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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients: SpringerBriefs in Probability and Mathematical Statistics

Autor Haesung Lee, Wilhelm Stannat, Gerald Trutnau
en Limba Engleză Paperback – 28 aug 2022
This book provides analytic tools to describe local and global behavior of solutions to Itô-stochastic differential equations with non-degenerate Sobolev diffusion coefficients and locally integrable drift. Regularity theory of partial differential equations is applied to construct such solutions and to obtain strong Feller properties, irreducibility, Krylov-type estimates, moment inequalities, various types of non-explosion criteria, and long time behavior, e.g., transience, recurrence, and convergence to stationarity. 
The approach is based on the realization of the transition semigroup associated with the solution of a stochastic differential equation as a strongly continuous semigroup in the Lp-space with respect to a weight that plays the role of a sub-stationary or stationary density. This way we obtain in particular a rigorous functional analytic description of the generator of the solution of a stochastic differential equation and its full domain. The existence of such a weight is shown under broad assumptions on the coefficients. A remarkable fact is that although the weight may not be unique, many important results are independent of it. 
Given such a weight and semigroup, one can construct and further analyze in detail a weak solution to the stochastic differential equation combining variational techniques, regularity theory for partial differential equations, potential, and generalized Dirichlet form theory. 
Under classical-like or various other criteria for non-explosion we obtain as one of our main applications the existence of a pathwise unique and strong solution with an infinite lifetime. These results substantially supplement the classical case of locally Lipschitz or monotone coefficients.
We further treat other types of uniqueness and non-uniqueness questions, such as uniqueness and non-uniqueness of the mentioned weights and uniqueness in law, in a certain sense, of the solution.

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Specificații

ISBN-13: 9789811938306
ISBN-10: 981193830X
Pagini: 126
Ilustrații: XV, 126 p. 1 illus.
Dimensiuni: 155 x 235 mm
Greutate: 0.23 kg
Ediția:1st ed. 2022
Editura: Springer Nature Singapore
Colecția Springer
Seria SpringerBriefs in Probability and Mathematical Statistics

Locul publicării:Singapore, Singapore

Cuprins

Chapter 1. Introduction.- Chapter 2. The abstract Cauchy problem in Lr-spaces with weights.- Chapter 3.Stochastic differential equations.- Chapter 4. Conclusion and outlook.

Recenzii

“The book is well and precisely written, with many details, comments, explanations, and historical remarks. In my opinion, the monograph should be very interesting for people working on stochastic differential equations and their applications.” (Anna Karczewska, zbMATH 1514.60003, 2023)

“It is a research monograph rich in new results; moreover, it is carefully written, many arguments are given in detail, comparison with available results is provided and attention is paid to motivating all steps well. So for a reader with a sufficient background in the theory of semigroups, Dirichlet forms and stochastic analysis, the book may serve as a welcome introduction to the field of analytic methods in stochastic analysis.” (Jan I. Seidler, Mathematical Reviews, July,2023)

Notă biografică

Dr. Haesung Lee is working at Department of Mathematics and Computer Science, Korea Science Academy of KAIST.
Professor Wilhelm Stannat is working at Institut für Mathematik, Technische Universität Berlin. 
Professor Gerald Trutnau is a full-professor at Department of Mathematical Sciences, Seoul National University. 

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Caracteristici

Presents local and global properties of stochastic differential equations under minimal assumptions (state of the art) Shows the missing link between regularity theory of partial differential equations and stochastic differential equations Provides the right framework for the analysis of stochastic differential equations with measurable coefficients