This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process.

Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Holder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations."

The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay--a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications.

After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications.

Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.

This book introduces optimal control problems for large families of deterministic and stochastic systems with discrete or continuous time parameter. These families include most of the systems studied in many disciplines, including Economics, Engineering, Operations Research, and Management Science, among many others. The main objective is to give a concise, systematic, and reasonably self contained presentation of some key topics in optimal control theory. To this end, most of the analyses are based on the dynamic programming (DP) technique. This technique is applicable to almost all control problems that appear in theory and applications. They include, for instance, finite and infinite horizon control problems in which the underlying dynamic system follows either a deterministic or stochastic difference or differential equation. In the infinite horizon case, it also uses DP to study undiscounted problems, such as the ergodic or long-run average cost. After a general introduction to control problems, the book covers the topic dividing into four parts with different dynamical systems: control of discrete-time deterministic systems, discrete-time stochastic systems, ordinary differential equations, and finally a general continuous-time MCP with applications for stochastic differential equations. The first and second part should be accessible to undergraduate students with some knowledge of elementary calculus, linear algebra, and some concepts from probability theory (random variables, expectations, and so forth). Whereas the third and fourth part would be appropriate for advanced undergraduates or graduate students who have a working knowledge of mathematical analysis (derivatives, integrals, ...) and stochastic processes.

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs.

This book provides a pedagogical examination of the way in which stochastic models are encountered in applied sciences and techniques such as physics, engineering, biology and genetics, economics and social sciences. It covers Markov and semi-Markov models, as well as their particular cases: Poisson, renewal processes, branching processes, Ehrenfest models, genetic models, optimal stopping, reliability, reservoir theory, storage models, and queuing systems. Given this comprehensive treatment of the subject, students and researchers in applied sciences, as well as anyone looking for an introduction to stochastic models, will find this title of invaluable use.

This book is an introductory guide to using Levy processes for credit risk modelling. It covers all types of credit derivatives: from the single name vanillas such as Credit Default Swaps (CDSs) right through to structured credit risk products such as Collateralized Debt Obligations (CDOs), Constant Proportion Portfolio Insurances (CPPIs) and Constant Proportion Debt Obligations (CPDOs) as well as new advanced rating models for Asset Backed Securities (ABSs). Jumps and extreme events are crucial stylized features, essential in the modelling of the very volatile credit markets - the recent turmoil in the credit markets has once again illustrated the need for more refined models. Readers will learn how the classical models (driven by Brownian motions and Black-Scholes settings) can be significantly improved by using the more flexible class of Levy processes. By doing this, extreme event and jumps can be introduced into the models to give more reliable pricing and a better assessment of the risks. The book brings in high-tech financial engineering models for the detailed modelling of credit risk instruments, setting up the theoretical framework behind the application of Levy Processes to Credit Risk Modelling before moving on to the practical implementation. Complex credit derivatives structures such as CDOs, ABSs, CPPIs, CPDOs are analysed and illustrated with market data.

Inequalities and Extremal Problems in Probability and Statistics: Selected Topics presents various kinds of useful inequalities that are applicable in many areas of mathematics, the sciences, and engineering. The book enables the reader to grasp the importance of inequalities and how they relate to probability and statistics. This will be an extremely useful book for researchers and graduate students in probability, statistics, and econometrics, as well as specialists working across sciences, engineering, financial mathematics, insurance, and mathematical modeling of large risks.

Aufbauend auf dem ersten Band, werden in diesem Buch weiterfuhrende Konzepte der Wahrscheinlichkeitstheorie ausfuhrlich und verstandlich diskutiert. Mit vielen exemplarisch durchgerechneten Aufgaben, einer Vielzahl weiterer Problemstellungen und ausfuhrlichen Loesungen bietet es dem Leser die Moeglichkeit, die eigenen Fahigkeiten standig zu erweitern und kritisch zu uberprufen und ein tieferes Verstandnis der Materie zu erlangen. Realitatsnahe Anwendungen ermoeglichen einen Ausblick in die breite Verwendbarkeit dieser Theorie.Auch in diesem Band wird auf die Entwicklung der Begriffsbildung und der mathematischen Konzepte besonderer Wert gelegt, sodass man ihre Bedeutung bei der Erzeugung wie auch standige Verbesserung von Forschungsinstrumenten fur die Untersuchung unserer Welt erleben kann. Gerichtet ist das Buch an Gymnasiasten, Studienanfanger an Hochschulen, Lehrer und Interessierte, die sich mit diesem Gebiet vertraut machen moechten.

Optimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using MATLAB.

Das Buch gibt eine Einfuhrung in weiterfuhrende Themengebiete der stochastischen Prozesse und der zugehoerigen stochastischen Analysis und verbindet diese mit einer fundierten Darstellung von Grundlagen der Finanzmathematik. Es ist inhaltlich weitreichend und legt gleichzeitig viel Wert auf gute Lesbarkeit, Motivation und Erklarung der behandelten Sachverhalte. Finanzmathematische Fragestellungen werden zunachst im Rahmen diskreter Modelle eingefuhrt und dann auf zeitstetige Modelle ubertragen. Die grundlegende Konstruktion des stochastischen Integrals und die zugehoerige Martingaltheorie liefern fundamentale Methoden der Theorie stochastischer Prozesse zur Konstruktion von geeigneten stochastischen Modellen der Finanzmathematik, z.B. mit Hilfe von stochastischen Differentialgleichungen. Zentrale Resultate der stochastischen Analysis wie Ito -Formel, Satz von Girsanov und Martingaldarstellungssatze erhalten in der Finanzmathematik grundlegende Bedeutung, z.B. fur die risiko-neutrale Bewertungsformel (Black-Scholes Formel) oder die Frage nach der Hedgebarkeit von Optionen und der Vollstandigkeit von Marktmodellen. Kapitel zur Bewertung von Optionen in vollstandigen und nichtvollstandigen Markten und zur Bestimmung optimaler Hedgingstrategien schliessen die Thematik ab. Vorausgesetzt werden fortgeschrittene Kenntnisse der Wahrscheinlichkeitstheorie, insbesondere zu zeitdiskreten Prozessen (Martingale, Markov-Ketten) sowie zeitstetigen Prozessen (Brownsche Bewegung, Levy-Prozesse, Prozesse mit unabhangigen Zuwachsen, Markovprozesse). Das Buch ist somit fur fortgeschrittene Studierende als begleitende Lekture sowie fur Dozenten als Grundlage fur eigene Lehrveranstaltungen geeignet.

Principles and Methods for Data Science, Volume 43 in the Handbook of Statistics series, highlights new advances in the field, with this updated volume presenting interesting and timely topics, including Competing risks, aims and methods, Data analysis and mining of microbial community dynamics, Support Vector Machines, a robust prediction method with applications in bioinformatics, Bayesian Model Selection for Data with High Dimension, High dimensional statistical inference: theoretical development to data analytics, Big data challenges in genomics, Analysis of microarray gene expression data using information theory and stochastic algorithm, Hybrid Models, Markov Chain Monte Carlo Methods: Theory and Practice, and more.

The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).

Dieses vierfarbige Lehrbuch wendet sich an Student(inn)en der Mathematik in Bachelor-Studiengangen. Es bietet eine fundierte, lebendige und mit diversen Erklarvideos audiovisuell erweiterte Einfuhrung sowohl in die Stochastik einschliesslich der Mathematischen Statistik als auch der Mass- und Integrationstheorie. Durch besondere didaktische Elemente eignet es sich insbesondere zum Selbststudium und als vorlesungsbegleitender Text. Herausragende Merkmale sind: durchgangig vierfarbiges Layout mit mehr als 140 Abbildungen pragnant formulierte Kerngedanken bilden die Abschnittsuberschriften Selbsttests ermoeglichen Lernkontrollen wahrend des Lesens farbige Merkkasten heben das Wichtigste hervor "Unter-der-Lupe"-Boxen zoomen in Beweise hinein, motivieren und erklaren Details "Hintergrund-und-Ausblick"-Boxen betrachten weiterfuhrende Gesichtspunkte Zusammenfassungen zu jedem Kapitel sowie UEbersichtsboxen mehr als 330 UEbungsaufgaben zahlreiche uber QR-Codes verlinkte Erklarvideos Die Inhalte dieses Buches basieren groesstenteils auf dem Werk "Grundwissen Mathematikstudium - Hoehere Analysis, Numerik und Stochastik", werden aber wegen der curricularen Bedeutung hiermit in vollstandig uberarbeiteter Form als eigenstandiges Werk veroeffentlicht.

Dieses Buch verschafft Ihnen einen UEberblick uber einige der bekanntesten Verfahren des maschinellen Lernens aus der Perspektive der mathematischen Statistik. Nach der Lekture kennen Sie die jeweils gestellten Forderungen an die Daten sowie deren Vor- und Nachteile und sind daher in der Lage, fur ein gegebenes Problem ein geeignetes Verfahren vorzuschlagen. Beweise werden nur dort ausfuhrlich dargestellt oder skizziert, wo sie einen didaktischen Mehrwert bieten - ansonsten wird auf die entsprechenden Fachartikel verwiesen. Fur die praktische Anwendung ist ein genaueres Studium des jeweiligen Verfahrens und der entsprechenden Fachliteratur noetig, zu der Sie auf Basis dieses Buchs aber schnell Zugang finden. Das Buch richtet sich an Studierende der Mathematik hoeheren Semesters, die bereits Vorkenntnisse in Wahrscheinlichkeitstheorie besitzen. Behandelt werden sowohl Methoden des Supervised Learning und Reinforcement Learning als auch des Unsupervised Learning. Der Umfang entspricht einer einsemestrigen vierstundigen Vorlesung. Die einzelnen Kapitel sind weitestgehend unabhangig voneinander lesbar, am Ende jedes Kapitels kann das erworbene Wissen anhand von UEbungsaufgaben und durch Implementierung der Verfahren uberpruft werden. Quelltexte in der Programmiersprache R stehen auf der Springer-Produktseite zum Buch zur Verfugung.

Focusing on recent advances in option pricing under the SABR model, this book shows how to price options under this model in an arbitrage-free, theoretically consistent manner. It extends SABR to a negative rates environment, and shows how to generalize it to a similar model with additional degrees of freedom, allowing simultaneous model calibration to swaptions and CMSs. Since the SABR model is used on practically every trading floor to construct interest rate options volatility cubes in an arbitrage-free manner, a careful treatment of it is extremely important. The book will be of interest to experienced industry practitioners, as well as to students and professors in academia. Aimed mainly at financial industry practitioners (for example quants and former physicists) this book will also be interesting to mathematicians who seek intuition in the mathematical finance.

This book focuses on quantitative approximation results for weak limit theorems when the target limiting law is infinitely divisible with finite first moment. Two methods are presented and developed to obtain such quantitative results. At the root of these methods stands a Stein characterizing identity discussed in the third chapter and obtained thanks to a covariance representation of infinitely divisible distributions. The first method is based on characteristic functions and Stein type identities when the involved sequence of random variables is itself infinitely divisible with finite first moment. In particular, based on this technique, quantitative versions of compound Poisson approximation of infinitely divisible distributions are presented. The second method is a general Stein's method approach for univariate selfdecomposable laws with finite first moment. Chapter 6 is concerned with applications and provides general upper bounds to quantify the rate of convergence in classical weak limit theorems for sums of independent random variables. This book is aimed at graduate students and researchers working in probability theory and mathematical statistics.

This book explains the notion of Brakke's mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke's mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke's existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard's regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

This is the first book to promote the use of stochastic, or random, processes to understand, model and predict our climate system. One of the most important applications of this technique is in the representation of comprehensive climate models of processes which, although crucial, are too small or fast to be explicitly modelled. The book shows how stochastic methods can lead to improvements in climate simulation and prediction, compared with more conventional bulk-formula parameterization procedures. Beginning with expositions of the relevant mathematical theory, the book moves on to describe numerous practical applications. It covers the complete range of time scales of climate variability, from seasonal to decadal, centennial, and millennial. With contributions from leading experts in climate physics, this book is invaluable to anyone working on climate models, including graduate students and researchers in the atmospheric and oceanic sciences, numerical weather forecasting, climate prediction, climate modelling, and climate change.

Josef Anton Strini analyzes a special stochastic optimal control problem. The problem under study arose from a dynamic cash management model in finance, where decisions about the dividend and financing policies of a firm have to be made. Additionally, using the dynamic programming approach, he extends the present discourse by the formal derivation of the Hamilton-Jacobi-Bellman equation and by examining the verification step carefully. Finally, the treatment is completed by solving the problem numerically.

This book provides a concise introduction to the behavior of mechanical structures and testing their stochastic stability under the influence of noise. It explains the physical effects of noise and in particular the concept of Gaussian white noise. In closing, the book explains how to model the effects of noise on mechanical structures, and how to nullify / compensate for it by designing effective controllers.

Das Lehrbuch vermittelt solides Basiswissen zu den thematischen Schwerpunkten Produktmasse, Fourier-Transformation, Transformationsformel, Konvergenzbegriffe, absolute Stetigkeit und Masse auf topologischen Raumen. Hoehepunkte sind die Herleitung des Riesz'schen Darstellungssatzes und der Beweis der Existenz und Eindeutigkeit des Haar'schen Masses. Der Band enthalt ferner mathematikhistorische Ausfluge und Kurzportrats von Mathematikern, die zum Thema des Buchs wichtige Beitrage geliefert haben, sowie zahlreiche UEbungsaufgaben zur Vertiefung des Stoffs.

In many areas of human endeavor, the systems involved are not available for direct measurement. Instead, by combining mathematical models for a system's evolution with partial observations of its evolving state, we can make reasonable inferences about it. The increasing complexity of the modern world makes this analysis and synthesis of high-volume data an essential feature in many real-world problems.
The celebrated Kalman-Bucy filter, designed for linear dynamical systems with linearly structured measurements, is the most famous Bayesian filter. Its generalizations to nonlinear systems and/or observations are collectively referred to as nonlinear filtering (NLF), an extension of the Bayesian framework to the estimation, prediction, and interpolation of nonlinear stochastic dynamics. NLF uses a stochastic model to make inferences about an evolving system and is a theoretically optimal algorithm.
The breadth of its applications, firmly established and still emerging, is simply astounding. Early uses such as cryptography, tracking, and guidance were mostly of a military nature. Since then, the scope has exploded. It includes the study of global climate, estimating the state of the economy, identifying tumors using non-invasive methods, and much more.
The Oxford Handbook of Nonlinear Filtering is the first comprehensive written resource for the subject. It contains classical and recent results and applications, with contributions from 58 authors. Collated into 10 parts, it covers the foundations of nonlinear filtering, connections to stochastic partial differential equations, stability and asymptotic analysis, estimation and control, approximation theory and numerical methods for solving the nonlinear filtering problem (including particle methods). It also contains a part dedicated to the application of nonlinear filtering to several problems in mathematical finance.

The activity of neurons in the brain is noisy in that their firing times are random when they are firing at a given mean rate. This introduces a random or stochastic property into brain processing which we show in this book is fundamental to understanding many aspects of brain function, including probabilistic decision making, perception, memory recall, short-term memory, attention, and even creativity. In The Noisy Brain we show that in many of these processes, the noise caused by the random neuronal firing times is useful. However, this stochastic dynamics can be unstable or overstable, and we show that the stability of attractor networks in the brain in the face of noise may help to understand some important dysfunctions that occur in schizophrenia, normal aging, and obsessive-compulsive disorder. The Noisy Brain provides a unifying computational approach to brain function that links synaptic and biophysical properties of neurons through the firing of single neurons to the properties of the noise in large connected networks of noisy neurons to the levels of functional neuroimaging and behaviour. The book describes integrate-and-fire neuronal attractor networks with noise, and complementary mean-field analyses using approaches from theoretical physics. The book shows how they can be used to understand neuronal, functional neuroimaging, and behavioural data on decision-making, perception, memory recall, short-term memory, attention, and brain dysfunctions that occur in schizophrenia, normal aging, and obsessive-compulsive disorder. The Noisy Brain will be valuable for those in the fields of neuroscience, psychology, cognitive neuroscience, and biology from advanced undergraduate level upwards. It will also be of interest to those interested in neuroeconomics, animal behaviour, zoology, psychiatry, medicine, physics, and philosophy. The book has been written with modular chapters and sections, making it possible to select particular Chapters for course work. Advanced material on the physics of stochastic dynamics in the brain is contained in the Appendix.

A fascinating and instructive guide to Markov chains for experienced users and newcomers alike This unique guide to Markov chains approaches the subject along the four convergent lines of mathematics, implementation, simulation, and experimentation. It introduces readers to the art of stochastic modeling, shows how to design computer implementations, and provides extensive worked examples with case studies. Markov Chains: From Theory to Implementation and Experimentation begins with a general introduction to the history of probability theory in which the author uses quantifiable examples to illustrate how probability theory arrived at the concept of discrete-time and the Markov model from experiments involving independent variables. An introduction to simple stochastic matrices and transition probabilities is followed by a simulation of a two-state Markov chain. The notion of steady state is explored in connection with the long-run distribution behavior of the Markov chain. Predictions based on Markov chains with more than two states are examined, followed by a discussion of the notion of absorbing Markov chains. Also covered in detail are topics relating to the average time spent in a state, various chain configurations, and n-state Markov chain simulations used for verifying experiments involving various diagram configurations. Fascinating historical notes shed light on the key ideas that led to the development of the Markov model and its variants Various configurations of Markov Chains and their limitations are explored at length Numerous examples from basic to complex are presented in a comparative manner using a variety of color graphics All algorithms presented can be analyzed in either Visual Basic, Java Script, or PHP Designed to be useful to professional statisticians as well as readers without extensive knowledge of probability theory Covering both the theory underlying the Markov model and an array of Markov chain implementations, within a common conceptual framework, Markov Chains: From Theory to Implementation and Experimentation is a stimulating introduction to and a valuable reference for those wishing to deepen their understanding of this extremely valuable statistical tool. Paul A. Gagniuc, PhD, is Associate Professor at Polytechnic University of Bucharest, Romania. He obtained his MS and his PhD in genetics at the University of Bucharest. Dr. Ganiuc s work has been published in numerous high profile scientific journals, ranging from the Public Library of Science to BioMed Central and Nature journals. He is the recipient of several awards for exceptional scientific results and a highly active figure in the review process for different scientific areas.

Stochastic processes have a wide range of applications ranging from image processing, neuroscience, bioinformatics, financial management, and statistics. Mathematical, physical, and engineering systems use stochastic processes for modeling and reasoning phenomena. While comparing AI-stochastic systems with other counterpart systems, we are able to understand their significance, thereby applying new techniques to obtain new real-time results and solutions. Stochastic Processes and Their Applications in Artificial Intelligence opens doors for artificial intelligence experts to use stochastic processes as an effective tool in real-world problems in computational biology, speech recognition, natural language processing, and reinforcement learning. Covering key topics such as social media, big data, and artificial intelligence models, this reference work is ideal for mathematicians, industry professionals, researchers, scholars, academicians, practitioners, instructors, and students.