Fractional Brownian motion and the Markov property.

Carmona, Philippe; Coutin, Laure

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Fractional Ornstein-Uhlenbeck processes.

Cheridito, Patrick, Kawaguchi, Hideyuki, Maejima, Makoto (2003)

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Approximation of the fractional Brownian sheet Ornstein-Uhlenbeck sheet

Laure Coutin, Monique Pontier (2007)

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A stochastic “Fubini” lemma and an approximation theorem for integrals on the plane are used to produce a simulation algorithm for an anisotropic fractional Brownian sheet. The convergence rate is given. These results are valuable for any value of the Hurst parameters $(α_{1}, α_{2}) \in {] 0, 1 [}^{2}, α_{i} \neq \frac{1}{2} .$ Finally, the approximation process is iterative on the quarter plane $ℝ_{+}^{2} .$ A sample of such simulations can be used to test estimators of the parameters = 1,2.

Further exponential generalization of Pitman's $2 M - X$ theorem.

Baudoin, Fabrice (2002)

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On the constructions of the skew Brownian motion.

Lejay, Antoine (2006)

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Canonical decompositions of certain generalized Brownian bridges.

Alili, Larbi (2002)

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Where did the Brownian particle go?

Pemantle, Robin, Peres, Yuval, Pitman, Jim, Yor, Marc (2001)

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Time reversal for drifted fractional Brownian motion with Hurst index $H > 1 / 2$ .

Darses, Sebastien, Saussereau, Bruno (2007)

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Linear filtering with fractional Brownian motion in the signal and observation processes.

Kleptsyna, M.L., Kloeden, P.E., Anh, V.V. (1999)

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Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems.

Bojdecki, Tomasz, Gorostiza, Luis G., Talarczyk, Anna (2007)

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Hitting properties of a random string.

Mueller, C., Tribe, R. (2002)

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The brownian burglar : conditioning brownian motion by its local time process

Jonathan Warren, Marc Yor (1998)

Séminaire de probabilités de Strasbourg

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Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Bernard Roynette, Marc Yor (2010)

ESAIM: Probability and Statistics

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We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_{t}^{-} : = \int_{0}^{t} 1_{X_{s} < 0} d s, t \geq 0)$ . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, (2006) 171–246]).