the usual square function. The methods (using a good lambda inequality) are close to 1404
Comment: Generalized by Takaoka 3313
Keywords: Exponential martingales, Continuous martingales
Nature: Original
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XIV: 07, 62-75, LNM 784 (1980)
BARLOW, Martin T.; YOR, Marc
Sur la construction d'une martingale continue de valeur absolue donnée (Martingale theory)
This paper consists of two notes on Gilat's theorem (Ann. Prob. 5, 1977, See also 1358). The problem consists in constructing, given a continuous positive submartingale $Y$, a continuous martingale $X$ (possibly on a different space) such that $|X|$ has the same law as $Y$. Let $A$ be the increasing process associated with $Y$; it is necessary for the existence of $X$ that $dA$ should be carried by $\{Y=0\}$. This is shown by the first note (Yor's) to be also sufficient---more precisely, in this case the solutions of Gilat's problem are all continuous. The second note (Barlow's) shows how to construct a Gilat martingale by ``putting a random $\pm$ sign in front of each excursion of $Y$'', a simple intuitive idea and a delicate proof
Keywords: Gilat's theorem
Nature: Original
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XIV: 08, 76-101, LNM 784 (1980)
SHARPE, Michael J.
Local times and singularities of continuous local martingales (Martingale theory)
This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh 1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$
Comment: See Calais-Genin 1717
Keywords: Local times, Local martingales, Semimartingales in an open interval
Nature: Original
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XIV: 09, 102-103, LNM 784 (1980)
MEYER, Paul-André
Sur un résultat de L. Schwartz (Martingale theory)
the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (Semimartingales dans les variétés..., Lecture Notes in M. 780): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$
Comment: The results are extended in Meyer-Stricker Stochastic Analysis and Applications, part B, Advances in M. Supplementary Studies, 1981
Keywords: Semimartingales in a random open set
Nature: Exposition, Original additions
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XIV: 27, 227-248, LNM 784 (1980)
JACOD, Jean; MÉMIN, Jean
Sur la convergence des semimartingales vers un processus à accroissements indépendants (General theory of processes, Stochastic calculus, Martingale theory)
A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)
Keywords: Convergence in law, Tightness
Nature: Original
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XIV: 30, 255-255, LNM 784 (1980)
REBOLLEDO, Rolando
Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (General theory of processes, Martingale theory)
Concerns 1311. For the definitive version, see Mém. Soc. Math. France, 62, 1979
Keywords: Central limit theorem, Skorohod topology, Local martingales, Jumps
Nature: Correction
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XIV: 48, 496-499, LNM 784 (1980)
COCOZZA-THIVENT, Christiane; YOR, Marc
Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (Martingale theory)
This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way
Comment: See 518 for an earlier proof
Keywords: Changes of time
Nature: Original
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XV: 10, 151-166, LNM 850 (1981)
MEYER, Paul-André
Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)
The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (J. Funct. Anal., 38, 1980)
Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)
Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ
Nature: Original
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XV: 17, 251-258, LNM 850 (1981)
PITMAN, James W.
A note on $L_2$ maximal inequalities (Martingale theory)
This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$
Keywords: Maximal inequality, Doob's inequality
Nature: Original
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XV: 18, 259-277, LNM 850 (1981)
BRU, Bernard; HEINICH, Henri; LOOTGIETER, Jean-Claude
Autour de la dualité $(H^1,BMO)$ (Martingale theory)
This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand
Keywords: $BMO$, $H^1$ space, Hardy spaces
Nature: Original
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XV: 19, 278-284, LNM 850 (1981)
ÉMERY, Michel
Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)
Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential
Keywords: $BMO$
Nature: Exposition, Original additions
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XV: 20, 285-289, LNM 850 (1981)
CHOU, Ching Sung
Une inégalité de martingales avec poids (Martingale theory)
Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)
Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions
Nature: Original
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XV: 25, 347-350, LNM 850 (1981)
MAISONNEUVE, Bernard
Surmartingales-mesures (Martingale theory)
Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu, Martingales à temps discret
Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (Zeit. für W-theorie, 1978) in continuous time. This topic is now called Kunita decomposition, see 1005 and the corresponding references
Keywords: Supermartingales, Kunita decomposition
Nature: Original
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XV: 28, 388-398, LNM 850 (1981)
SPILIOTIS, Jean
Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)
The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see Izvestija Akad Nauk, 38, 1974, and Krylov's book Controlled Diffusion processes, Springer 1980) dealt with the existence of densities for several dimensional stochastic integrals with measurable bounded integrands, satisfying an ellipticity condition. It is a puzzling fact that nobody ever succeeded in unifying these results. Krylov's method depends on results of the Russian school on Monge-Ampère equations (see Pogorelov The Minkowski Multidimensional Problem, 1978). This exposition attempts, rather modestly, to explain in the seminar's language what it is all about, and in particular to show the place where a crucial lemma on convex functions is used
Keywords: Stochastic integrals, Existence of densities
Nature: Exposition
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XV: 32, 493-498, LNM 850 (1981)
STRICKER, Christophe
Quasi-martingales et variations (Martingale theory)
This paper contains remarks on quasimartingales, the most useful of which being perhaps the fact that, for a right-continuous process, the stochastic variation is the same with respect to the filtrations $({\cal F}_{t})$ and $({\cal F}_{t-})$
Keywords: Quasimartingales
Nature: Original
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XV: 40, 590-603, LNM 850 (1981)
STROOCK, Daniel W.; YOR, Marc
Some remarkable martingales (Martingale theory)
This is a sequel to a well-known paper by the authors (Ann. ENS, 13, 1980) on the subject of pure martingales. A continuous martingale $(M_t)$ with $<M,M>_{\infty}=\infty$ is pure if the time change which reduces it to a Brownian motion $(B_t)$ entails no loss of information, i.e., if $M$ is measurable w.r.t. the $\sigma$-field generated by $B$. The first part shows the purity of certain stochastic integrals. Among the striking examples considered, the stochastic integrals $\int_0^t B^n_sdB_s$ are extremal for every integer $n$, pure for $n$ odd, but nothing is known for $n$ even. A beautiful result unrelated to purity is the following: complex Brownian motion $Z_t$ starting at $z_0$ and its (Lévy) area integral generate the same filtration if and only if $z_0\neq0$
Keywords: Pure martingales, Previsible representation
Nature: Original
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XV: 41, 604-617, LNM 850 (1981)
LÉPINGLE, Dominique; MEYER, Paul-André; YOR, Marc
Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)
This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case
Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps
Nature: Original
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XV: 45, 643-668, LNM 850 (1981)
AUERHAN, J.; LÉPINGLE, Dominique
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)
This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given
Comment: Further results are given by Malric Ann. Inst. H. Poincaré 26 (1990)
Nature: Original
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XVI: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique; MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (Trans. Amer Math. Soc., 210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials
Nature: Original
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XVI: 10, 151-152, LNM 920 (1982)
MEYER, Paul-André
Sur une inégalité de Stein (Applications of martingale theory)
In his book Topics in harmonic analysis related to the Littlewood-Paley theory (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation
Keywords: Littlewood-Paley theory, Martingale inequalities
Nature: Exposition, Original additions
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XVI: 18, 219-220, LNM 920 (1982)
STRICKER, Christophe
Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)
For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property
Keywords: Quadratic variation, Previsible representation
Nature: Original
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XVI: 19, 221-233, LNM 920 (1982)
YOR, Marc
Application de la relation de domination à certains renforcements des inégalités de martingales (Martingale theory)
The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper 1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman 1517
Comment: See an earlier paper of the author on this subject, Stochastics, 3, 1979. The author mentions that part of the results were discovered slightly earlier by R.~Gundy
Keywords: Martingale inequalities, Domination inequalities
Nature: Original
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XVI: 25, 285-297, LNM 920 (1982)
UPPMAN, Are
Un théorème de Helly pour les surmartingales fortes (Martingale theory)
Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences
Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory
Keywords: Supermartingales, Strong supermartingales
Nature: Original
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XVI: 29, 338-347, LNM 920 (1982)
YAN, Jia-An
À propos de l'intégrabilité uniforme des martingales exponentielles (Martingale theory)
Sufficient conditions are given for the uniform integrability of the exponential ${\cal E}(M)$, where $M$ is a local martingale with jumps $\ge-1$, refining older results of Lépingle and Mémin, and of the author. They involve the Lévy measure of the martingale
Comment: In the lemma p.339 delete the assumption $0<\beta$
Keywords: Exponential martingales
Nature: Original
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XVIII: 32, 500-500, LNM 1059 (1984)
ÉMERY, Michel
Sur l'exponentielle d'une martingale de $BMO$ (Martingale theory)
This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale
Keywords: Stochastic exponentials, $BMO$
Nature: Original
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XX: 13, 162-185, LNM 1204 (1986)
BOULEAU, Nicolas; LAMBERTON, Damien
Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)
Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $R_+$ defined by $M(\lambda)=\lambda\int_0^\infty r(y)e^{-y\lambda}dy,$ where $r(y)$ is bounded and Borel on $R_+$, then the operator $T_M=\int_{[0,\infty)}M(\lambda)dE_{\lambda},$ which is obviously bounded on $L^2$, is actually bounded on all $L^p$ spaces of the invariant measure, $1<p<\infty$. The method also leads to new Littlewood-Paley inequalities for semigroups admitting a carré du champ operator
Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ
Nature: Original
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XXII: 14, 147-154, LNM 1321 (1988)
ÉMERY, Michel
En cherchant une caractérisation variationnelle des martingales (Martingale theory)
Let $\mu$ be a probability on $R_+$ and $\cal H$ the Hilbert space of all measurable and adapted processes $X$ such that $E[\int_0^\infty X_s^2\mu(ds)$ is finite. Martingales in $\cal H$ are characterized as minimizers of the $\cal H$-norm among all $X$ such that $\int_0^\infty X_s\mu(ds)$ is a given random variable
Comment: There is a large overlap with Pliska, Springer LN in Control and Information Theory 43, 1983
Keywords: Martingales
Nature: Well-known
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XXXI: 12, 113-125, LNM 1655 (1997)
ELWORTHY, Kenneth David; LI, Xu-Mei; YOR, Marc
On the tails of the supremum and the quadratic variation of strictly local martingales (Martingale theory)
The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given
Comment: In discrete time, see the following article 3113. Related results are due to Takaoka 3313
Keywords: Continuous martingales, Local martingales, Quadratic variation, Maximal process
Nature: Original
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XXXI: 16, 176-189, LNM 1655 (1997)
ÉMERY, Michel
Closed sets supporting a continuous divergent martingale (Martingale theory)
This note gives a characterization of all closed subsets $F$ of $R^d$ such that, for every $F$-valued continuous martingale $X$, the limit $X_\infty$ exists in $F$ (or $R^d$) with non-zero probability. The criterion is as follows: To each $F$ is associated a smaller closed set $F'$ obtained, roughly speaking, by chopping off all prominent parts of $F$; this map $F\mapsto F'$ is iterated, giving a decreasing sequence $(F^n)$ with limit $F^\infty$; the condition is that $F^\infty$ is empty. (If $d=2$, $F^\infty$ is also the largest closed subset of $F$ such that all connected components of its complementary are convex)
Comment: Two similar problems are discussed in 1485
Keywords: Continuous martingales, Asymptotic behaviour of processes
Nature: Original
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XXXIII: 13, 327-333, LNM 1709 (1999)
TAKAOKA, Koichiro
Some remarks on the uniform integrability of continuous martingales (Martingale theory)
For a continuous local martingale which converges a.s., a general relation links the asymptotic tails of the maximal variable and the quadratic variation. This unifies previous results by Azéma-Gundy-Yor 1406, Elworthy-Li-Yor 3112 and Probab. Theory Related Fields 115 (1999)
Keywords: Uniform integrability, Continuous martingales, Local martingales
Nature: Original
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XXXIII: 16, 342-348, LNM 1709 (1999)
GRANDITS, Peter
Some remarks on L$^\infty $, H$^\infty $, and $BMO$ (Martingale theory)
It is known from 1212 that neither $L^\infty$ nor $H^\infty$ is dense in $BMO$. This article answers a question raised by Durrett (Brownian Motion and Martingales in Analysis, Wadworth 1984): Does there exist a $BMO$-martingale which has a best approximation in $L^\infty$? The answer is negative, but becomes positive if $L^\infty$ is replaced with $H^\infty$
Keywords: $BMO$, Hardy spaces
Nature: Original
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XLIII: 17, 395-412, LNM 2006 (2011)
CZICHOWSKY, Christoph; WESTRAY, Nicholas; ZHENG, Harry
Convergence in the semimartingale topology and constrained portfolios (Martingale theory, Mathematical finance)
Nature: Original

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XLIII: 19, 437-439, LNM 2006 (2011)
BAKER, David; YOR, Marc
On martingales with given marginals and the scaling property (Martingale theory, Theory of Brownian motion)
Nature: Original

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XLIII: 20, 441-449, LNM 2006 (2011)
BAKER, David; DONATI-MARTIN, Catherine; YOR, Marc
A sequence of Albin type continuous martingales with Brownian marginals and scaling (Martingale theory)
Keywords: Martingales, Brownian marginals
Nature: Original

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XLIII: 21, 451-503, LNM 2006 (2011)
HIRSCH, Francis; PROFETA, Christophe; ROYNETTE, Bernard; YOR, Marc
Constructing self-similar martingales via two Skorokhod embeddings (Martingale theory)
Keywords: Skorokhod embeddings, Hardy-Littlewood functions, Convex order, Schauder fixed point theorem, Self-similar martingales, Karamata's representation theorem
Nature: Original

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