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Brownian Motion

The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform

David Nualart – 1st expert on this subject based on the ideXlab platform

  • stochastic differential equations driven by fractional Brownian Motion and standard Brownian Motion
    arXiv: Probability, 2008
    Co-Authors: Joao Guerra, David Nualart


    We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian Motion with Hurst parameter H>1/2 and a multidimensional standard Brownian Motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito stochastic calculus. The existence result is based on the Yamada-Watanabe theorem.

  • Brownian Motion reflected on Brownian Motion
    Probability Theory and Related Fields, 2002
    Co-Authors: Krzysztof Burdzy, David Nualart


    We study Brownian Motion reflected on an “independent” Brownian path. We prove results on the joint distribution of both processes and the support of the parabolic measure in the space-time domain bounded by a Brownian path. We show that there exist two different natural local times for a Brownian path reflected on a Brownian path.

José Luís Da Silva – 2nd expert on this subject based on the ideXlab platform

  • Local times for grey Brownian Motion
    International Journal of Modern Physics: Conference Series, 2020
    Co-Authors: José Luís Da Silva


    In this paper we study the grey Brownian Motion, namely its representation and local time. First it is shown that grey Brownian Motion may be represented in terms of a standard Brownian Motion and then using a criterium of S. Berman, Trans. Amer. Math. Soc., 137, 277–299 (1969), we show that grey Brownian Motion admits a λ-square integrable local time almost surely (λ denotes the Lebesgue measure). As a consequence we obtain the occupation formula and state possible generalizations of these results.

  • Singularity of Generalized Grey Brownian Motion and Time-Changed Brownian Motion
    arXiv: Probability, 2018
    Co-Authors: José Luís Da Silva, Mohamed Erraoui


    The generalized grey Brownian Motion is a time continuous self-similar with stationary increments stochastic process whose one dimensional distributions are the fundamental solutions of a stretched time fractional differential equation. Moreover, the distribution of the time-changed Brownian Motion by an inverse stable process solves the same equation, hence both processes have the same one dimensional distribution. In this paper we show the mutual singularity of the probability measures on the path space which are induced by generalized grey Brownian Motion and the time-changed Brownian Motion though they have the same one dimensional distribution. This singularity property propagates to the probability measures of the processes which are solutions to the stochastic differential equations driven by these processes.

Mounir Zili – 3rd expert on this subject based on the ideXlab platform

  • Mixed sub-fractional Brownian Motion
    Random Operators and Stochastic Equations, 2014
    Co-Authors: Mounir Zili


    Abstract: A new extension of the sub-fractional Brownian Motion, and thus of the Brownian Motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian Motions, that we have chosen to call the mixed sub-fractional Brownian Motion. In this paper, we study some basic properties of this process, its non-Markovian and non-stationarity characteristics, the conditions under which it is a semimartingale, and the main features of its sample paths. We also show that this process could serve to get a good model of certain phenomena, taking not only the sign (like in the case of the sub-fractional Brownian Motion), but also the strength of dependence between the increments of this phenomena into account.

  • On the sub-mixed fractional Brownian Motion
    arXiv: Probability, 2012
    Co-Authors: Charles El-nouty, Mounir Zili


    Let ${S_t^H, t \geq 0} $ be a linear combination of a Brownian Motion and of an independent sub-fractional Brownian Motion with Hurst index $0 < H < 1$. Its main properties are studied and it is shown that $S^H $ can be considered as an intermediate process between a sub-fractional Brownian Motion and a mixed fractional Brownian Motion. Finally, we determine the values of $H$ for which $S^H$ is not a semi-martingale.