Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients (1st ed. 2022) (SpringerBriefs in Probability and Mathematical Statistics)
By: and and
Sign Up Now!
Already a Member? Log In
You must be logged into UK education collection to access this title.
Learn about membership options,
or view our freely available titles.
- Synopsis
- 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.
- Copyright:
- 2022
Book Details
- Book Quality:
- Publisher Quality
- ISBN-13:
- 9789811938313
- Related ISBNs:
- 9789811938306
- Publisher:
- Springer Nature Singapore
- Date of Addition:
- 09/02/22
- Copyrighted By:
- The Author
- Adult content:
- No
- Language:
- English
- Has Image Descriptions:
- No
- Categories:
- Nonfiction, Mathematics and Statistics
- Submitted By:
- Bookshare Staff
- Usage Restrictions:
- This is a copyrighted book.
Reviews
Other Books
- by Haesung Lee
- by Wilhelm Stannat
- by Gerald Trutnau
- in Nonfiction
- in Mathematics and Statistics