The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform
David Nualart - One of the best experts on this subject based on the ideXlab platform.
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stochastic differential equations driven by fractional Brownian Motion and standard Brownian Motion
arXiv: Probability, 2008Co-Authors: Joao Guerra, David NualartAbstract: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.
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Brownian Motion reflected on Brownian Motion
Probability Theory and Related Fields, 2002Co-Authors: Krzysztof Burdzy, David NualartAbstract: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 - One of the best experts on this subject based on the ideXlab platform.
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Local times for grey Brownian Motion
International Journal of Modern Physics: Conference Series, 2020Co-Authors: José Luís Da SilvaAbstract: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.
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Singularity of Generalized Grey Brownian Motion and Time-Changed Brownian Motion
arXiv: Probability, 2018Co-Authors: José Luís Da Silva, Mohamed ErraouiAbstract: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 - One of the best experts on this subject based on the ideXlab platform.
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Mixed sub-fractional Brownian Motion
Random Operators and Stochastic Equations, 2014Co-Authors: Mounir ZiliAbstract: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.
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On the sub-mixed fractional Brownian Motion
arXiv: Probability, 2012Co-Authors: Charles El-nouty, Mounir ZiliAbstract: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.
Danyu Yang - One of the best experts on this subject based on the ideXlab platform.
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G -Brownian Motion as Rough Paths and Differential Equations Driven by G -Brownian Motion
arXiv: Probability, 2014Co-Authors: Xi Geng, Zhongmin Qian, Danyu YangAbstract:The present article is devoted to the study of sample paths of G-Brownian Motion and stochastic differential equations (SDEs) driven by G-Brownian Motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian Motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian Motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian Motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian Motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian Motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian Motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian Motion of independent interest.
Peter Mörters - One of the best experts on this subject based on the ideXlab platform.
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The average density of super-Brownian Motion
Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2020Co-Authors: Peter MörtersAbstract:Abstract In this paper we prove the existence of average densities for the support of a super-Brownian Motion at a fixed time. Our result establishes a dimension-dependent fractal parameter for super-Brownian Motion, which enables us to compare the local mass density of the super-Brownian Motion at a fixed time with the local mass density of the occupation measure of a standard Brownian Motion.
