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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 $({\cal E}^x,\ x\in**R**)$, 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 $(\tilde{\cal E}^x,\ x\in**R**)$ 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

Retrieve article from Numdam

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,

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XX: 31, 465-502, LNM 1204 (1986)

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\in

Comment: Another filtration $(\tilde{\cal E}^x,\ x\in

Keywords: Previsible representation, Martingales, Filtrations

Nature: Original

Retrieve article from Numdam

XXII: 16, 163-165, LNM 1321 (1988)

Extending Lévy's characterisation of Brownian motion

Retrieve article from Numdam

XXVI: 21, 234-248, LNM 1526 (1992)

Generalised transforms, quasi-diffusions, and Désiré André's equation

Retrieve article from Numdam

XXXII: 27, 412-425, LNM 1686 (1998)

Brownian motion, excursions, and matrix factors

Retrieve article from Numdam