Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients

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Haesung Lee Wilhelm Stannat

Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients

Haesung Lee, Wilhelm Stannat, Gerald Trutnau

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English

Springer Verlag, Singapore
28 August 2022

Calculus & mathematical analysis; Real analysis, real variables; Functional analysis & transforms; Probability & statistics; Stochastics

Series: SpringerBriefs in Probability and Mathematical Statistics

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.

By: Haesung Lee, Wilhelm Stannat, Gerald Trutnau
Imprint: Springer Verlag, Singapore
Country of Publication: Singapore
Edition: 1st ed. 2022
Dimensions: Height: 235mm, Width: 155mm,
Weight: 232g
ISBN: 9789811938306
ISBN 10: 981193830X
Series: SpringerBriefs in Probability and Mathematical Statistics
Pages: 126
Publication Date: 28 August 2022
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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

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.

Reviews for Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients

“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)