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Tyrone E. Duncan - One of the best experts on this subject based on the ideXlab platform.
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Some topics in Fractional Brownian Motion
Risk and Decision Analysis, 2009Co-Authors: Tyrone E. DuncanAbstract:Fractional Brownian Motion denotes a family of Gaussian processes whose applicability has been demonstrated empirically for a wide variety of physical phenomena. For more than five decades, these processes have described risky outcomes or physical uncertainties and in particular for more than four decades these processes have been used to model fluctuations in economic data. These processes have a selfsimilarity or fractal property in probability law. They can provide a model for long range dependence, rare events and bursty behavior. Kolmogorov [15] initially defined the family of Fractional Brownian Motions motivated by his study of turbulence [16,17], though Schonberg [25] predated Kolmogorov’s construction by showing that the covariance functions for these processes are positive definite and thereby Gaussian processes can be constructed. Hurst [14] empirically identified an index parameter of these processes in his modeling of rainfall in the Nile River region. The rainfall in this region had a bursty behavior with often long periods of either excessive rainfall or meager rainfall. Mandelbrot [19] motivated by his study of fractals studied Fractional Brownian Motions to model economic data as well as turbulence [20]. He coined the name Fractional Brownian Motion and noted that Hurst empirically computed an index for these processes which is now called the Hurst index. Mandelbrot and van Ness [21] investigated some basic properties of Fractional Brownian Motion. The empirical evidence for models with a Fractional Brownian Motion has continued in hydrology, finance and turbulence. More recently the empirical evidence for the use of Fractional Brownian Motions (FBMs) as random models broadened to internet traffic [4,18], medicine [23] and cognition [3,12]. 2. Fractional Brownian Motion
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Mutual Information for Stochastic Signals and Fractional Brownian Motion
IEEE Transactions on Information Theory, 2008Co-Authors: Tyrone E. DuncanAbstract:The mutual information between a stochastic signal and this signal plus a Fractional Brownian Motion (described as an additive Fractional Gaussian noise channel) is expressed as the error of an estimation problem that can be naturally associated with this model. If the stochastic signal with the additive Fractional Brownian Motion occurs multiplied by a scalar parameter, then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem. These results generalize some results for a model where the Fractional Brownian Motion is a Brownian Motion to a model with an arbitrary Fractional Brownian Motion.
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ACC - Some Stochastic Systems with a Fractional Brownian Motion and Applications to Control
2007 American Control Conference, 2007Co-Authors: Tyrone E. DuncanAbstract:In this paper some stochastic systems are considered that are described by stochastic differential equations with a Fractional Brownian Motion. The notion of a weak solution is introduced and obtained by a transformation of the measure for a Fractional Brownian Motion by a Radon-Nikodym derivative. This weak solution approach is used to solve a control problem for a controlled stochastic differential equation with a Fractional Brownian Motion and to verify the existence of an optimal control. An estimation problem for a stochastic signal observed with an additive Fractional Brownian Motion is formulated and solved. The conditional expectation which solves this problem is exhibited explicitly.
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Prediction for some processes related to a Fractional Brownian Motion
Statistics & Probability Letters, 2006Co-Authors: Tyrone E. DuncanAbstract:In this paper, explicit expressions are given for some conditional expectations for the prediction of some stochastic processes that are obtained from a Fractional Brownian Motion with the Hurst parameter in the interval (0,1). These processes are constructed as solutions of stochastic differential equations with a Fractional Brownian Motion or as solutions of multiple stochastic integrals.
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Fractional Brownian Motion and stochastic equations in hilbert spaces
Stochastics and Dynamics, 2002Co-Authors: Tyrone E. Duncan, B Pasikduncan, Bohdan MaslowskiAbstract:In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical Fractional Brownian Motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures are verified. The equivalence of the probability laws for the solution evaluated at different times and different initial conditions and the convergence of these probability laws to the limiting probability are verified. These results are applied to specific stochastic parabolic and hyperbolic differential equations. The solution of a specific parabolic equation with the Fractional Brownian Motion only in the boundary condition is shown to have many results that are analogues of the results for a Fractional Brownian Motion in the domain.
B. Pasik-duncan - One of the best experts on this subject based on the ideXlab platform.
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Stochastic calculus for Fractional Brownian Motion. I. Theory
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000Co-Authors: Tyrone E. Duncan, Y.z. Hu, B. Pasik-duncanAbstract:Describes some of the results in Duncan et al. (2000) for a stochastic calculus for a Fractional Brownian Motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals are defined with explicit expressions for their first two moments. Multiple and iterated integrals of a Fractional Brownian Motion are defined and various properties of these integrals are given. A square integrable functional on a probability space of a Fractional Brownian Motion is expressed as an infinite series of multiple integrals.
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Some methods of stochastic calculus for Fractional Brownian Motion
Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), 1999Co-Authors: Tyrone E. Duncan, Y.z. Hu, B. Pasik-duncanAbstract:Some results for stochastic calculus for a Fractional Brownian Motion are described and an application to identification is given. A stochastic integral is defined that has mean zero and an explicit expression is given for the second moment. Another stochastic integral is defined and the two stochastic integrals are explicitly related. An Ito formula is given for a smooth function of a Fractional Brownian Motion. A parameter identification problem is described for a linear stochastic differential equation with Fractional Brownian Motion and a family of strongly consistent estimates is given.
Jf Coeurjolly - One of the best experts on this subject based on the ideXlab platform.
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identification of the multivariate Fractional Brownian Motion
IEEE Transactions on Signal Processing, 2011Co-Authors: Po Amblard, Jf CoeurjollyAbstract:This paper deals with the identification of the multivariate Fractional Brownian Motion, a recently developed extension of the Fractional Brownian Motion to the multivariate case. This process is a p-multivariate self-similar Gaussian process parameterized by p different Hurst exponents Hi, p scaling coefficients σi (of each component) and also by p(p-1) coefficients ρij,ηij (for i, j=1, ..., p with j >; i ) allowing two components to be more or less strongly correlated and allowing the process to be time reversible or not. We investigate the use of discrete filtering techniques to estimate jointly or separately the different parameters and prove the efficiency of the methodology with a simulation study and the derivation of asymptotic results.
Po Amblard - One of the best experts on this subject based on the ideXlab platform.
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identification of the multivariate Fractional Brownian Motion
IEEE Transactions on Signal Processing, 2011Co-Authors: Po Amblard, Jf CoeurjollyAbstract:This paper deals with the identification of the multivariate Fractional Brownian Motion, a recently developed extension of the Fractional Brownian Motion to the multivariate case. This process is a p-multivariate self-similar Gaussian process parameterized by p different Hurst exponents Hi, p scaling coefficients σi (of each component) and also by p(p-1) coefficients ρij,ηij (for i, j=1, ..., p with j >; i ) allowing two components to be more or less strongly correlated and allowing the process to be time reversible or not. We investigate the use of discrete filtering techniques to estimate jointly or separately the different parameters and prove the efficiency of the methodology with a simulation study and the derivation of asymptotic results.
Ely Merzbach - One of the best experts on this subject based on the ideXlab platform.
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A Set-indexed Fractional Brownian Motion
Journal of Theoretical Probability, 2006Co-Authors: Erick Herbin, Ely MerzbachAbstract:We define and prove the existence of a Fractional Brownian Motion in- dexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian Motion, when the condition of independance is relaxed. Relations with the Levy Fractional Brownian Motion and with the Fractional Brownian sheet are studied. We prove sta- tionarity of the increments and a property of self-similarity with respect to the action of solid Motions. Moreover, we show that there no "really nice" set indexed Fractional Brownian Motion other than set-indexed Brownian Motion. Finally, behavior of the set-indexed Fractional Brownian Motion along increasing paths is analysed.
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The Multiparameter Fractional Brownian Motion
2006Co-Authors: Erick Herbin, Ely MerzbachAbstract:We define and study the multiparameter Fractional Brownian Motion. This process is a generalization of both the classical Fractional Brownian Motion and the multiparameter Brownian Motion, when the condition of independence is relaxed. Relations with the Lévy Fractional Brownian Motion and with the Fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the Fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter Fractional Brownian Motion along increasing paths is analysed.
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The Multiparameter Fractional Brownian Motion
Math Everywhere, 1Co-Authors: Erick Herbin, Ely MerzbachAbstract:We define and study the multiparameter Fractional Brownian Motion. This process is a generalization of both the classical Fractional Brownian Motion and the multiparameter Brownian Motion, when the condition of independence is relaxed. Relations with the Levy Fractional Brownian Motion and with the Fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the Fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter Fractional Brownian Motion along increasing paths is analysed.
