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 functionsNature: Original
Retrieve article from Numdam
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 convergenceNature: Original Retrieve article from Numdam
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
401Keywords: Two-parameter martingalesNature: Original Retrieve article from Numdam
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
1119Keywords: Holomorphic processes,
Brownian sheetNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
XII: 19, 170-179, LNM 649 (1978)
MÉTRAUX, C.
Quelques inégalités pour martingales à paramètre bidimensionnel (
Several parameter processes)
This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities
Keywords: Burkholder inequalitiesNature: Original Retrieve article from Numdam
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 transformsNature: Original Retrieve article from Numdam
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 pathsNature: Original Retrieve article from Numdam
XIII: 16, 204-215, LNM 721 (1979)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Un petit théorème de projection pour processus à deux indices (
Several parameter processes)
This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of
1315 to the two coordinates
Keywords: Previsible processes (several parameters),
Previsible projections,
Random measuresNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
XV: 47, 671-672, LNM 850 (1981)
BAKRY, Dominique
Une remarque sur les semi-martingales à deux indices (
Several parameter processes)
Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general
Keywords: Two-parameter semimartingalesNature: Original Retrieve article from Numdam
XV: 48, 673-688, LNM 850 (1981)
MAZZIOTTO, Gérald;
SZPIRGLAS, Jacques
Un exemple de processus à deux indices sans l'hypothèse F4 (
Several parameter processes)
A natural two-parameter filtration is associated with a random point in the positive quadrant $
R^2_+$. Though it does not satisfy in general the Cairoli-Walsh commutation property called F4, it is possible to develop for this filtration a reasonable theory of optional and previsible processes, projection theorems, etc
Keywords: Two-parameter processesNature: Original Retrieve article from Numdam
XVI: 08, 134-137, LNM 920 (1982)
BAKRY, Dominique
Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (
Malliavin's calculus,
Several parameter processes)
A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (
1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true
Keywords: Ornstein-Uhlenbeck processNature: Original Retrieve article from Numdam
