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Toward the late 1990s, several research groups independently began
developing new, related theories in mathematical finance. These
theories did away with the standard stochastic geometric diffusion
"Samuelson" market model (also known as the Black-Scholes model
because it is used in that most famous theory), instead opting for
models that allowed minimax approaches to complement or replace
stochastic methods. Among the most fruitful models were those
utilizing game-theoretic tools and the so-called interval market
model. Over time, these models have slowly but steadily gained
influence in the financial community, providing a useful
alternative to classical methods. A self-contained monograph, The
Interval Market Model in Mathematical Finance: Game-Theoretic
Methods assembles some of the most important results, old and new,
in this area of research. Written by seven of the most prominent
pioneers of the interval market model and game-theoretic finance,
the work provides a detailed account of several closely related
modeling techniques for an array of problems in mathematical
economics. The book is divided into five parts, which successively
address topics including: * probability-free Black-Scholes theory;
* fair-price interval of an option; * representation formulas and
fast algorithms for option pricing; * rainbow options; * tychastic
approach of mathematical finance based upon viability theory. This
book provides a welcome addition to the literature, complementing
myriad titles on the market that take a classical approach to
mathematical finance. It is a worthwhile resource for researchers
in applied mathematics and quantitative finance, and has also been
written in a manner accessible to financially-inclined readers with
a limited technical background.
Toward the late 1990s, several research groups independently began
developing new, related theories in mathematical finance. These
theories did away with the standard stochastic geometric diffusion
"Samuelson" market model (also known as the Black-Scholes model
because it is used in that most famous theory), instead opting for
models that allowed minimax approaches to complement or replace
stochastic methods. Among the most fruitful models were those
utilizing game-theoretic tools and the so-called interval market
model. Over time, these models have slowly but steadily gained
influence in the financial community, providing a useful
alternative to classical methods. A self-contained monograph, The
Interval Market Model in Mathematical Finance: Game-Theoretic
Methods assembles some of the most important results, old and new,
in this area of research. Written by seven of the most prominent
pioneers of the interval market model and game-theoretic finance,
the work provides a detailed account of several closely related
modeling techniques for an array of problems in mathematical
economics. The book is divided into five parts, which successively
address topics including: * probability-free Black-Scholes theory;
* fair-price interval of an option; * representation formulas and
fast algorithms for option pricing; * rainbow options; * tychastic
approach of mathematical finance based upon viability theory. This
book provides a welcome addition to the literature, complementing
myriad titles on the market that take a classical approach to
mathematical finance. It is a worthwhile resource for researchers
in applied mathematics and quantitative finance, and has also been
written in a manner accessible to financially-inclined readers with
a limited technical background.
This advanced book focuses on ordinary differential equations
(ODEs) in Banach and more general locally convex spaces, most
notably the ODEs on measures and various function spaces. It
briefly discusses the fundamentals before moving on to the cutting
edge research in linear and nonlinear partial and
pseudo-differential equations, general kinetic equations and
fractional evolutions. The level of generality chosen is suitable
for the study of the most important nonlinear equations of
mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov,
Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman,
nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal
extensions, mass-action-law kinetics from chemistry. It also covers
nonlinear evolutions arising in evolutionary biology and mean-field
games, optimization theory, epidemics and system biology, in
general models of interacting particles or agents describing
splitting and merging, collisions and breakage, mutations and the
preferential-attachment growth on networks. The book is intended
mainly for upper undergraduate and graduate students, but is also
of use to researchers in differential equations and their
applications. It particularly highlights the interconnections
between various topics revealing where and how a particular result
is used in other chapters or may be used in other contexts, and
also clarifies the links between the languages of
pseudo-differential operators, generalized functions, operator
theory, abstract linear spaces, fractional calculus and path
integrals.
Complexity science is the study of systems with many interdependent
components. Such systems - and the self-organization and emergent
phenomena they manifest - lie at the heart of many challenges of
global importance. This book is a coherent introduction to the
mathematical methods used to understand complexity, with plenty of
examples and real-world applications. It starts with the crucial
concepts of self-organization and emergence, then tackles
complexity in dynamical systems using differential equations and
chaos theory. Several classes of models of interacting particle
systems are studied with techniques from stochastic analysis,
followed by a treatment of the statistical mechanics of complex
systems. Further topics include numerical analysis of PDEs, and
applications of stochastic methods in economics and finance. The
book concludes with introductions to space-time phases and selfish
routing. The exposition is suitable for researchers, practitioners
and students in complexity science and related fields at advanced
undergraduate level and above.
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