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X: 18, 401-413, LNM 511 (1976)

**PRATELLI, Maurizio**

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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X: 19, 414-421, LNM 511 (1976)

**PRATELLI, Maurizio**

Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)

This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)

Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals

Nature: Original

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XII: 01, 1-19, LNM 649 (1978)

**PRATELLI, Maurizio**

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,*Comm Pure Appl. Math.*, **26**, 1973

Keywords: Interpolation, $BMO$

Nature: Original

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XIII: 09, 126-131, LNM 721 (1979)

**PRATELLI, Maurizio**

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

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XIV: 04, 26-48, LNM 784 (1980)

**LENGLART, Érik**; **LÉPINGLE, Dominique**; **PRATELLI, Maurizio**

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,*Probabilités et Potentiel E *; see also 1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.} **49**,1982

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

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XVII: 14, 125-131, LNM 986 (1983)

**PRATELLI, Maurizio**

Majoration dans $L^p$ du type Métivier-Pellaumail pour les semimartingales

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XVII: 31, 311-320, LNM 986 (1983)

**PRATELLI, Maurizio**

La classe des semimartingales qui permettent d'intégrer les processus optionnels

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XXII: 10, 129-137, LNM 1321 (1988)

**PRATELLI, Maurizio**

Intégration stochastique et géométrie des espaces de Banach

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XXXIII: 14, 334-338, LNM 1709 (1999)

**PRATELLI, Maurizio**

An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales

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XXXIX: 05, 119-135, LNM 1874 (2006)

**DE DONNO, Marzia**; **PRATELLI, Maurizio**

Stochastic integration with respect to a sequence of semimartingales

XL: 22, 411-414, LNM 1899 (2007)

**DE DONNO, Marzia**; **PRATELLI, Maurizio**

On a lemma by Ansel and Stricker

XLIII: 08, 215-219, LNM 2006 (2011)

**PRATELLI, Maurizio**

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

Retrieve article from Numdam

X: 19, 414-421, LNM 511 (1976)

Espaces fortement stables de martingales de carré intégrable (Martingale theory, Stochastic calculus)

This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see 1018)

Keywords: Stable subpaces, Square integrable martingales, Stochastic integrals, Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

XII: 01, 1-19, LNM 649 (1978)

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,

Keywords: Interpolation, $BMO$

Nature: Original

Retrieve article from Numdam

XIII: 09, 126-131, LNM 721 (1979)

Le support exact du temps local d'une martingale continue (Martingale theory)

It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$

Keywords: Local times

Nature: Original

Retrieve article from Numdam

XIV: 04, 26-48, LNM 784 (1980)

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

Retrieve article from Numdam

XVII: 14, 125-131, LNM 986 (1983)

Majoration dans $L^p$ du type Métivier-Pellaumail pour les semimartingales

Retrieve article from Numdam

XVII: 31, 311-320, LNM 986 (1983)

La classe des semimartingales qui permettent d'intégrer les processus optionnels

Retrieve article from Numdam

XXII: 10, 129-137, LNM 1321 (1988)

Intégration stochastique et géométrie des espaces de Banach

Retrieve article from Numdam

XXXIII: 14, 334-338, LNM 1709 (1999)

An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales

Retrieve article from Numdam

XXXIX: 05, 119-135, LNM 1874 (2006)

Stochastic integration with respect to a sequence of semimartingales

XL: 22, 411-414, LNM 1899 (2007)

On a lemma by Ansel and Stricker

XLIII: 08, 215-219, LNM 2006 (2011)

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition