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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 time

Nature: Exposition, Original additions

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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 decomposition

Nature: Original

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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 1404

Comment: Generalized by Takaoka 3313

Keywords: Exponential martingales, Continuous martingales

Nature: Original

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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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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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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 3126

Keywords: Continuous martingales, Bessel processes, Pitman's theorem

Nature: Original

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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 3125

Comment: See also chap. 12 of Yor,*Some Aspects of Brownian Motion Part~II*, Birkhäuser (1997)

Keywords: Continuous martingales, Bessel processes, Diffusion processes, Pitman's theorem

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

Retrieve article from Numdam

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

Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor

Keywords: Continuous martingales, Changes of time

Nature: Exposition, Original additions

Retrieve article from Numdam

X: 13, 209-215, LNM 511 (1976)

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

Keywords: Continuous martingales, Krickeberg decomposition

Nature: Original

Retrieve article from Numdam

XIV: 06, 53-61, LNM 784 (1980)

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 1404

Comment: Generalized by Takaoka 3313

Keywords: Exponential martingales, Continuous martingales

Nature: Original

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

XXXI: 16, 176-189, LNM 1655 (1997)

Closed sets supporting a continuous divergent martingale (Martingale theory)

This note gives a characterization of all closed subsets $F$ of $

Comment: Two similar problems are discussed in 1485

Keywords: Continuous martingales, Asymptotic behaviour of processes

Nature: Original

Retrieve article from Numdam

XXXI: 25, 256-265, LNM 1655 (1997)

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

Comment: Extended to more general diffusions in the next article 3126

Keywords: Continuous martingales, Bessel processes, Pitman's theorem

Nature: Original

Retrieve article from Numdam

XXXI: 26, 266-271, LNM 1655 (1997)

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 3125

Comment: See also chap. 12 of Yor,

Keywords: Continuous martingales, Bessel processes, Diffusion processes, Pitman's theorem

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