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＜電子ブック＞
Jump SDEs and the Study of Their Densities : A Self-Study Book / by Arturo Kohatsu-Higa, Atsushi Takeuchi
(Universitext. ISSN:21916675)

版	1st ed. 2019.
出版者	(Singapore : Springer Nature Singapore : Imprint: Springer)
出版年	2019
大きさ	XIX, 355 p. 6 illus : online resource
著者標目	*Kohatsu-Higa, Arturo author Takeuchi, Atsushi author SpringerLink (Online service)
件　名	LCSH:Probabilities LCSH:Functional analysis LCSH:Differential equations FREE:Probability Theory FREE:Functional Analysis FREE:Differential Equations
一般注記	Review of some basic concepts of probability theory -- Simple Poisson process and its corresponding SDEs -- Compound Poisson process and its associated stochastic calculus -- Construction of Lévy processes and their corresponding SDEs: The finite variation case -- Construction of Lévy processes and their corresponding SDEs: The infinite variation case -- Multi-dimensional Lévy processes and their densities -- Flows associated with stochastic differential equations with jumps -- Overview -- Techniques to study the density -- Basic ideas for integration by parts formulas -- Sensitivity formulas -- Integration by parts: Norris method -- A non-linear example: The Boltzmann equation -- Further hints for the exercises 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 HTTP:URL=https://doi.org/10.1007/978-981-32-9741-8

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