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5 matches found
XV: 14, 206-209, LNM 850 (1981)
McGILL, Paul
A direct proof of the Ray-Knight theorem (Brownian motion)
The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor, Continuous Martingales and Brownian Motion, Chapter XI, exercice (2.7)
Keywords: Local times, Ray-Knight theorems, Bessel processes
Nature: New proof of known results
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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 $({\cal E}^x,\ x\inR)$, 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 $(\tilde{\cal E}^x,\ x\inR)$ 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, Filtrations
Nature: Original
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XXII: 16, 163-165, LNM 1321 (1988)
McGILL, Paul; RAJEEV, Bhaskaran; RAO, B.V.
Extending Lévy's characterisation of Brownian motion
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XXVI: 21, 234-248, LNM 1526 (1992)
McGILL, Paul
Generalised transforms, quasi-diffusions, and Désiré André's equation
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XXXII: 27, 412-425, LNM 1686 (1998)
McGILL, Paul
Brownian motion, excursions, and matrix factors
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