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
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
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
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
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
