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 sheetNature: Original
Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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
1709Comment: 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 sheetNature: Original Retrieve article from Numdam
XLIII: 01, 3-70, LNM 2006 (2011)
PICARD, Jean
Representation formulae for the fractional Brownian motion (
Theory of processes)
Keywords: Fractional Brownian motion,
Brownian motionNature: Original,
Survey
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 theoremNature: Exposition
XLIII: 09, 221-239, LNM 2006 (2011)
MAROUBY, Matthieu
Simulation of a Local Time Fractional Stable Motion (
Theory of processes)
Keywords: Stable processes,
Self-similar processes,
Shot noise series,
Local times,
Fractional Brownian motion,
SimulationNature: Original
XLV: 15, 365-400, LNM 2078 (2013)
PAGÈS, Gilles
Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (
Theory of processes)
Keywords: Co-monotony,
antithetic simulation method,
processes with independent increments,
Liouville processes,
fractional Brownian motion,
Asian options,
sensitivity,
barrier optionsNature: Original
