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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients (SpringerBriefs in Probability and Mathematical Statistics) - Softcover
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.
"synopsis" may belong to another edition of this title.
About the Author
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.
From the Back Cover
"About this title" may belong to another edition of this title.
- PublisherSpringer
- Publication date2022
- ISBN 10 981193830X
- ISBN 13 9789811938306
- BindingPaperback
- LanguageEnglish
- Edition number1
- Number of pages141
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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients
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Springer Nature Singapore, Springer Nature Singapore, 2022
ISBN 10: 981193830X
ISBN 13: 9789811938306
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Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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 theLp-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. Seller Inventory # 9789811938306
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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients
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Springer Nature Singapore, Springer Nature Singapore Aug 2022, 2022
ISBN 10: 981193830X
ISBN 13: 9789811938306
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Taschenbuch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -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 theLp-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. 144 pp. Englisch. Seller Inventory # 9789811938306
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Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients (SpringerBriefs in Probability and Mathematical Statistics)
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