Most branches of science involving random fluctuations can be approached by Stochastic Calculus. These include, but are not limited to, signal processing, noise filtering, stochastic control, optimal stopping, electrical circuits, financial markets, molecular chemistry, population dynamics, etc. All these applications assume a strong mathematical background, which in general takes a long time to develop. Stochastic Calculus is not an easy to grasp theory, and in general, requires acquaintance with the probability, analysis and measure theory.The goal of this book is to present Stochastic Calculus at an introductory level and not at its maximum mathematical detail. The author's goal was to capture as much as possible the spirit of elementary deterministic Calculus, at which students have been already exposed. This assumes a presentation that mimics similar properties of deterministic Calculus, which facilitates understanding of more complicated topics of Stochastic Calculus.The second edition contains several new features that improved the first edition both qualitatively and quantitatively. First, two more chapters have been added, Chapter 12 and Chapter 13, dealing with applications of stochastic processes in Electrochemistry and global optimization methods.This edition contains also a final chapter material containing fully solved review problems and provides solutions, or at least valuable hints, to all proposed problems. The present edition contains a total of about 250 exercises.This edition has also improved presentation from the first edition in several chapters, including new material.

Technical analysis points out that the best source of information to beat the market is the price itself. Introducing readers to technical analysis in a more succinct and practical way, Ramlall focuses on the key aspects, benefits, drawbacks, and the main tools of technical analysis. Chart Patterns, Point & Figure, Stochastics, Sentiment indicators, Elliot Wave Theory, RSI, R, Candlesticks and more are covered, including both the concepts and the practical applications. Also including programming technical analysis tools, this book is a valuable tool for both researchers and practitioners.

This volume is based on talks given at a conference celebrating Stanislav Molchanov's 65th birthday held in June of 2005 at the Centre de Recherches Mathematiques (Montreal, QC, Canada). The meeting brought together researchers working in an exceptionally wide range of topics reflecting the quality and breadth of Molchanov's past and present research accomplishments. This collection of survey and research papers gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics. Information for our distributors: Titles in this series are co-published with the Centre de Recherches Mathematiques.

Through examples of large complex graphs in realistic networks, research in graph theory has been forging ahead into exciting new directions. Graph theory has emerged as a primary tool for detecting numerous hidden structures in various information networks, including Internet graphs, social networks, biological networks, or, more generally, any graph representing relations in massive data sets. How will we explain from first principles the universal and ubiquitous coherence in the structure of these realistic but complex networks? In order to analyze these large sparse graphs, we use combinatorial, probabilistic, and spectral methods, as well as new and improved tools to analyze these networks. The examples of these networks have led us to focus on new, general, and powerful ways to look at graph theory.The book, based on lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs, presents new perspectives in graph theory and helps to contribute to a sound scientific foundation for our understanding of discrete networks that permeate this information age.

This text introduces engineering students to probability theory and stochastic processes. Along with thorough mathematical development of the subject, the book presents intuitive explanations of key points in order to give students the insights they need to apply math to practical engineering problems. The first seven chapters contain the core material that is essential to any introductory course. In one-semester undergraduate courses, instructors can select material from the remaining chapters to meet their individual goals. Graduate courses can cover all chapters in one semester.

An interdisciplinary approach to understanding queueing and graphical networks

In today's era of interdisciplinary studies and research activities, network models are becoming increasingly important in various areas where they have not regularly been used. Combining techniques from stochastic processes and graph theory to analyze the behavior of networks, "Fundamentals of Stochastic Networks" provides an interdisciplinary approach by including practical applications of these stochastic networks in various fields of study, from engineering and operations management to communications and the physical sciences.

The author uniquely unites different types of stochastic, queueing, and graphical networks that are typically studied independently of each other. With balanced coverage, the book is organized into three succinct parts:

Part I introduces basic concepts in probability and stochastic processes, with coverage on counting, Poisson, renewal, and Markov processes

Part II addresses basic queueing theory, with a focus on Markovian queueing systems and also explores advanced queueing theory, queueing networks, and approximations of queueing networks

Part III focuses on graphical models, presenting an introduction to graph theory along with Bayesian, Boolean, and random networks

The author presents the material in a self-contained style that helps readers apply the presented methods and techniques to science and engineering applications. Numerous practical examples are also provided throughout, including all related mathematical details.

Featuring basic results without heavy emphasis on proving theorems, "Fundamentals of Stochastic Networks" is a suitable book for courses on probability and stochastic networks, stochastic network calculus, and stochastic network optimization at the upper-undergraduate and graduate levels. The book also serves as a reference for researchers and network professionals who would like to learn more about the general principles of stochastic networks.

The book provides a sound mathematical base for life insurance mathematics and applies the underlying concepts to concrete examples. Moreover the models presented make it possible to model life insurance policies by means of Markov chains. Two chapters covering ALM and abstract valuation concepts on the background of Solvency II complete this volume. Numerous examples and a parallel treatment of discrete and continuous approaches help the reader to implement the theory directly in practice.

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."

This friendly guide is the companion you need to convert pure mathematics into understanding and facility with a host of probabilistic tools. The book provides a high-level view of probability and its most powerful applications. It begins with the basic rules of probability and quickly progresses to some of the most sophisticated modern techniques in use, including Kalman filters, Monte Carlo techniques, machine learning methods, Bayesian inference and stochastic processes. It draws on thirty years of experience in applying probabilistic methods to problems in computational science and engineering, and numerous practical examples illustrate where these techniques are used in the real world. Topics of discussion range from carbon dating to Wasserstein GANs, one of the most recent developments in Deep Learning. The underlying mathematics is presented in full, but clarity takes priority over complete rigour, making this text a starting reference source for researchers and a readable overview for students.

This volume in the Mastering Mathematical Finance series strikes just the right balance between mathematical rigour and practical application. Existing books on the challenging subject of stochastic interest rate models are often too advanced for Master's students or fail to include practical examples. Stochastic Interest Rates covers practical topics such as calibration, numerical implementation and model limitations in detail. The authors provide numerous exercises and carefully chosen examples to help students acquire the necessary skills to deal with interest rate modelling in a real-world setting. In addition, the book's webpage at www.cambridge.org/9781107002579 provides solutions to all of the exercises as well as the computer code (and associated spreadsheets) for all numerical work, which allows students to verify the results.

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.

This book presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering. In finance, the stochastic calculus is applied to pricing options by no arbitrage. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition. This book aims to present the theory of stochastic calculus and its applications to an audience which possesses only a basic knowledge of calculus and probability. It may be used as a textbook by graduate and advanced undergraduate students in stochastic processes, financial mathematics and engineering. It is also suitable for researchers to gain working knowledge of the subject. It contains many solved examples and exercises making it suitable for self study. In the book many of the concepts are introduced through worked-out examples, eventually leading to a complete, rigorous statement of the general result, and either a complete proof, a partial proof or a reference. Using such structure, the text will provide a mathematically literate reader with rapid introduction to the subject and its advanced applications. This book covers models in mathematical finance, biology and engineering. For mathematicians, this book can be used as a first text on stochastic calculus or as a companion to more rigorous texts by a way of examples and exercises.

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.

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

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 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.

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.

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.

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.

The goal of this textbook is to introduce students to the stochastic analysis tools that play an increasing role in the probabilistic approach to optimization problems, including stochastic control and stochastic differential games. While optimal control is taught in many graduate programs in applied mathematics and operations research, the author was intrigued by the lack of coverage of the theory of stochastic differential games. This is the first title in SIAM's Financial Mathematics book series and is based on the author's lecture notes. It will be helpful to students who are interested in stochastic differential equations (forward, backward, forward-backward); the probabilistic approach to stochastic control (dynamic programming and the stochastic maximum principle); and mean field games and control of McKean-Vlasov dynamics. The theory is illustrated by applications to models of systemic risk, macroeconomic growth, flocking/schooling, crowd behavior, and predatory trading, among others.

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 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.

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.

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.

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.