Overview
- 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
Part of the book series: SpringerBriefs in Probability and Mathematical Statistics (SBPMS)
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Table of contents (4 chapters)
Keywords
About this book
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.
Reviews
“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)
Authors and Affiliations
About the authors
Professor Gerald Trutnau is a full-professor at Department of Mathematical Sciences, Seoul National University.
Bibliographic Information
Book Title: Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients
Authors: Haesung Lee, Wilhelm Stannat, Gerald Trutnau
Series Title: SpringerBriefs in Probability and Mathematical Statistics
DOI: https://doi.org/10.1007/978-981-19-3831-3
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
Softcover ISBN: 978-981-19-3830-6Published: 28 August 2022
eBook ISBN: 978-981-19-3831-3Published: 27 August 2022
Series ISSN: 2365-4333
Series E-ISSN: 2365-4341
Edition Number: 1
Number of Pages: XV, 126
Number of Illustrations: 1 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Partial Differential Equations, Real Functions, Functional Analysis