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Cambridge Mathematical Library

Now available in paperback for the first time; essential reading for all students of probability theory.

Rogers L. C. G.: Diffusions, Markov
Processes, and Martingales: Volume 1, F

Diffusions, Markov Processes, and Martingales: Volume 1, Foundations

Some frequently used notation; 1. Brownian motion; Part I. Introduction: 2. Basics about Brownian motion; 3. Brownian motion in higher dimensions; 4. Gaussian processes and Lévy processes; Part II. Some Classical Theory: 5. Basic measure theory; 6. Basic probability theory; 7. Stochastic processes; 8. Discrete-parameter martingale theory; 9. Continuous-parameter martingale theory; 10. Probability measure on Lusin spaces; Part III. Markov Processes: 11. Transition functions and resolvents; 12. Feller-Dynkin processes; 13. Additive functionals; 14. Approach to ray processes: the Martin boundary; 15. Ray processes; 16. Applications; References; Index.

Oxford Finance Series

The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Bjork has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model. Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action. This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.

Arbitrage Theory in Continuous Time

Beginnend mit einer ausführlichen Diskussion der Primfaktorzerlegung natürlicher Zahlen werden die grundlegenden algebraischen Strukturen, also Gruppen, Ringe und Körper, eingeführt und anhand von Beispielen veranschaulicht. Mit fortschreitender algebraischer Entwicklung kehrt der Text zu zahlentheoretischen Fragestellungen zurück, wobei weitere Themen der Schulmathematik in den abstrakten Rahmen eingebettet werden, der dadurch eine greifbarere Bedeutung erhält. Einige Erkenntnisse, die zuvor aufwendig erarbeitet wurden, werden durch den Einsatz abstrakterer Argumente besser strukturiert und einfacher begründet. Im abschließenden Kapitel werden Körpererweiterungen und die Galoistheorie behandelt, mit deren Hilfe dann insbesondere die Konstruierbarkeit geometrischer Figuren mit Zirkel und Lineal sowie die Frage nach Lösungsformeln für Polynomgleichungen erneut untersucht und zu einem gewissen Abschluss gebracht werden.

Das Buch wendet sich an Leser und Leserinnen, die Interesse an mathematischen Fragestellungen mitbringen und mit mengentheoretischer Notation, mit Beweistechniken sowie idealerweise mit elementaren Aspekten der Vektorraumtheorie vertraut sind.

Ausgewählte Fragestellungen und wichtige Konzepte

Algebra und Zahlentheorie

A fascinating look into how the transformative joys of mathematical experience are available to&nbsp;everyone, not just specialists &nbsp; Math has a reputation for being inaccessible. People think that it requires a special gift or that comprehension is a matter of genes. Yet the greatest mathematicians throughout history, from René Descartes to Alexander Grothendieck, have insisted that this is not the case. Like Albert Einstein, who famously claimed to have “no special talent,” they said that they had accomplished what they did using ordinary human doubts, weaknesses, curiosity, and imagination. &nbsp; David Bessis guides us on an illuminating path toward deeper mathematical comprehension, reconnecting us with the mental plasticity we experienced as children. With simple, concrete examples, Bessis shows how mathematical comprehension is integral to the great learning milestones of life, such as learning to see, to speak, to walk, and to eat with a spoon. &nbsp; Focusing on the deeply human roots of mathematics, Bessis dispels the myths of mathematical genius. He offers an engaging initiation into the experience of math not as a series of discouragingly incomprehensible logic problems but as a physical activity akin to yoga, meditation, or a martial art. This perspective will change the way you think not only about math but also about intelligence, intuition, and everything that goes on inside your head.

A Secret World of Intuition and Curiosity

Mathematica

Springer Nature Proceedings excluding Computer Science

Melvyn B. Nathanson

CANT, New York, USA, 2022 and 2023

Combinatorial and Additive Number Theory VI

A one-stop-shop for all the math you should have learned for your programming career. Every great programming challenge has mathematical principles at its heart. Whether you’re optimizing search algorithms, building physics engines for games, or training neural networks, success depends on your grasp of core mathematical concepts.&nbsp; In Math for Programming, you’ll master the essential mathematics that will take you from basic coding to serious software development. You’ll discover how vectors and matrices give you the power to handle complex data, how calculus drives optimization and machine learning, and how graph theory leads to advanced search algorithms. Through clear explanations and practical examples, you’ll learn to: <ul><li>Harness linear algebra to manipulate data with unprecedented efficiency</li><li>Apply calculus concepts to optimize algorithms and drive simulations</li><li>Use probability and statistics to model uncertainty and analyze data</li><li>Master the discrete mathematics that powers&nbsp;modern data structures</li><li>Solve dynamic problems through differential equations</li></ul> Whether you’re seeking to fill gaps in your mathematical foundation or looking to refresh your understanding of core concepts, Math for Programming will turn complex math into a practical tool you’ll use every day.

Math for Programming

Mathematics and Statistics (R0)

Anima Nagar

Riddhi Shah

Shrihari Sridharan

Elements of Dynamical Systems

Science, Technology, and Management

Optimizing Engineering Problems through Heuristic Techniques

How do you remember more and forget less?How can you earn more and become more creative just by moving house? &nbsp; And how do you pack a car boot most efficiently?This is your shortcut to the art of the shortcut.

The Art of the Shortcut

Thinking Better

Springer Undergraduate Texts in Mathematics and Technology

This text introduces upper division undergraduate/beginning graduate students in mathematics, finance, or economics, to the core topics of a beginning course in finance/financial engineering. Particular emphasis is placed on exploiting the power of the Monte Carlo method to illustrate and explore financial principles. Monte Carlo is the uniquely appropriate tool for modeling the random factors that drive financial markets and simulating their implications.The Monte Carlo method is introduced early and it is used in conjunction with the geometric Brownian motion model (GBM) to illustrate and analyze the topics covered in the remainder of the text. Placing focus on Monte Carlo methods allows for students to travel a short road from theory to practical applications. Coverage includes investment science, mean-variance portfolio theory, option pricing principles, exotic options, option trading strategies, jump diffusion and exponential Lévy alternative models, and the Kelly criterion for maximizing investment growth.Novel features:inclusion of both portfolio theory and contingent claim analysis in a single textpricing methodology for exotic optionsexpectation analysis of option trading strategiespricing models that transcend the Black–Scholes frameworkoptimizing investment allocationsconcepts thoroughly explored through numerous simulation exercisesnumerous worked examples and illustrationsThe mathematical background required is a year and one-half course in calculus, matrix algebra covering solutions of linear systems, and a knowledge of probability including expectation, densities and the normal distribution. A refresher for these topics is presented in the Appendices. The programming background needed is how to code branching, loops and subroutines in some mathematical or general purpose language.The mathematical background required is a year and one-half course in calculus, matrix algebra covering solutions of linear systems, and a knowledge of probability including expectation, densities and the normal distribution. A refresher for these topics is presented in the Appendices. The programming background needed is how to code branching, loops and subroutines in some mathematical or general purpose language.Also by the author: (with F. Mendivil) Explorations in Monte Carlo, ©2009, ISBN: 978-0-387-87836-2; (with J. Herod) Mathematical Biology: An Introduction with Maple and Matlab, Second edition, ©2009, ISBN: 978-0-387-70983-3.

Finance with Monte Carlo

An accessible, motivated introduction to one of the most dynamic areas of mathematics

Decades ago, Mumford wrote that algebraic geometry "seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics." The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of "geometric spaces" before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections.Provides a comprehensive introduction certain to become the standard on the subject
Features a wealth of exercises that enable students to learn by doing
Requires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebra
Uses an example-driven approach that builds mathematical intuition
Is a self-contained textbook for graduate students and an essential reference for researchers

Foundations of Algebraic Geometry

The Rising Sea

Elements in Behavioural and Experimental Economics

The branch of psychology that studies how physical objects are perceived by subjects is known as psychophysics. A feature of the experimental design is that the experimenter presents objectively measurable objects that are imperfectly perceived by subjects. The responses are stochastic in that a subject might respond differently in otherwise identical situations. These stochastic choices can be compared to the objectively measurable properties. This Element offers a brief introduction to the topic, explains how psychophysics insights are already present in economics, and describes experimental techniques with the goal that they are useful in the design of economics experiments. Noise is a ubiquitous feature of experimental economics and there is a large strand of economics literature that carefully considers the noise. However, the authors view the psychophysics experimental techniques as uniquely suited to helping experimental economists uncover what is hiding in the noise.

Imperfect Perception and Stochastic Choice in Experiments

Chapman & Hall/CRC Texts in Statistical Science

Building on the author's more than 35 years of teaching experience, Modeling and Analysis of Stochastic Systems, Third Edition, covers the most important classes of stochastic processes used in the modeling of diverse systems. For each class of stochastic process, the text includes its definition, characterization, applications, transient and limiting behavior, first passage times, and cost/reward models. The third edition has been updated with several new applications, including the Google search algorithm in discrete time Markov chains, several examples from health care and finance in continuous time Markov chains, and square root staffing rule in Queuing models. More than 50 new exercises have been added to enhance its use as a course text or for self-study. The sequence of chapters and exercises has been maintained between editions, to enable those now teaching from the second edition to use the third edition.Rather than offer special tricks that work in specific problems, this book provides thorough coverage of general tools that enable the solution and analysis of stochastic models. After mastering the material in the text, readers will be well-equipped to build and analyze useful stochastic models for real-life situations.

Modeling and Analysis of Stochastic Systems

Diffusions, Markov Processes, and Martingales: Volume 1, Foundations

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