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
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
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
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: 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
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
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
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
