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
312Keywords: Square integrable martingales,
Angle bracket,
Stochastic integralsNature: Exposition,
Original additions
Retrieve article from Numdam
IV: 01, 1-27, LNM 124 (1970)
CAIROLI, Renzo
Une inégalité pour martingales à indices multiples et ses applications (
Several parameter processes)
This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions
Comment: The next landmark in the theory is Cairoli-Walsh,
Acta. Math.,
134, 1975. For the modern results, see Imkeller,
Two Parameter Processes and their Quadratic Variation, Lect. Notes in M.
1308, 1989
Keywords: Two-parameter martingales,
Maximal inequality,
Almost sure convergenceNature: Original Retrieve article from Numdam
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 martingalesNature: Exposition,
Original additions Retrieve article from Numdam
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
1017Keywords: Local martingales,
Stochastic integrals,
Change of variable formulaNature: Original Retrieve article from Numdam
V: 04, 37-57, LNM 191 (1971)
CAIROLI, Renzo
Décomposition de processus à indices doubles (
Several parameter processes)
A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (
Acta Math.,
134, 1975)
Comment: See Cairoli
401Keywords: Two-parameter martingalesNature: Original Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
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 timeNature: Exposition,
Original additions Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
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 decompositionNature: Original Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
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 decompositionNature: Original Retrieve article from Numdam
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 integralsNature: Original Retrieve article from Numdam
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 $ėlta M_t$ 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
1335Keywords: Local martingales,
JumpsNature: Original Retrieve article from Numdam
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
1335Keywords: Local martingales,
Jumps,
Optional stochastic integralsNature: Original Retrieve article from Numdam
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 integralsNature: Original Retrieve article from Numdam
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
1249Keywords: Stochastic integrals,
Hilbert space valued martingales,
Operator stochastic integrals,
$BMO$Nature: Original Retrieve article from Numdam
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
1408Keywords: Conformal martingalesNature: Original Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
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 logarithmNature: Original Retrieve article from Numdam
XII: 18, 162-169, LNM 649 (1978)
CAIROLI, Renzo
Une représentation intégrale pour les martingales fortes (
Several parameter processes)
This paper uses the results of Cairoli-Walsh,
Ann. Prob. 5, 1977, to prove a stochastic integral representation of the strong martingales of the Brownian sheet filtration, without assuming they are square integrable
Keywords: Strong martingales,
Brownian sheetNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 integralsNature: Original Retrieve article from Numdam
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
1404Comment: Generalized by Takaoka
3313Keywords: Exponential martingales,
Continuous martingalesNature: Original Retrieve article from Numdam
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
1717Keywords: Local times,
Local martingales,
Semimartingales in an open intervalNature: 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: 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,
JumpsNature: Correction Retrieve article from Numdam
XV: 05, 44-102, LNM 850 (1981)
MEYER, Paul-André
Géométrie stochastique sans larmes (
Stochastic differential geometry)
Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is
continuous semimartingales in manifolds, following L.~Schwartz (LN
780, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous
martingales in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions
Comment: A short introduction by the same author can be found in
Stochastic Integrals, Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery,
Stochastic Calculus on Manifolds Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Transfer principle,
Stochastic differential equations,
Stochastic integrals,
Stratonovich integralsNature: Original Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
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,
JumpsNature: Original Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
XVI-S: 59, 217-236, LNM 921 (1982)
DARLING, Richard W.R.
Martingales in manifolds - Definition, examples and behaviour under maps (
Stochastic differential geometry)
Martingales in manifolds have been introduced independently by Meyer
1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions
Comment: More details are given in
Bull. L.M.S. 15 (1983),
Publ R.I.M.S. Kyoto~
19 (1983) and
Zeit. für W-theorie 65 (1984). Characterizating of manifold-valued martingales by convex functions has become a powerful tool: see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989) and his St-Flour lectures (Springer LNM 1738)
Keywords: Martingales in manifolds,
Semimartingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XVII: 18, 179-184, LNM 986 (1983)
HE, Sheng-Wu;
YAN, Jia-An;
ZHENG, Wei-An
Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (
Stochastic calculus,
Stochastic differential geometry)
Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent
Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk
1719 is a small remark on perfect convergence
Keywords: Semimartingales,
Martingales in manifoldsNature: Original Retrieve article from Numdam
XVII: 20, 187-193, LNM 986 (1983)
MEYER, Paul-André
Le théorème de convergence des martingales dans les variétés riemanniennes, d'après R.W. Darling et W.A. Zheng (
Stochastic differential geometry)
Exposition of two results on the asymptotic behaviour of martingales in a Riemannian manifold: First, Darling's theorem says that on the event where the Riemannian quadratic variation $<X,X>_\infty$ of a martingale $X$ is finite, $X_\infty$ exists in the Aleksandrov compactification of $V$. Second, Zheng's theorem asserts that on the event where $X_\infty$ exists in $V$, the Riemannian quadratic variation $<X,X>_\infty$ is finite
Comment: Darling's result is in
Publ. R.I.M.S. Kyoto 19 (1983) and Zheng's in
Zeit. für W-theorie 63 (1983). As observed in He-Yan-Zheng
1718, a stronger version of Zheng's theorem holds (with the same argument): On the event where $X_\infty$ exists in $V$, $X$ is a semimartingale up to infinity (so for instance solutions to good SDE's driven by $X$ also have a limit at infinity)
Keywords: Martingales in manifoldsNature: Exposition 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
XX: 23, 352-374, LNM 1204 (1986)
HAKIM-DOWEK, M.;
LÉPINGLE, Dominique
L'exponentielle stochastique des groupes de Lie (
Stochastic differential geometry)
Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$
Comment: See also Karandikar
Ann. Prob. 10 (1982) and
1722. For a sequel, see Arnaudon
2612Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XX: 31, 465-502, LNM 1204 (1986)
McGILL, Paul
Integral representation of martingales in the Brownian excursion filtration (
Brownian motion,
Stochastic calculus)
An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in
R)$, where ${\cal E}^x$ denotes the Brownian excursion $\sigma$-field below $x$ introduced by D. Williams
1343, who also showed that every $({\cal E}^x)$ martingale is continuous
Comment: Another filtration $(\tilde{\cal E}^x,\ x\in
R)$ of Brownian excursions below $x$ has been proposed by Azéma; the structure of martingales is quite diffferent: they are discontinuous. See Y. Hu's thesis (Paris VI, 1996), and chap.~16 of Yor,
Some Aspects of Brownian Motion, Part~II, Birkhäuser, 1997
Keywords: Previsible representation,
Martingales,
FiltrationsNature: Original 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
XXV: 18, 196-219, LNM 1485 (1991)
PICARD, Jean
Calcul stochastique avec sauts sur une variété (
Stochastic differential geometry)
It is known from Meyer
1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
JumpsNature: Original 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
XXVI: 12, 146-154, LNM 1526 (1992)
ARNAUDON, Marc
Connexions et martingales dans les groupes de Lie (
Stochastic differential geometry)
The stochastic exponential of Hakim-Dowek-Lépingle
2023 is interpreted in terms of second-order geometry, studied in details and generalized
Keywords: Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXVI: 13, 155-156, LNM 1526 (1992)
ARNAUDON, Marc;
MATTHIEU, Pierre
Appendice : Décomposition en produit de deux browniens d'une martingale à valeurs dans un groupe muni d'une métrique bi-invariante (
Stochastic differential geometry)
Using
2612, it is shown that in a Lie group with a bi-invariant Riemannian structure, every martingale is a time-changed product of two Brownian motions
Keywords: Martingales in manifolds,
Lie groupNature: Original 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
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
3313Keywords: Continuous martingales,
Local martingales,
Quadratic variation,
Maximal processNature: Original 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
XXXI: 25, 256-265, LNM 1655 (1997)
TAKAOKA, Koichiro
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem (
Stochastic calculus)
Martingales involving the future minimum of a transient Bessel process are studied, and shown to satisfy a non Markovian SDE. In dimension $>3$, uniqueness in law does not hold for this SDE. This generalizes Saisho-Tanemura
Tokyo J. Math. 13 (1990)
Comment: Extended to more general diffusions in the next article
3126Keywords: Continuous martingales,
Bessel processes,
Pitman's theoremNature: Original Retrieve article from Numdam
XXXI: 26, 266-271, LNM 1655 (1997)
RAUSCHER, Bernhard
Some remarks on Pitman's theorem (
Stochastic calculus)
For certain transient diffusions $X$, local martingales which are functins of $X_t$ and the future infimum $\inf_{u\ge t}X_u$ are constructed. This extends the preceding article
3125Comment: See also chap. 12 of Yor,
Some Aspects of Brownian Motion Part~II, Birkhäuser (1997)
Keywords: Continuous martingales,
Bessel processes,
Diffusion processes,
Pitman's theoremNature: 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
XXXIII: 04, 217-220, LNM 1709 (1999)
DE MEYER, Bernard
Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (
Brownian motion,
Filtrations)
Barlow's conjecture is proved with a simpler argument than in
3219Keywords: Filtrations,
Spider martingalesNature: New proof of known results Retrieve article from Numdam
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 martingalesNature: Original Retrieve article from Numdam
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 marginalsNature: Original
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 theoremNature: Original
XLIV: 02, 41-59, LNM 2046 (2012)
MIJATOVIĆ, Aleksandar;
NOVAK, Nika;
URUSOV, Mikhail
Martingale property of generalized stochastic exponentials (
Theory of martingales)
Keywords: Generalized stochastic exponentials,
Local martingales vs. true martingales,
One-dimensional diffusionsNature: Original
XLIV: 04, 75-103, LNM 2046 (2012)
QIAN, Zhongmin;
YING, Jiangang
Martingale representations for diffusion processes and backward stochastic differential equations (
Stochastic calculus)
Keywords: Backward Stochastic Differential equations,
Dirichlet forms,
Hunt processes,
Martingales,
Natural filtration,
Non-linear equationsNature: Original
