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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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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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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 1335

Keywords: Local martingales, Jumps

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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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: 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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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: 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: 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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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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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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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 diffusions

Nature: Original

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

Retrieve article from Numdam

VI: 06, 98-100, LNM 258 (1972)

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,

Keywords: Changes of time, Local martingales, Weak martingales

Nature: Original

Retrieve article from Numdam

VI: 07, 101-104, LNM 258 (1972)

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

Retrieve article from Numdam

XI: 21, 356-361, LNM 581 (1977)

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 1335

Keywords: Local martingales, Jumps

Nature: Original

Retrieve article from Numdam

XI: 29, 418-434, LNM 581 (1977)

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

Retrieve article from Numdam

XIII: 10, 132-137, LNM 721 (1979)

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

Retrieve article from Numdam

XIII: 22, 250-252, LNM 721 (1979)

Caractérisation d'une classe de semimartingales (Martingale theory, Stochastic calculus)

The class of semimartingales $X$ such that the stochastic integral $J\,

Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)

Keywords: Local martingales, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIV: 08, 76-101, LNM 784 (1980)

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

Retrieve article from Numdam

XIV: 18, 152-160, LNM 784 (1980)

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

Retrieve article from Numdam

XIV: 30, 255-255, LNM 784 (1980)

Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (General theory of processes, Martingale theory)

Concerns 1311. For the definitive version, see

Keywords: Central limit theorem, Skorohod topology, Local martingales, Jumps

Nature: Correction

Retrieve article from Numdam

XXXI: 12, 113-125, LNM 1655 (1997)

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

Retrieve article from Numdam

XXXIII: 13, 327-333, LNM 1709 (1999)

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

Keywords: Uniform integrability, Continuous martingales, Local martingales

Nature: Original

Retrieve article from Numdam

XLIV: 02, 41-59, LNM 2046 (2012)

Martingale property of generalized stochastic exponentials (Theory of martingales)

Keywords: Generalized stochastic exponentials, Local martingales vs. true martingales, One-dimensional diffusions

Nature: Original