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BL
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| Record Number
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890667
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| Main Entry
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Kohatsu-Higa, Arturo
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| Title & Author
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Jump SDEs and the study of their densities : : a self-study book /\ Arturo Kohatsu-Higa, Atsushi Takeuchi.
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| Publication Statement
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Singapore :: Springer,, 2019.
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| Series Statement
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Universitext,
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| Page. NO
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1 online resource (xix, 355 pages) :: illustrations
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| ISBN
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9789813297401
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: 9789813297418
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: 9789813297425
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: 9813297409
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: 9813297417
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: 9813297425
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9789813297401
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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; Acknowledgements; Contents; Notations; 1 Review of Some Basic Concepts of Probability Theory; 1.1 Characteristic Function; 1.2 Conditional Expectation; Part I Construction of Lévy Processes and Their Stochastic Calculus; 2 Simple Poisson Process and Its Corresponding SDEs; 2.1 Introduction and Poisson Process; 2.1.1 Preliminaries: The Poisson and the Exponential Distribution; 2.1.2 Definition of the Poisson Process; 3 Compound Poisson Process and Its Associated Stochastic Calculus; 3.1 Compound Poisson Process; 3.2 Lévy Process; 3.3 Poisson Random Measure
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10.2.1 Example: Diffusion Case10.3 The Change of Variables Method; 10.4 IBP with Change of Weak Representation Formula; 10.4.1 The Plain Change of Variables Formula; 10.4.2 The Change of Representation Argument by Sum and Convolution; 10.5 The Case of Jump Processes; 10.5.1 The Picard Method; 11 Sensitivity Formulas; 11.1 Models Driven by Gamma Processes; 11.2 Stable Processes; 11.2.1 Alternative Probabilistic Representation for Stable Random Variables; 11.2.2 Sensitivity Analysis on Stable Processes; 11.3 Sensitivity Analysis for Truncated Stable Processes
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3.4 Stochastic Calculus for Compound Poisson Processes3.4.1 Compensated Poisson Random Measure; 3.4.2 The Itô Formula; 3.5 Stochastic Integrals; 3.6 Stochastic Differential Equations; 4 Construction of Lévy Processes and Their Corresponding SDEs: The Finite Variation Case; 4.1 Construction of Lévy Processes: The Finite Variation Case; 4.1.1 Itô Formula; 4.2 Differential Equations; 5 Construction of Lévy Processes and Their Corresponding SDEs: The Infinite Variation Case; 5.1 Construction of Lévy Processes: The Infinite Variation Case; 5.2 Itô Formula for the Lévy Process Z
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5.3 Stochastic Integrals with Respect to Infinite Variation ... 5.4 Some Extensions of Jump Processes; 5.5 Non-homogeneous Poisson Process; 5.5.1 Stochastic Equation Driven by a Poisson Process; 5.5.2 Generator: Time Homogeneous Case; 5.5.3 Generator: Non-homogeneous Case; 5.5.4 Generator for the Solution to the SDE (5.5); 5.5.5 SDE Driven by Poisson Process; 5.6 Subordinated Brownian Motion; 6 Multi-dimensional Lévy Processes and Their Densities; 6.1 Infinitely Divisible Processes in mathbbRd; 6.2 Classification of Probability Measures; 6.3 Densities for Lévy Processes
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6.4 Stable Laws in Dimension Two7 Flows Associated with Stochastic Differential Equations with Jumps; 7.1 Stochastic Differential Equations; 7.2 Stochastic Flows; 7.3 Remark; Part II Densities of Jump SDEs; 8 Overview; 8.1 Introduction; 8.2 Explaining the Methods in Few Words; 9 Techniques to Study the Density; 9.1 On an Approximation Argument; 9.2 Using Characteristic Functions; 9.3 An Application for Stable Laws and a Probabilistic Representation; 10 Basic Ideas for Integration by Parts Formulas; 10.1 Basic Set-Up; 10.1.1 The Unstable IBP Formula; 10.2 Stability by Summation
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| Abstract
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The present book deals with a streamlined presentation of Lévy processes and their densities. It is directed at advanced undergraduates who have already completed a basic probability course. Poisson random variables, exponential random variables, and the introduction of Poisson processes are presented first, followed by the introduction of Poisson random measures in a simple case. With these tools the reader proceeds gradually to compound Poisson processes, finite variation Lévy processes and finally one-dimensional stable cases. This step-by-step progression guides the reader into the construction and study of the properties of general Lévy processes with no Brownian component. In particular, in each case the corresponding Poisson random measure, the corresponding stochastic integral, and the corresponding stochastic differential equations (SDEs) are provided. The second part of the book introduces the tools of the integration by parts formula for jump processes in basic settings and first gradually provides the integration by parts formula in finite-dimensional spaces and gives a formula in infinite dimensions. These are then applied to stochastic differential equations in order to determine the existence and some properties of their densities. As examples, instances of the calculations of the Greeks in financial models with jumps are shown. The final chapter is devoted to the Boltzmann equation.
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| Subject
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Lévy processes.
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| Subject
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Stochastic differential equations.
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| Subject
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Lévy processes.
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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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| Added Entry
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Takeuchi, Atsushi
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