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 spacesNature: Exposition
Retrieve article from Numdam
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 Retrieve article from Numdam
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) processesNature: Original Retrieve article from Numdam
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
1325Keywords: Semimartingales,
Spaces of semimartingalesNature: Original Retrieve article from Numdam
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,
StabilityNature: Original Retrieve article from Numdam
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 inequalityNature: Original Retrieve article from Numdam
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 semimartingalesNature: Original Retrieve article from Numdam
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,
CompensatorsNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 Retrieve article from Numdam
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
Comment: See also
1506,
1507 (for less general s.d.e.'s), and
1624Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Original Retrieve article from Numdam
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
1657Keywords: Semimartingales in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
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: SemimartingalesNature: Original additions Retrieve article from Numdam
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 Retrieve article from Numdam
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 functionsNature: Original Retrieve article from Numdam
XIX: 12, 177-206, LNM 1123 (1985)
BAKRY, Dominique;
ÉMERY, Michel
Diffusions hypercontractives Retrieve article from Numdam
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 inequalitiesNature: New proof of known results Retrieve article from Numdam
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: MartingalesNature: Well-known Retrieve article from Numdam
XXIII: 06, 66-87, LNM 1372 (1989)
ÉMERY, Michel
On the Azéma martingales Retrieve article from Numdam
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
1505Keywords: Stochastic differential equations,
Semimartingales in manifolds,
Transfer principleNature: Original Retrieve article from Numdam
XXIV: 29, 442-447, LNM 1426 (1990)
ÉMERY, Michel
Sur les martingales d'Azéma (suite) Retrieve article from Numdam
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 processesNature: Exposition,
Original additions Retrieve article from Numdam
XXV: 02, 10-23, LNM 1485 (1991)
ÉMERY, Michel
Quelques cas de représentation chaotique Retrieve article from Numdam
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
2650Keywords: Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
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 Retrieve article from Numdam
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 manifoldsNature: Correction Retrieve article from Numdam
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 manifoldsNature: Correction Retrieve article from Numdam
XXVII: 14, 122-132, LNM 1557 (1993)
DUBINS, Lester E.;
ÉMERY, Michel;
YOR, Marc
On the Lévy transformation of Brownian motions and continuous martingalesComment: An erratum is given in
4421 in Volume XLIV.
Nature: Original Retrieve article from Numdam
XXVIII: 21, 256-278, LNM 1583 (1994)
ATTAL, Stéphane;
ÉMERY, Michel
Équations de structure pour des martingales vectorielles Retrieve article from Numdam
XXVIII: 27, 334-334, LNM 1583 (1994)
ÉMERY, Michel
Correction au volume XXV Retrieve article from Numdam
XXIX: 07, 56-69, LNM 1613 (1995)
ATTAL, Stéphane;
BURDZY, Krzysztof;
ÉMERY, Michel;
HU, Yue-Yun
Sur quelques filtrations et transformations browniennes Retrieve article from Numdam
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
1485Keywords: Continuous martingales,
Asymptotic behaviour of processesNature: Original Retrieve article from Numdam
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
3325Keywords: Filtrations,
Spider martingales,
Walsh's Brownian motion,
Cosiness,
Slutsky's lemmaNature: New exposition of known results,
Original additions Retrieve article from Numdam
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 Retrieve article from Numdam
XXXIII: 06, 240-256, LNM 1709 (1999)
BEGHDADI-SAKRANI, Samia;
ÉMERY, Michel
On certain probabilities equivalent to coin-tossing, d'après Schachermayer Retrieve article from Numdam
XXXIII: 08, 267-276, LNM 1709 (1999)
ÉMERY, Michel;
SCHACHERMAYER, Walter
Brownian filtrations are not stable under equivalent time-changes Retrieve article from Numdam
XXXIII: 10, 291-303, LNM 1709 (1999)
ÉMERY, Michel;
SCHACHERMAYER, Walter
A remark on Tsirelson's stochastic differential equation Retrieve article from Numdam
XXXV: 07, 123-138, LNM 1755 (2001)
ÉMERY, Michel
A discrete approach to the chaotic representation property Retrieve article from Numdam
XXXV: 20, 265-305, LNM 1755 (2001)
ÉMERY, Michel;
SCHACHERMAYER, Walter
On Vershik's standardness criterion and Tsirelson's notion of cosinessComment: An erratum is given in
4421 in vol. XLIV.
Retrieve article from Numdam
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 filtrationNature: Original
XLIV: 21, 467-467, LNM 2046 (2012)
ÉMERY, Michel;
YOR, Marc
Erratum to Séminaire XXVIIComment: This is an erratum to
2714.
Keywords: Brownian motion,
Continuous martingaleNature: Correction
XLIV: 22, 468-468, LNM 2046 (2012)
ÉMERY, Michel;
SCHACHERMAYER, Walter
Erratum to Séminaire XXXVComment: This is an erratum to
3520.
Keywords: Vershik's standardness criterion,
CosinessNature: Correction
XLV: 04, 141-157, LNM 2078 (2013)
DELLACHERIE, Claude;
ÉMERY, Michel
Filtrations Indexed by Ordinals; Application to a Conjecture of S. LaurentKeywords: FiltrationNature: Original
XLV: 05, 159-165, LNM 2078 (2013)
ÉMERY, Michel
A Planar Borel Set Which Divides Every Non-negligible Borel ProductKeywords: FiltrationNature: Original
XLVI: 15, 377-394, LNM 2123 (2014)
BROSSARD, Jean;
ÉMERY, Michel;
LEURIDAN, Christophe
Skew-product decomposition of planar Brownian motion and complementabilityNature: Original
XLVII: 01, xi-xxxi, LNM 2137 (2015)
AZÉMA, Jacques;
BARRIEU, Pauline;
BERTOIN, Jean;
CABALLERO, Maria Emilia;
DONATI-MARTIN, Catherine;
ÉMERY, Michel;
HIRSCH, Francis;
HU, Yueyun;
LEDOUX, Michel;
NAJNUDEL, Joseph;
MANSUY, Roger;
MICLO, Laurent;
SHI, Zhan;
WILLIAMS, David
TémoignagesNature: Tribute
