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I: 02, 18-33, LNM 39 (1967)

**CAIROLI, Renzo**

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

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III: 03, 34-92, LNM 88 (1969)

**CAIROLI, Renzo**

Étude probabiliste d'un problème de Dirichlet (Several parameter processes)

This paper aims at a better understanding of separately harmonic functions with respect to a product of two Markov processes, which provide one of the main examples of two parameter martingales (the other one being the Brownian sheet). Here the recently published work of J.~Walsh (*Ann. Inst. Fourier,* **18**, 1968) on the Dirichlet problem for biharmonic functions was discussed and reinterpreted

Comment: An early step in the theory of two parameter martingales. See also Cairoli,*Publ. Inst. Stat. Univ. Paris,* **15**, 1966

Keywords: Dirichlet problem, Biharmonic functions

Nature: Original

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IV: 01, 1-27, LNM 124 (1970)

**CAIROLI, Renzo**

Une inégalité pour martingales à indices multiples et ses applications (Several parameter processes)

This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions

Comment: The next landmark in the theory is Cairoli-Walsh,*Acta. Math.*, **134**, 1975. For the modern results, see Imkeller, *Two Parameter Processes and their Quadratic Variation,* Lect. Notes in M. **1308**, 1989

Keywords: Two-parameter martingales, Maximal inequality, Almost sure convergence

Nature: Original

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V: 04, 37-57, LNM 191 (1971)

**CAIROLI, Renzo**

Décomposition de processus à indices doubles (Several parameter processes)

A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (*Acta Math.*, **134**, 1975)

Comment: See Cairoli 401

Keywords: Two-parameter martingales

Nature: Original

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XI: 18, 327-339, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in*Acta Math.* **4**, 1975. These processes have stochastic integral representations along (increasing) paths, with a common kernel called their derivative. Under an integrability restriction, a process holomorphic in a region of the plane is shown to be extendable as a holomorphic process to a larger region of a canonical shape (intersection of a rectangle and a disk centered at the origin)

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 19, 340-348, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in*Ann. Prob.* **5**, 1971 for additional results

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 20, 349-355, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

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XII: 18, 162-169, LNM 649 (1978)

**CAIROLI, Renzo**

Une représentation intégrale pour les martingales fortes (Several parameter processes)

This paper uses the results of Cairoli-Walsh,*Ann. Prob.* 5, 1977, to prove a stochastic integral representation of the strong martingales of the Brownian sheet filtration, without assuming they are square integrable

Keywords: Strong martingales, Brownian sheet

Nature: Original

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XIII: 13, 162-173, LNM 721 (1979)

**CAIROLI, Renzo**

Sur la convergence des martingales indexées par ${\bf N}\times{\bf N}$ (Several parameter processes)

For two parameter (discrete) martingales, it is known that uniform integrability does not imply a.s. convergence. But if all (discrete) martingale transforms by indicators of previsible sets are uniformly integrable, then a.s. convergence obtains

Keywords: Almost sure convergence, Martingale transforms

Nature: Original

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XIII: 14, 174-198, LNM 721 (1979)

**CAIROLI, Renzo**; **GABRIEL, Jean-Pierre**

Arrêt de certaines suites multiples de variables aléatoires indépendantes (Several parameter processes, Independence)

Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $**N**^d$ ($d>1$) replacing $**N**$. Then a stopping time $T$ becomes a stopping point, of which two definitions can be given (the past at time $n$ being defined either as the past rectangle, or the complement of the future rectangle), and $|T|$ being defined as the product of the coordinates). The appropriate space then is $L\log L$ or $L\log^d L$ depending on the kind of stopping times involved. Also the integrability of the supremum of the processes along random increasing paths is considered

Keywords: Stopping points, Random increasing paths

Nature: Original

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XIV: 03, 18-25, LNM 784 (1980)

**CAIROLI, Renzo**

Sur l'extension de la définition d'intégrale stochastique (Several parameter processes)

A result of Wong-Zakai (*Ann. Prob.* **5**, 1977) extending the definition of the two kinds of stochastic integrals relative to the Brownian sheet is generalized to cover the case of stochastic integration relative to martingales, or strong martingales

Comment: A note at the end of the paper suggests some improvements

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

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XIV: 47, 489-495, LNM 784 (1980)

**CAIROLI, Renzo**

Intégrale stochastique curviligne le long d'une courbe rectifiable (Several parameter processes)

The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

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Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

Retrieve article from Numdam

III: 03, 34-92, LNM 88 (1969)

Étude probabiliste d'un problème de Dirichlet (Several parameter processes)

This paper aims at a better understanding of separately harmonic functions with respect to a product of two Markov processes, which provide one of the main examples of two parameter martingales (the other one being the Brownian sheet). Here the recently published work of J.~Walsh (

Comment: An early step in the theory of two parameter martingales. See also Cairoli,

Keywords: Dirichlet problem, Biharmonic functions

Nature: Original

Retrieve article from Numdam

IV: 01, 1-27, LNM 124 (1970)

Une inégalité pour martingales à indices multiples et ses applications (Several parameter processes)

This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions

Comment: The next landmark in the theory is Cairoli-Walsh,

Keywords: Two-parameter martingales, Maximal inequality, Almost sure convergence

Nature: Original

Retrieve article from Numdam

V: 04, 37-57, LNM 191 (1971)

Décomposition de processus à indices doubles (Several parameter processes)

A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (

Comment: See Cairoli 401

Keywords: Two-parameter martingales

Nature: Original

Retrieve article from Numdam

XI: 18, 327-339, LNM 581 (1977)

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

Retrieve article from Numdam

XI: 19, 340-348, LNM 581 (1977)

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

Retrieve article from Numdam

XI: 20, 349-355, LNM 581 (1977)

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

Retrieve article from Numdam

XII: 18, 162-169, LNM 649 (1978)

Une représentation intégrale pour les martingales fortes (Several parameter processes)

This paper uses the results of Cairoli-Walsh,

Keywords: Strong martingales, Brownian sheet

Nature: Original

Retrieve article from Numdam

XIII: 13, 162-173, LNM 721 (1979)

Sur la convergence des martingales indexées par ${\bf N}\times{\bf N}$ (Several parameter processes)

For two parameter (discrete) martingales, it is known that uniform integrability does not imply a.s. convergence. But if all (discrete) martingale transforms by indicators of previsible sets are uniformly integrable, then a.s. convergence obtains

Keywords: Almost sure convergence, Martingale transforms

Nature: Original

Retrieve article from Numdam

XIII: 14, 174-198, LNM 721 (1979)

Arrêt de certaines suites multiples de variables aléatoires indépendantes (Several parameter processes, Independence)

Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $

Keywords: Stopping points, Random increasing paths

Nature: Original

Retrieve article from Numdam

XIV: 03, 18-25, LNM 784 (1980)

Sur l'extension de la définition d'intégrale stochastique (Several parameter processes)

A result of Wong-Zakai (

Comment: A note at the end of the paper suggests some improvements

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

Retrieve article from Numdam

XIV: 47, 489-495, LNM 784 (1980)

Intégrale stochastique curviligne le long d'une courbe rectifiable (Several parameter processes)

The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

Retrieve article from Numdam