Series: Probability Theory and Stochastic Modelling, Vol. 83
1st ed. 2017, X, 700 p. 8 illus.
Printed book
Hardcover
ISBN 978-3-319-56437-1
* First comprehensive presentation of state of the art theory of mean
field games with special emphasis on the probabilistic approach
* Numerous applications with explicit examples including numerical solutions
* Self-contained treatment of related topics such as analysis on
Wasserstein space and mean field control problems
This two-volume book offers a comprehensive treatment of the probabilistic approach to
mean field game models and their applications. The book is self-contained in nature and
includes original material and applications with explicit examples throughout, including
numerical solutions.
Volume I of the book is entirely devoted to the theory of mean field games without a
common noise. The first half of the volume provides a self-contained introduction to
mean field games, starting from concrete illustrations of games with a finite number of
players, and ending with ready-for-use solvability results. Readers are provided with the
tools necessary for the solution of forward-backward stochastic differential equations of
the McKean-Vlasov type at the core of the probabilistic approach. The second half of this
volume focuses on the main principles of analysis on the Wasserstein space. It includes
Lions' approach to the Wasserstein differential calculus, and the applications of its results
to the analysis of stochastic mean field control problems.
Together, both Volume I and Volume II will greatly benefit mathematical graduate
students and researchers interested in mean field games. The authors provide a detailed
road map through the book allowing different access points for different readers and
building up the level of technical detail. The accessible approach and overview will allow
interested researchers in the applied sciences to obtain a clear overview of the state of the
art in mean field games.
Series: Probability Theory and Stochastic Modelling, Vol. 84
1st ed. 2017, X, 704 p. 4 illus. in color.
Printed book
Hardcover
ISBN 978-3-319-56435-7
* First comprehensive presentation of state of the art theory of mean
field games with special emphasis on the probabilistic approach
* Numerous applications with explicit examples including
numerical solutions
* Self-contained treatment of related topics
This two-volume book offers a comprehensive treatment of the probabilistic approach
to mean field game models and their applications. The book is self-contained in nature
and includes original material and applications with explicit examples throughout,
including numerical solutions. Volume II tackles the analysis of mean field games in
which the players are affected by a common source of noise. The first part of the volume
introduces and studies the concepts of weak and strong equilibria, and establishes
general solvability results. The second part is devoted to the study of the master equation,
a partial differential equation satisfied by the value function of the game over the space of
probability measures. Existence of viscosity and classical solutions are proven and used to
study asymptotics of games with finitely many players.
Together, both Volume I and Volume II will greatly benefit mathematical graduate
students and researchers interested in mean field games. The authors provide a detailed
road map through the book allowing different access points for different readers and
building up the level of technical detail. The accessible approach and overview will allow
interested researchers in the applied sciences to obtain a clear overview of the state of the
art in mean field games.
Series: Developments in Mathematics, Vol. 50
1st ed. 2017, X, 192 p. 2 illus. in color.
Printed book
Hardcover
ISBN 978-3-319-58794-3
* Overviews approximation by linear combinations of positive linear
operators
* Enriches understanding of fundamental and recent results from the
past decade
* Serves as a reference for graduate courses and a basis for future
study and development
This book presents a systematic overview of approximation by linear combinations
of positive linear operators, a useful tool used to increase the order of approximation.
Fundamental and recent results from the past decade are described with their
corresponding proofs. The volume consists of eight chapters that provide detailed
insight into the representation of monomials of the operators Ln , direct and inverse
estimates for a broad class of positive linear operators, and case studies involving finite
and unbounded intervals of real and complex functions. Strong converse inequalities
of Type A in terminology of Ditzian*Ivanov for linear combinations of Bernstein and
Bernstein-Kantorovich operators and various Voronovskaja-type estimates for some
linear combinations are analyzed and explained. Graduate students and researchers
in approximation theory will find the list of open problems in approximation of linear
combinations useful. The book serves as a reference for graduate and postgraduate
courses as well as a basis for future study and development.
Series: Applied Mathematical Sciences, Vol. 196
1st ed. 2017, VIII, 336 p. 27 illus., 25 illus. in
color. With online files/update.
Printed book
Hardcover
ISBN 978-3-319-57510-0
* Includes both theoretical and computational exercises, allowing for
use with mixed-level classes
* Provides Matlab codes for examples
* The first book to emphasizes the Wong-Zakai approximation
* Offers an approach to stochastic modeling other than the common
Monte Carlo methods
This book covers numerical methods for stochastic partial differential equations with
white noise using the framework of Wong-Zakai approximation. The book begins with
some motivational and background material in the introductory chapters and is divided
into three parts. Part I covers numerical stochastic ordinary differential equations.
Here the authors start with numerical methods for SDEs with delay using the Wong-
Zakai approximation and finite difference in time. Part II covers temporal white noise.
Here the authors consider SPDEs as PDEs driven by white noise, where discretization
of white noise (Brownian motion) leads to PDEs with smooth noise, which can then
be treated by numerical methods for PDEs. In this part, recursive algorithms based on
Wiener chaos expansion and stochastic collocation methods are presented for linear
stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations
are exploited as an application of stochastic collocation methods, where a numerical
comparison with other integration methods in random space is made. Part III covers
spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic
equations as well as other equations with additive noise. Numerical methods for SPDEs
with multiplicative noise are also discussed using the Wiener chaos expansion method. In
addition, some SPDEs driven by non-Gaussian white noise are discussed and some model
reduction methods (based on Wick-Malliavin calculus) are presented for generalized
polynomial chaos expansion methods. Powerful techniques are provided for solving
stochastic partial differential equations.
This book can be considered as self-contained. Necessary background knowledge is
presented in the appendices. Basic knowledge of probability theory and stochastic
calculus is presented in Appendix A. In Appendix B some semi-analytical methods for
SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided.
In Appendix D, all the conclusions which are needed for proofs are presented, and in
Appendix E a method to compute the convergence rate empirically is included.
Series: Graduate Texts in Mathematics, Vol. 276
1st ed. 2017, XII, 614 p. 33 illus.
Printed book
Hardcover
ISBN 978-3-319-58539-0
* Presents core material in functional analysis alongside several
advanced topics
* Includes over 400 exercises, with essential exercises marked as such
* Gives a careful introduction to amenability, property (T), and
expander graphs
* Develops relatively advanced material in spectral theory, including
a connection of the spectral theory of Banach algebras to the prime
number theorem
This textbook provides a careful treatment of functional analysis and some of its
applications in analysis, number theory, and ergodic theory.
In addition to discussing core material in functional analysis, the authors cover more
recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace
operator, amenability and property (T), the measurable functional calculus, spectral
theory for unbounded operators, and an account of Tao’s approach to the prime number
theorem using Banach algebras. The book further contains numerous examples and
exercises, making it suitable for both lecture courses and self-study.
Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and
advanced undergraduate students with some background in analysis and algebra, but will
also appeal to everyone with an interest in seeing how functional analysis can be applied
to other parts of mathematics.