Due 2019-04-14
3rd ed. 2019, Approx. 400p. 23 illus.
Softcover
ISBN 978-3-030-02779-7
Contains recent developments within stochastic control and its applications
Discusses both the dynamic programming method and the stochastic
maximum principle method
Comprehensively presents financial markets modelled by jump diffusions,
backward stochastic differential equations and convex risk measures
Includes optimal control of mean-field systems and stochastic differential
games in the expanded and updated chapters about optimal stopping and
stochastic control
The main purpose of the book is to give a rigorous introduction to the most important and
useful solution methods of various types of stochastic control problems for jump diffusions
and their applications. Both the dynamic programming method and the stochastic maximum
principle method are discussed, as well as the relation between them. Corresponding
verification theorems involving the Hamilton?Jacobi?Bellman equation and/or (quasi-)variational
inequalities are formulated. The text emphasises applications, mostly to finance. All the main
results are illustrated by examples and exercises appear at the end of each chapter with
complete solutions. This will help the reader understand the theory and see how to apply it.
The book assumes some basic knowledge of stochastic analysis, measure theory and partial
differential equations. The3rdedition is an expanded and updated version of the2ndedition,
containing recent developments within stochastic control and its applications. Specifically, there
is a new chapter devoted to a comprehensive presentation of financial markets modelled by
jump diffusions, and one on backward stochastic differential equations and convex risk
measures. Moreover, the authors have expanded the optimal stopping and the stochastic
control chapters to include optimal control of mean-field systems and stochastic differential
games.
Due 2019-04-12
2019, Approx. 475 p.
Hardcover
ISBN 978-3-030-03429-0
Introduces the basic methods used in the qualitative mathematical analysis
of nonlinear models
Reveals a number of surprising interactions between several fields of
mathematics, including topology, functional analysis, mathematical physics,
and potential theory
Can be used as supplementary reading in any course on elliptic PDEs at the
graduate level or as a seminar text
This book emphasizes those basic abstract methods and theories that are useful in the study
of nonlinear boundary value problems. The content is developed over six chapters, providing a
thorough introduction to the techniques used in the variational and topological analysis of
nonlinear boundary value problems described by stationary differential operators. The authors
give a systematic treatment of the basic mathematical theory and constructive methods for
these classes of nonlinear equations as well as their applications to various processes arising
in the applied sciences. They show how these diverse topics are connected to other important
parts of mathematics, including topology, functional analysis, mathematical physics, and
potential theory. Throughout the book a nice balance is maintained between rigorous
mathematics and physical applications. The primary readership includes graduate students and
researchers in pure and applied nonlinear analysis.
January 2019
Pages: 452
ISBN: 978-981-3276-61-1 (hardcover)
This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:
ODE theory;
Maximum Principles (weak, strong and propagation principles);
Lie groups (with an emphasis on the construction of Lie groups).
This book also provides an introduction to the basic theory of Geometrical Analysis, with a new foundational presentation based on Ordinary Differential Equation techniques, in a unitary and self-contained way.
The book also contains:
58 figures;
182 exercises;
3 Appendices;
a Further Reading section.
Part 1. Geometrical Analysis of Flows:
Flows of Vector Fields in Space
The Exponential Theorem
The Composition of Flows of Vector Fields
Hadamard's Theorem for Flows
The CBHD Operation on Finite Dimensional Lie Algebras
The Connectivity Theorem
The Carnot-Caratheodory Distance
Part 2. Geometrical Analysis of Maximum Principles:
The Weak Maximum Principle
Corollaries of the Weak Maximum Principle
The Maximum Propagation Principle
The Maximum Propagation along the Drift
Part 3. Applications to ODEs and Lie Groups:
The Differential of the Flow wrt its Parameters
The Exponential Theorem for ODEs
The Exponential Theorem for Lie Groups
The Local Third Theorem of Lie
Construction of Carnot Groups
Exponentiation of Vector Field Algebras into Lie Groups
On the Convergence of the CBHD Series
Part 4. Appendices:
Some prerequisites of Linear Algebra
Dependence Theory for ODEs
A Brief Review of Lie Group Theory
Further Readings
Bibliography
Index
Graduate students and researchers in geometrical analysis.
January 2019
Pages: 224
ISBN: 978-981-3276-46-8 (hardcover)
A mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.
The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.
Introduction
Dynamical Systems and Mathematical Billiards
Billiard in Elementary Mathematics
Billiard and Geometry
Billiard and Physics
Graduate students, young scientists and researchers interested in mathematical billiards and dynamical systems.