1st ed. 2016, XIV, 250 p.
Hardcover
ISBN 978-3-319-49111-0
Series: Probability and Its Applications
* Provides a thorough discussion of the state-of-the art in the area with
a special emphasis on the methods employed
* Gives results in a final form and poses a number of open questions at
the same time
* Discusses numerous examples and applications
This book offers a detailed review of perturbed random walks, perpetuities, and random
processes with immigration. Being of major importance in modern probability theory,
both theoretical and applied, these objects have been used to model various phenomena
in the natural sciences as well as in insurance and finance. The book also presents the
many significant results and efficient techniques and methods that have been worked out
in the last decade.
The first chapter is devoted to perturbed random walks and discusses their asymptotic
behavior and various functionals pertaining to them, including supremum and firstpassage
time. The second chapter examines perpetuities, presenting results on continuity
of their distributions and the existence of moments, as well as weak convergence of
divergent perpetuities. Focusing on random processes with immigration, the third chapter
investigates the existence of moments, describes long-time behavior and discusses
limit theorems, both with and without scaling. Chapters four and five address branching
random walks and the Bernoulli sieve, respectively, and their connection to the results of
the previous chapters.
With many motivating examples, this book appeals to both theoretical and applied
probabilists.
1st ed. 2017, XIII, 507 p. 182 illus., 59 illus. in
color.
Hardcover
ISBN 978-3-319-49312-1
Series: Undergraduate Texts in Mathematics
* Concentrates on n-dimensional Euclidean space
* Uses multivariable calculus to teach mathematics as a blend of
reasoning, computing, and problem-solving, doing justice to the
structure, the details, and the scope of the ideas
* Contains figures, formulas, and words to guide the reader to do
mathematics resourcefully by marshaling the skills of geometric
intuition, algebraic manipulation, and incisive use of natural
language
The graceful role of analysis in underpinning calculus is often lost to their separation in
the curriculum. This book entwines the two subjects, providing a conceptual approach to
multivariable calculus closely supported by the structure and reasoning of analysis. The
setting is Euclidean space, with the material on differentiation culminating in the inverse
and implicit function theorems, and the material on integration culminating in the general
fundamental theorem of integral calculus. More in-depth than most calculus books but
less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space
offers a rich blend of content to students outside the traditional mathematics major, while
also providing transitional preparation for those who will continue on in the subject.
The writing in this book aims to convey the intent of ideas early in discussion. The
narrative proceeds through figures, formulas, and text, guiding the reader to do
mathematics resourcefully by marshaling the skills of
* geometric intuition (the visual cortex being quickly instinctive)
* algebraic manipulation (symbol-patterns being precise and robust)
* incisive use of natural language (slogans that encapsulate central ideas enabling a
large-scale grasp of the subject).
Thinking in these ways renders mathematics coherent, inevitable, and fluid.
The prerequisite is single-variable calculus, including familiarity with the foundational
theorems and some experience with proofs.
1st ed. 2016, XX, 308 p. 64 illus., 62 illus. in
color.
Softcover
ISBN 978-1-4939-6790-2
Series: Pseudo-Differential Operators, Vol. 12
* Introduction of a modern finite-difference approach
* Presents few new results on FD schemes for PDEs, including schemes
which preserve positivity
* Gives the reader a detailed description of the new method,
including the whole theory and real practical examples so it can be
immediately used for building reader's own applications
This monograph presents a novel numerical approach to solving partial integrodifferential
equations arising in asset pricing models with jumps, which greatly exceeds
the efficiency of existing approaches. The method, based on pseudo-differential operators
and several original contributions to the theory of finite-difference schemes, is new as
applied to the Levy processes in finance, and is herein presented for the first time in a
single volume. The results within, developed in a series of research papers, are collected
and arranged together with the necessary background material from Levy processes, the
modern theory of finite-difference schemes, the theory of M-matrices and EM-matrices,
etc., thus forming a self-contained work that gives the reader a smooth introduction to
the subject. For readers with no knowledge of finance, a short explanation of the main
financial terms and notions used in the book is given in the glossary.
The latter part of the book demonstrates the efficacy of the method by solving some
typical problems encountered in computational finance, including structural default
models with jumps, and local stochastic volatility models with stochastic interest rates
and jumps. The author also adds extra complexity to the traditional statements of these
problems by taking into account jumps in each stochastic component while all jumps
are fully correlated, and shows how this setting can be efficiently addressed within the
framework of the new method.
Written for non-mathematicians, this book will appeal to financial engineers and analysts,
econophysicists, and researchers in applied numerical analysis. It can also be used as an
advance course on modern finite-difference methods or computational finance.
1st ed. 2016, XIV, 294 p. 1 illus.
Softcover
ISBN 978-3-319-50447-6
Series: Lecture Notes in Mathematics, Vol. 2178
* A long-awaited detailed account of an ordinary equivariant
(co)homology theory for compact Lie Group actions that is fully
stable and has Poincare Duality for all G-manifolds
* Extends all the commonly used ordinary equivariant theories,
including Borel (co)homology, Bredon-Illman (co)homology,
and RO(G)-graded ordinary (co)homology
* Includes numerous motivating examples and calculations
* Includes guides to reading on several levels, from a first reading
avoiding technicalities to more in-depth readings
* The first published results on the theory for actions by general
compact Lie groups
Filling a gap in the literature, this book takes the reader to the frontiers of equivariant
topology, the study of objects with specified symmetries. The discussion is motivated
by reference to a list of instructive gtoyh examples and calculations in what is a relatively
unexplored field. The authors also provide a reading path for the first-time reader less
interested in working through sophisticated machinery but still desiring a rigorous
understanding of the main concepts.
The subjectfs classical counterparts, ordinary
homology and cohomology, dating back to the work of Henri Poincare in topology, are
calculational and theoretical tools which are important in many parts of mathematics
and theoretical physics, particularly in the study of manifolds. Similarly powerful tools
have been lacking, however, in the context of equivariant topology. Aimed at advanced
graduate students and researchers in algebraic topology and related fields, the book
assumes knowledge of basic algebraic topology and group actions.
1st ed. 2016, XV, 175 p. 16 illus., 13 illus. in
color.
Softcover
ISBN 978-3-319-50486-5
Ecole d'Ete de Probabilites de Saint-Flour XLVI 2016
Series: Ecole d'Ete de Probabilites de Saint-Flour, Vol. 2175
* The first book to be devoted to this active field of research in
probability and statistical physicsAimed at experienced researchers,
but also accessible to masters and Ph.D. studentsAuthored by a
leading expert in the subject
Analyzing the phase transition from diffusive to localized behavior in a model of directed
polymers in a random environment, this volume places particular emphasis on the
localization phenomenon. The main question
is: What does the path of a random walk look like if rewards and penalties are spatially
randomly distributed?
This model, which provides a simplified version of stretched elastic chains pinned by
random impurities, has attracted much research activity, but it (and its relatives) still
holds many secrets, especially in high dimensions. It has non-gaussian scaling limits and
it belongs to the so-called KPZ universality class when the space is one-dimensional.
Adopting a Gibbsian approach, using general and powerful tools from probability theory,
the discrete model is studied in full generality.
Presenting the state-of-the art from different perspectives, and written in the form of
a first course on the subject, this monograph is aimed at researchers in probability or
statistical physics, but is also accessible to masters and Ph.D. students.