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

1. Auflage 2022
ISBN: 978-981-19-3830-6
Verlag: Springer Nature Singapore

Buch, Englisch, 126 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 230 g

Reihe: 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 L-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.

Zielgruppe

Research

Autoren/Hrsg.

Lee, Haesung

Trutnau, Gerald

Stannat, Wilhelm

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Introduction- main questions to be answered- our approach- what is meant with  analytical- why Lp-measure spaces with weights? Lebesgue measure too restrictive (... from the perspective of stochastics), e.g. there are unique invariant measures different to Lebesgue measure- orientation towards weighted measure spaces (pre-invariant measures)
1. The Cauchy problem in Lp-spaces with weights1.1 The abstract setting, existence1.2 Existence and regularity of pre-invariant densities (class of admissible coefficients)1.3 Uniqueness (Lp-uniqueness), regularity and analytic irreducibility of solutions to the CP
2. Stochastic Differential Equations2.1 Existence2.1.1 Construction of a Markov process corresponding to a regularized version of the solution to the Cauchy problem2.1.2. Main tools: Krylov type estimate of additive functionals $\mathbb{E}_x[\int_0^t f(X_s)ds]$2.1.3. Identification of weak solutions to SDEs (or identification of the SDE weakly solved by …)
2.2 Global properties2.2.1 Non-explosion and moment inequalities2.2.2 Irreducibility, transience and recurrence 2.2.3 Long time behavior: Ergodicity, existence and uniqueness of invariant measures, examples/counterexamples
2.3 Uniqueness 2.3.1 Pathwise uniqueness and strong solutions 2.3.2 Uniqueness in law (via the martingale problem)
2.4 Further topics (convergence, approximation)
Outlook

Über Autor(innen)

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.