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9 matches found
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)
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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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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XXV: 33, 407-424, LNM 1485 (1991)
ROSEN, Jay S.
Second order limit laws for the local times of stable processes (Limit theorems)
Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable
Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi Ann. Prob. 28 (2000). A good reference on this subject is Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet
Nature: Original
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XXIX: 26, 266-289, LNM 1613 (1995)
EISENBAUM, Nathalie
Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)
After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (Proc. Conf. Probability in Banach Spaces~8, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended
Comment: A general reference on the subject is Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet
Nature: Original
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XXXI: 20, 216-224, LNM 1655 (1997)
EISENBAUM, Nathalie
Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)
Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709
Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi Ann. Prob. 28 (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006) is a general reference on the subject
Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet
Nature: Original
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