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BL
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| Record Number
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890627
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| Main Entry
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Umarov, Sabir
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| Title & Author
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Beyond the triangle : : Brownian motion, Ito calculus, and Fokker-Planck equation : fractional generalizations /\ by Sabir Umarov (University of New Haven, USA), Marjorie Hahn (Tufts University, USA), Kei Kobayashi (Fordham University, USA).
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| Publication Statement
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New Jersey :: World Scientific,, 2017.
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| Page. NO
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1 online resource
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| ISBN
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9789813230927
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: 9813230924
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9789813230910
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9813230916
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Contents; Preface; Acknowledgments; 1. Introduction; 1.1 Why fractional generalizations of the Fokker-Planck equation?; 1.2 The problem formulation; 2. The original triangle: Brownian motion, Ito stochastic calculus, and Fokker-Planck-Kolmogorov equation; 2.1 Introduction; 2.2 Brownian motion; 2.3 Ito calculus; 2.4 FPK equations for stochastic processes driven by Brownian motion; 2.4.1 FPK equation associated with Brownian motion; 2.4.2 FPK equations associated with SDEs driven by Brownian motion; 2.4.3 Connection with semigroup theory.
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2.4.4 Markovian processes and the Chapman-Kolmogorov equation 2.4.5 FPK equations associated with SDEs driven by Brownian motion in bounded domains; 3. Fractional Calculus; 3.1 The Riemann-Liouville fractional derivative; 3.2 The Caputo-Djrbashian fractional derivative; 3.3 Laplace transform of fractional derivatives; 3.4 Distributed order differential operators; 3.5 The Liouville-Weyl fractional derivatives and the Fourier transform; 3.6 The Riesz potential and the Riesz-Feller fractional derivative; 3.7 Multi-dimensional Riesz potentials and hyper-singular integrals.
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4. Pseudo-differential operators associated with Levy processesIntroduction; 4.1 Pseudo-differential operators; 4.2 Pseudo-differential operators with singular symbols; 4.3 Pseudo-differential operators associated with Levy processes; 4.4 Some abstract facts on semigroups and linear operators; 4.5 Pseudo-differential operators on manifolds; 4.6 Pseudo-differential operators associated with stochastic processes in bounded domains; 5. Stochastic processes and time-changes; Introduction; 5.1 The Skorokhod space and its relevant topologies; 5.2 Semimartingales and time-changes; 5.3 Levy processes.
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5.4 Subordinators and their inverses5.5 Gaussian processes; 6. Stochastic calculus for time-changed semimartingales and its applications to SDEs; Introduction; 6.1 Stochastic calculus for time-changed semimartingales; 6.2 SDEs driven by time-changed semimartingales; 6.3 CTRW approximations of time-changed processes in the Skorokhod spaces; 6.4 CTRW approximations of time-changed processes in the sense of finite-dimensional distributions; 6.5 Approximations of stochastic integrals driven by time-changed processes; 6.6 Numerical approximations of SDEs driven by a time-changed Brownian motion.
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7. Fractional Fokker-Planck-Kolmogorov equationsIntroduction; 7.1 FPK and FPK equations associated with SDEs driven by Brownian motion and Levy processes; 7.2 TFFPK /TDFPK equations associated with SDEs driven by time-changed Levy processes; 7.2.1 Theory; 7.2.2 Applications; 7.3 FPK equations associated with SDEs driven by fractional Brownian motion; 7.3.1 An operator approach to derivation of fractional FPK equations; 7.4 Fractional FPK equations associated with stochastic processes which are time changes of solutions of SDEs driven by fractional Brownian motion; 7.4.1 Theory.
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| Subject
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Brownian motion processes.
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Fokker-Planck equation.
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Stochastic differential equations.
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Brownian motion processes.
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Fokker-Planck equation.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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Stochastic differential equations.
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| Dewey Classification
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515/.353
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| LC Classification
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QA274.75.U43 2017
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| Added Entry
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Hahn, Marjorie G.
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Kobayashi, Kei, (Mathematics professor)
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