Browse by: Author name - Classification - Keywords - Nature

44 matches found
IX: 21, 425-436, LNM 465 (1975)
ÉMERY, Michel
Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)
It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz, Ann. Inst. Fourier, 6, 1955-56) that the distribution functions of measures are characterized by simple algebraic properties and right continuity
Comment: This elegant theorem apparently never had applications
Keywords: Distribution functions on ordered spaces
Nature: Exposition
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XI: 39, 566-573, LNM 581 (1977)
ÉMERY, Michel
Information associée à un semigroupe (Markov processes)
This paper contains the proof of two important theorems of Donsker and Varadhan (Comm. Pure and Appl. Math., 1975)
Nature: Exposition
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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: 24, 260-280, LNM 721 (1979)
ÉMERY, Michel
Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)
The stability theory for stochastic differential equations was developed independently by Émery (Zeit. für W-Theorie, 41, 1978) and Protter (same journal, 44, 1978). However, these results were stated in the language of convergent subsequences instead of true topological results. Here a linear topology (like convergence in probability: metrizable, complete, not locally convex) is defined on the space of semimartingales. Side results concern the Banach spaces $H^p$ and $S^p$ of semimartingales. Several useful continuity properties are proved
Comment: This topology has become a standard tool. For its main application, see the next paper 1325
Keywords: Semimartingales, Spaces of semimartingales
Nature: Original
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XIII: 25, 281-293, LNM 721 (1979)
ÉMERY, Michel
Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (Stochastic calculus)
This is the main application of the topologies on processes and semimartingales introduced in 1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools
Comment: For all of this subject, the book of Protter Stochastic Integration and Differential Equations, Springer 1989, is a useful reference
Keywords: Stochastic differential equations, Stability
Nature: Original
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XIV: 13, 118-124, LNM 784 (1980)
ÉMERY, Michel
Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (Stochastic calculus)
Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see 1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales
Comment: See 1352. A general reference on the Métivier-Pellaumail method can be found in their book Stochastic Integration, Academic Press 1980. See also He-Wang-Yan, Semimartingale Theory and Stochastic Calculus, CRC Press 1992
Keywords: Semimartingales, Spaces of semimartingales, Stochastic differential equations, Doob's inequality, Métivier-Pellaumail inequality
Nature: Original
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XIV: 16, 140-147, LNM 784 (1980)
ÉMERY, Michel
Métrisabilité de quelques espaces de processus aléatoires (General theory of processes, Stochastic calculus)
As a sequel to the main work of 1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete
Keywords: Spaces of semimartingales
Nature: Original
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XIV: 18, 152-160, LNM 784 (1980)
ÉMERY, Michel
Compensation de processus à variation finie non localement intégrables (General theory of processes, Stochastic calculus)
First an example is given of a local martingale $M$ and an unbounded previsible process $H$ such that $H.M$ exists in the sense of 1126 and 1415, but is not a local martingale. This leads to a natural enlargement of the class of local martingales, which turns out to be the same suggested by Chou in 1322 under the name of class $(\Sigma_m)$. Once the class has been so extended, the operation of previsible compensation can be extended to a class of processes with finite variation, but not locally integrable variation, and the class of special semimartingales can be also enlarged
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997). Using the language of L. Schwartz 1530, it is the intersection of the set of (usual) semimartingales with the set of formal martingales
Keywords: Local martingales, Stochastic integrals, Compensators
Nature: Original
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XIV: 34, 316-317, LNM 784 (1980)
ÉMERY, Michel
Une propriété des temps prévisibles (General theory of processes)
The idea is to prove that the general theory of processes on a random interval $[0,T[$, where $T$ is previsible, is essentially the same as on $[0,\infty[$. To this order, a continuous, strictly increasing adapted process $(A_t)$ is constructed, such that $A_0=0$, $A_T=1$
Keywords: Previsible times
Nature: Original
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XIV: 35, 318-323, LNM 784 (1980)
ÉMERY, Michel
Annonçabilité des temps prévisibles. Deux contre-exemples (General theory of processes)
It is shown that two standard results on previsible stopping times on a probability space, namely that every previsible time $T$ can be foretold'' by a strictly increasing sequence $T_n\uparrow T$, and that the $T_n$ can themselves be taken previsible, become false if exceptional sets of measure zero are not allowed
Keywords: Previsible times
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: $BMO$
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XV: 39, 587-589, LNM 850 (1981)
ÉMERY, Michel
Non-confluence des solutions d'une équation stochastique lipschitzienne (Stochastic calculus)
This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet
Keywords: Stochastic differential equations, Flow of a s.d.e.
Nature: Original
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XVI-S: 58, 208-216, LNM 921 (1982)
ÉMERY, Michel
En marge de l'exposé de Meyer : Géométrie différentielle stochastique'' (Stochastic differential geometry)
Marginal remarks to Meyer 1657
Keywords: Semimartingales in manifolds, Stochastic differential equations
Nature: Original
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XVII: 19, 185-186, LNM 986 (1983)
ÉMERY, Michel
Note sur l'exposé précédent (Stochastic calculus)
A small remark on 1718: The event where a semimartingale converges perfectly is also the smallest (modulo negligibility) event where it is a semimartingale up to infinity
Keywords: Semimartingales
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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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XVIII: 33, 501-518, LNM 1059 (1984)
ÉMERY, Michel; ZHENG, Wei-An
Fonctions convexes et semimartingales dans une variété (Stochastic differential geometry)
On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling 1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales
Keywords: Semimartingales in manifolds, Martingales in manifolds, Convex functions
Nature: Original
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XIX: 12, 177-206, LNM 1123 (1985)
BAKRY, Dominique; ÉMERY, Michel
Diffusions hypercontractives
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XXI: 09, 173-175, LNM 1247 (1987)
ÉMERY, Michel; YUKICH, Joseph E.
A simple proof of the logarithmic Sobolev inequality on the circle (Real analysis)
The same kind of semi-group argument as in Bakry-Émery 1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle
Keywords: Logarithmic Sobolev inequalities
Nature: New proof of known results
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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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XXIII: 06, 66-87, LNM 1372 (1989)
ÉMERY, Michel
On the Azéma martingales
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XXIV: 28, 407-441, LNM 1426 (1990)
ÉMERY, Michel
On two transfer principles in stochastic differential geometry (Stochastic differential geometry)
Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer 1657, Schwartz 1655 or Emery Stochastic calculus in manifolds). The first one gives a Stratonovich SDE and needs coefficients regular enough; the second one gives an Ito equation and needs a connection on the manifold. Discretizing time and smoothly interpolating the driving semimartingale is known to give an approximation to the Stratonovich transfer; it is shown here that another discretized-time procedure converges to the Ito transfer. As an application, if the ODE makes geodesics to geodesics, then the Ito and Stratonovich SDE's are the same
Comment: An error is corrected in 2649. The term transfer principle'' was coined by Malliavin, Géométrie Différentielle Stochastique, Presses de l'Université de Montréal (1978); see also Bismut, Principes de Mécanique Aléatoire (1981) and 1505
Keywords: Stochastic differential equations, Semimartingales in manifolds, Transfer principle
Nature: Original
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XXIV: 29, 442-447, LNM 1426 (1990)
ÉMERY, Michel
Sur les martingales d'Azéma (suite)
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XXIV: 30, 448-452, LNM 1426 (1990)
ÉMERY, Michel; LÉANDRE, Rémi
Sur une formule de Bismut (Markov processes, Stochastic differential geometry)
This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group
Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes
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XXV: 02, 10-23, LNM 1485 (1991)
ÉMERY, Michel
Quelques cas de représentation chaotique
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XXV: 19, 220-233, LNM 1485 (1991)
ÉMERY, Michel; MOKOBODZKI, Gabriel
Sur le barycentre d'une probabilité dans une variété (Stochastic differential geometry)
In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$
Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (J. London Math. Soc. 46, 1992), as pointed out in 2650
Keywords: Martingales in manifolds, Convex functions
Nature: Original
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XXV: 23, 284-290, LNM 1485 (1991)
DUBINS, Lester E.; ÉMERY, Michel; YOR, Marc
A continuous martingale in the plane that may spiral away to infinity
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XXVI: 49, 633-633, LNM 1526 (1992)
ÉMERY, Michel
Correction au Séminaire~XXIV (Stochastic differential geometry)
An error in 2428 is pointed out; it is corrected by Cohen (Stochastics Stochastics Rep. 56, 1996)
Keywords: Stochastic differential equations, Semimartingales in manifolds
Nature: Correction
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XXVI: 50, 633-633, LNM 1526 (1992)
ÉMERY, Michel; MOKOBODZKI, Gabriel
Correction au Séminaire~XXV (Stochastic differential geometry)
Points out that the conjecture (due to Émery) at the bottom of page 232 in 2519 is refuted by Kendall (J. London Math. Soc. 46, 1992)
Keywords: Martingales in manifolds
Nature: Correction
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XXVII: 14, 122-132, LNM 1557 (1993)
DUBINS, Lester E.; ÉMERY, Michel; YOR, Marc
On the Lévy transformation of Brownian motions and continuous martingales
An erratum is given in 4421 in Volume XLIV.
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XXVIII: 21, 256-278, LNM 1583 (1994)
ATTAL, Stéphane; ÉMERY, Michel
Équations de structure pour des martingales vectorielles
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XXVIII: 27, 334-334, LNM 1583 (1994)
ÉMERY, Michel
Correction au volume XXV
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XXIX: 07, 56-69, LNM 1613 (1995)
ATTAL, Stéphane; BURDZY, Krzysztof; ÉMERY, Michel; HU, Yue-Yun
Sur quelques filtrations et transformations browniennes
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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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XXXII: 19, 264-305, LNM 1686 (1998)
BARLOW, Martin T.; ÉMERY, Michel; KNIGHT, Frank B.; SONG, Shiqi; YOR, Marc
Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)
Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA 7, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional, Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays
Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery, Astérisque 282 (2002). A simplified proof of Barlow's conjecture is given in 3304. For more on Théorème 1 (Slutsky's lemma), see 3221 and 3325
Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma
Nature: New exposition of known results, Original additions
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XXXII: 20, 306-312, LNM 1686 (1998)
ÉMERY, Michel; YOR, Marc
Sur un théorème de Tsirelson relatif à des mouvements browniens corrélés et à la nullité de certains temps locaux
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XXXIII: 06, 240-256, LNM 1709 (1999)
On certain probabilities equivalent to coin-tossing, d'après Schachermayer
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XXXIII: 08, 267-276, LNM 1709 (1999)
ÉMERY, Michel; SCHACHERMAYER, Walter
Brownian filtrations are not stable under equivalent time-changes
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XXXIII: 10, 291-303, LNM 1709 (1999)
ÉMERY, Michel; SCHACHERMAYER, Walter
A remark on Tsirelson's stochastic differential equation
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XXXV: 07, 123-138, LNM 1755 (2001)
ÉMERY, Michel
A discrete approach to the chaotic representation property
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XXXV: 20, 265-305, LNM 1755 (2001)
ÉMERY, Michel; SCHACHERMAYER, Walter
On Vershik's standardness criterion and Tsirelson's notion of cosiness
Comment: An erratum is given in 4421 in vol. XLIV.
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XXXIX: 10, 197-208, LNM 1874 (2006)
ÉMERY, Michel
Sandwiched filtrations and Lévy processes

XLII: 14, 383-396, LNM 1978 (2009)
ÉMERY, Michel
Recognising whether a filtration is Brownian: a case study (Theory of Brownian motion)
Keywords: Brownian filtration
Nature: Original
XLIV: 21, 467-467, LNM 2046 (2012)
ÉMERY, Michel; YOR, Marc
Erratum to Séminaire XXVII
This is an erratum to 2714.
Keywords: Brownian motion, Continuous martingale
Nature: Erratum
XLIV: 22, 468-468, LNM 2046 (2012)
ÉMERY, Michel; SCHACHERMAYER, Walter
Erratum to Séminaire XXXV
This is an erratum to 3520.
Keywords: Vershik's standardness criterion, Cosiness
Nature: Erratum