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BL
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| Record Number
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891245
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| Main Entry
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Kunita, H.
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| Title & Author
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Stochastic flows and jump-diffusions /\ Hiroshi Kunita.
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| Publication Statement
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Singapore :: Springer,, 2019.
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| Series Statement
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Probability theory and stochastic modelling,; volume 92
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| Page. NO
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1 online resource (xvii, 352 pages) :: illustrations
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| ISBN
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9789811338007
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: 9789811338014
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: 9789811338021
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: 9811338000
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: 9811338019
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: 9811338027
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9789811338007
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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; Preface; Introduction; Contents; 1 Probability Distributions and Stochastic Processes; 1.1 Probability Distributions and Characteristic Functions; 1.2 Gaussian, Poisson and Infinitely Divisible Distributions; 1.3 Random Fields and Stochastic Processes; 1.4 Wiener Processes, Poisson Random Measures and LévyProcesses; 1.5 Martingales and Backward Martingales; 1.6 Quadratic Variations of Semi-martingales; 1.7 Markov Processes and Backward Markov Processes; 1.8 Kolmogorov's Criterion for the Continuity of Random Field; 2 Stochastic Integrals
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2.1 Itô's Stochastic Integrals by Continuous Martingale and Wiener Process2.2 Itô's Formula and Applications; 2.3 Regularity of Stochastic Integrals Relative to Parameters; 2.4 Fisk-Stratonovitch Symmetric Integrals; 2.5 Stochastic Integrals with Respect to Poisson Random Measure; 2.6 Jump Processes and Related Calculus; 2.7 Backward Integrals and Backward Calculus; 3 Stochastic Differential Equations and Stochastic Flows; 3.1 Geometric Property of Solutions I; Case of Continuous SDE; 3.2 Geometric Property of Solutions II; Case of SDE with Jumps; 3.3 Master Equation
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3.4 Lp-Estimates and Regularity of Solutions C∞-Flows; 3.5 Backward SDE, Backward Stochastic Flow; 3.6 Forward-Backward Calculus for Continuous C∞-Flows; 3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE; 3.8 Forward-Backward Calculus for C∞-Flows of Jumps; 3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps; 3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift; 4 Diffusions, Jump-Diffusions and Heat Equations; 4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov Equations; 4.2 Exponential Transformation and Backward Heat Equation
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4.3 Backward Diffusions and Heat Equations4.4 Dual Semigroup, Inverse Flow and Backward Diffusion; 4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps; 4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of Diffeomorphic Jumps; 4.7 Volume-Preserving Flows; 4.8 Jump-Diffusion on Subdomain of Euclidean Space; 5 Malliavin Calculus; 5.1 Derivative and Its Adjoint on Wiener Space; 5.2 Sobolev Norms for Wiener Functionals; 5.3 Nondegenerate Wiener Functionals; 5.4 Difference Operator and Adjoint on Poisson Space; 5.5 Sobolev Norms for Poisson Functionals
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5.6 Estimations of Two Poisson Functionals by Sobolev Norms5.7 Nondegenerate Poisson Functionals; 5.8 Equivalence of Nondegenerate Conditions; 5.9 Product of Wiener Space and Poisson Space; 5.10 Sobolev Norms for Wiener-Poisson Functionals; 5.11 Nondegenerate Wiener-Poisson Functionals; 5.12 Compositions with Generalized Functions; 6 Smooth Densities and Heat Kernels; 6.1 H-Derivatives of Solutions of Continuous SDE; 6.2 Nondegenerate Diffusions; 6.3 Density and Fundamental Solution for Nondegenerate Diffusion; 6.4 Solutions of SDE on Wiener-Poisson Space; 6.5 Nondegenerate Jump-Diffusions
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| Abstract
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This monograph presents a modern treatment of (1) stochastic differential equations and (2) diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and jump processes in this book is unique: Each chapter starts from continuous processes and then proceeds to processes with jumps. In the first part of the book, it is shown that solutions of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the relation between stochastic flows and heat equations is discussed. The latter part investigates fundamental solutions of these heat equations (heat kernels) through the study of the Malliavin calculus. The author obtains smooth densities for transition functions of various types of diffusions and jump-diffusions and shows that these density functions are fundamental solutions for various types of heat equations and backward heat equations. Thus, in this book fundamental solutions for heat equations and backward heat equations are constructed independently of the theory of partial differential equations. Researchers and graduate student in probability theory will find this book very useful.
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| Subject
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Stochastic differential equations.
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| Subject
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Stochastic differential equations.
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| Dewey Classification
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519.2/2
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| LC Classification
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QA274.23
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