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# Mathematics and Statistics

# Fundamentals of Advanced Mathematics 1

## Categories, Algebraic Structures, Linear and Homological Algebra

The three volumes of this series of books, of which this is the first, put forward the mathematical elements that make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering.

This first volume focuses primarily on algebraic questions: categories and functors, groups, rings, modules and algebra. Notions are introduced in a general frame...

# Chi-squared Goodness-of-fit Tests for Censored Data

## Stochastic Models in Survival Analysis and Reliability Set – Volume 3

This book is devoted to the problems of construction and application of chi-squared goodness-of-fit tests for complete and censored data. Classical chi-squared tests assume that unknown distribution parameters are estimated using grouped data, but in practice this assumption is often forgotten. In this book, we consider modified chi-squared tests, which do not suffer from such a drawback. The auth...

# Banach, Fréchet, Hilbert and Neumann Spaces

## Analysis for PDEs Set – Volume 1

This book is the first of a set dedicated to the mathematical tools used in partial differential equations derived from physics.

Its focus is on normed or semi-normed vector spaces, including the spaces of Banach, Fréchet and Hilbert, with new developments on Neumann spaces, but also on extractable spaces.

The author presents the main properties of these spaces, which are useful for the construc...

# Semi-Markov Migration Models for Credit Risk

## Stochastic Models for Insurance Set – Volume 1

Credit risk is one of the most important contemporary problems for banks and insurance companies. Indeed, for banks, more than forty percent of the equities are necessary to cover this risk. Though this problem is studied by large rating agencies with substantial economic, social and financial tools, building stochastic models is nevertheless necessary to complete this descriptive orientation.

T...

# Exterior Algebras

## Elementary Tribute to Grassmann's Ideas

Exterior algebras have two key characteristics: they are relatively unknown and essential to linear algebra. Since being theorized by Grassmann in 1844, they have remained tragically forgotten.

This book presents the construction and use of exterior algebra calculations in an elementary and progressive manner. Aimed at students, teachers as well as other users of algebraic calculations – physic...

# Engineering Investment Process

## Making Value Creation Repeatable

This book explores the quantitative steps of a financial investment process.

The authors study how these steps are articulated in order to make any value creation, whatever the asset class, consistent and robust.

The discussion includes factors, portfolio allocation, statistical and economic backtesting, but also the influence of negative rates, dynamical trading, state-space models, stylized fa...

# Interpolation and Extrapolation Optimal Designs 2

## Finite Dimensional General Models

This book considers various extensions of the topics treated in the first volume of this series, in relation to the class of models and the type of criterion for optimality. The regressors are supposed to belong to a generic finite dimensional Haar linear space, which substitutes for the classical polynomial case. The estimation pertains to a general linear form of the coefficients of the model, e...

# Portfolio Optimization with Different Information Flow

## Optimization in Insurance and Finance Set

This book presents recent progress made in stochastic portfolio optimization with asymmetry information, modeled through different filtrations.

The authors apply the theory of the enlargement of filtrations and solve the optimization problem. Two main types of enlargement of filtration are discussed: initial and progressive, using tools from various fields, such as from stochastic calculus and c...

# Stochastic Analysis of Risk and Management

## Stochastic Models in Survival Analysis and Reliability Set – Volume 2

This book confronts a growing business problem. How plausible is it for a business to control the dynamic of their capital? Central to this book is the theory of the probability of ruin and how this may not necessarily mean the end of business activity. In addition to this, there are many probabilistic recommendations made by successful businessmen and women who have made their fortunes and are no...

# Simulation of Stochastic Processes with Given Accuracy and Reliability

Simulation has now become an integral part of research and development across many fields of study. Despite the large amounts of literature in the field of simulation and modeling, one recurring problem is the issue of accuracy and confidence level of constructed models.

By outlining the new approaches and modern methods of simulation of stochastic processes, this book provides methods and tools...

# Systemic Risk Tomography

## Signals, Measurement and Transmission Channels

In April 2010 Europe was shocked by the Greek financial turmoil. At that time, the global financial crisis, which started in the summer of 2007 and reached systemic dimensions in September 2008 with the Lehman Brothers’ crash, took a new course. An adverse feedback loop between sovereign and bank risks reflected into bubble-like spreads, as if financial markets had received a wake-up call concerni...

# Mathematics for Modeling and Scientific Computing

The book provides the mathematical basis for investigating numerical equations from physics, life sciences or engineering. Tools for analysis and algorithms are applied to a large set of relevant examples to show the difficulties and the limitations of the most naïve approaches. Not only do these examples give the opportunity to put into practice mathematical statements, but modeling issues are al...

# Ruin Probabilities

## Smoothness, Bounds, Supermartingale Approach

The book is an original monograph on risk theory, which is traditionally considered a branch of insurance mathematics. It deals with different continuous-time risk models and mainly provides results obtained by the authors recently.

The book covers several aspects of risk theory. It provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizo...

# Stochastic Models of Financial Mathematics

## Optimization in Insurance and Finance Set

This book presents a short introduction to continuous-time financial models. An overview of the basics of stochastic analysis precedes a focus on the Black–Scholes and interest rate models. Other topics covered include self-financing strategies, option pricing, exotic options and risk-neutral probabilities. Vasicek, Cox Ingersoll-Ross, and Heath–Jarrow–Morton interest rate models are also explored...

# Factorization of Boundary Value Problems Using the Invariant Embedding Method

This book presents a new “factorized” formulation, for boundary value problems for linear elliptic partial differential equations.

Based on the invariant embedding method of Richard Bellman, well-known for the synthesis of closed loop optimal control, and here applied to solving boundary value problems, this formulation is comprised of two decoupled Cauchy problems and a Riccati equation for Dir...

# Reliability of Engineering Systems and Technological Risks

## Stochastic Models in Survival Analysis and Reliability Set – Volume 1

This book is based on a lecture course to students specializing in the safety of technological processes and production.

The author focuses on three main problems in technological risks and safety: elements of reliability theory, the basic notions, models and methods of general risk theory and some aspects of insurance in the context of risk management.

Although the material in this book is aime...

# Degradation Processes in Reliability

## Mathematical Models and Methods in Reliability Set – Volume 3

“Degradation process” refers to many types of reliability models, which correspond to various kinds of stochastic processes used for deterioration modeling.

This book focuses on the case of a univariate degradation model with a continuous set of possible outcomes. The envisioned univariate models have one single measurable quantity which is assumed to be observed over time.

The first three cha...

# Analytical Modeling of Wireless Communication Systems

## Stochastic Models in Computer Science and Telecommunication Networks Set – Volume 1

Wireless networks represent an inexpensive and convenient way to connect to the Internet. However, despite their applications across several technologies, one challenge still remains: to understand the behavior of wireless sensor networks and assess their performance in large-scale scenarios.

When a large number of network nodes need to interact, developing suitable analytical models is essenti...

# Numerical Methods for Inverse Problems

This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system.

The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse probl...

# Interpolation and Extrapolation Optimal Designs 1

## Polynomial Regression and Approximation Theory

This book is the first of a series which focuses on the interpolation and extrapolation of optimal designs, an area with significant applications in engineering, physics, chemistry and most experimental fields.

In this volume, the authors emphasize the importance of problems associated with the construction of design. After a brief introduction on how the theory of optimal designs meets the theor...

# Financial Mathematics

## Optimization in Insurance and Finance Set

Devoted to financial markets both with discrete and continuous time, this book also describes how to make the transition from discrete to continuous time in option pricing. Chapter 1 presents the dynamic model of a financial market with discrete time. The general notions of an investor’s portfolio, self-financing strategy, arbitrage opportunity, and completeness are presented. An efficient market ...

# Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes

## Application to Reliability

Mark H.A. Davis introduced the Piecewise-Deterministic Markov Process (PDMP) class of stochastic hybrid models in an article in 1984. Today it is used to model a variety of complex systems in the fields of engineering, economics, management sciences, biology, Internet traffic, networks and many more. Yet, despite this, there is very little in the way of literature devoted to the development of num...

# Recurrent Event Modeling Based on the Yule Process

## Application to Water Network Asset Managment

This book presents research work into the reliability of drinking water pipes.

The infrastructure of water pipes is susceptible to routine failures, namely leakage or breakage, which occur in an aggregative manner in pipeline networks. Creating strategies for infrastructure asset management requires accurate modeling tools and first-hand experience of what repeated failures can mean in terms of s...

# Risk-Based and Factor Investing

## Quantitative Finance Set

This book is a collection of exclusive new articles written by leading academics and practitioners in the area of risk-based and factor investing (RBFI).

The articles introduce readers to some of the latest, cutting-edge research encountered by academics and professionals dealing with alternative non-return based portfolio construction techniques and quantitative investment risk premia strategies...

# Stochastic Calculus for Quantitative Finance

## Optimization in Insurance and Finance Set

In 1994 and 1998 F. Delbaen and W. Schachermayer published two breakthrough papers in which they proved continuous-time versions of the Fundamental Theorem of Asset Pricing. This is one of the most remarkable achievements in modern Mathematical Finance, which led to intensive investigations in many applications of the arbitrage theory on a mathematically rigorous basis of stochastic calculus.

Thi...

# Contagion Phenomena with Applications in Finance

## Quantitative Finance Set

Much of the previous literature on financial contagion and systematic risks has been motivated by the finding that cross-market correlations (resp. co-exceedances) between asset returns increase significantly during crisis periods. Is this increase due to an exogenous shock common to all markets (interdependence) or due to certain types of transmission of shocks between markets (contagion)?

Chap...

# Basic Stochastic Processes

In this book, the authors focus on two big families of stochastic processes: stochastic calculus, including Lévy processes, and Markov and semi-Markov models. From a financial point of view, essential concepts such as the Black and Scholes model, VaR indicators, actuarial evaluation, market values and fair pricing play a central role and will be presented.

The first chapter presents the essentia...

# Fat-tailed Distributions

## Data, Diagnostics and Dependence

This book has been written for numerate non-specialists and serves three purposes.

Firstly, it gathers together mathematical material from diverse but related fields of order statistics, records, extreme value theory, majorization, regular variation and subexponentiality. Secondly, it presents a new measure of obesity. Thirdly, and most importantly, the authors hope to convince readers that fat-t...

# Integral and Measure

## From Rather Simple to Rather Complex

This book is devoted to integration, one of the two main operations in calculus.

In Part 1, the definition of the integral of a one-variable function is different (not essentially, but rather methodically) from traditional definitions of Riemann or Lebesgue integrals. Such an approach allows us, on the one hand, to quickly develop the practical skills of integration as well as, on the other hand...

# Applications of Combinatorial Optimization

## 2nd Edition Revised and Updated

This updated and revised 2nd edition of the three-volume Combinatorial Optimization series covers a very large set of topics in this area, dealing with fundamental notions and approaches as well as several classical applications of Combinatorial Optimization.

Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: applied mathematics, the...

# Paradigms of Combinatorial Optimization

## Problems and New Approaches – 2nd Edition Revised and Updated

This updated and revised 2nd edition of the three-volume Combinatorial Optimization series covers a very large set of topics in this area, dealing with fundamental notions and approaches as well as several classical applications of Combinatorial Optimization.

Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: applied mathematics, the...

# Concepts of Combinatorial Optimization

## 2nd Edition Revised and Updated

This updated and revised 2nd edition of the three-volume Combinatorial Optimization series covers a very large set of topics in this area, dealing with fundamental notions and approaches as well as several classical applications of Combinatorial Optimization.

Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: applied mathematics, the...

# Statistical Models and Methods for Reliability and Survival Analysis

## In honor of M.S. Nikulin

Statistical Models and Methods for Reliability and Survival Analysis brings together contributions by specialists in statistical theory as they discuss their applications providing up-to-date developments in methods used in survival analysis, statistical goodness of fit, stochastic processes for system reliability, amongst others. Many of these are related to the work of Professor M. Nikulin in st...

# Markov Chains

## Theory, Algorithms and applications

Markov chains are a fundamental class of stochastic processes. They are widely used to solve problems in a large number of domains such as operational research, computer science, communication networks and manufacturing systems. The success of Markov chains is mainly due to their simplicity of use, the large number of available theoretical results and the quality of algorithms developed for the nu...

# Applied Diffusion Processes from Engineering to Finance

The aim of this book is to promote interaction between engineering, finance and insurance, as these three domains have many models and methods of solution in common for solving real-life problems. The authors point out the strict inter-relations that exist among the diffusion models used in engineering, finance and insurance. In each of the three fields, the basic diffusion models are presented an...

# Rasch Models in Health

The family of statistical models known as Rasch models started with a simple model for responses to questions in educational tests presented together with a number of related models that the Danish mathematician Georg Rasch referred to as models for measurement. Since the beginning of the 1950s the use of Rasch models has grown and has spread from education to the measurement of health status. Thi...

# Mathematical Statistics and Stochastic Processes

Generally, books on mathematical statistics are restricted to the case of independent identically distributed random variables. In this book however, both this case AND the case of dependent variables, i.e. statistics for discrete and continuous time processes, are studied. This second case is very important for today’s practitioners. Mathematical Statistics and Stochastic Processes is based on de...

# Stochastic Modeling and Analysis of Telecom Networks

This book addresses the stochastic modeling of telecommunication networks, introducing the main mathematical tools for that purpose, such as Markov processes, real and spatial point processes and stochastic recursions, and presenting a wide list of results on stability, performances and comparison of systems. The authors propose a comprehensive mathematical construction of the foundations of stoch...

# Stochastic Methods for Pension Funds

The book will present all the stochastic models that can be used for the study of pension schemes and the management of pension funds. Advanced stochastic tools will be used for the construction of the models presented.

Quantitative finance has become an extraordinary field of research over recent years and has been of interest from both an academic point of view as well as for practical applicat...

# Managerial Logic

The publication of the first book by Kenneth Arrow and Hervé Raynaud, in 1986, led to an important wave of research in the field of axiomatic approach applied to managerial logic. Managerial Logic summarizes the prospective results of this research and offers consultants, researchers, and decision makers a unified framework for handling the difficult decisions they face.

Based on confirmed result...

# Introduction to Stochastic Analysis

## Integrals and Differential Equations

This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for...

# Non-parametric Tests for Complete Data

Statistical analysis of data sets usually involves construction of a statistical model of the distribution of data within the available sample – and by extension the distribution of all data of the same category in the world. Statistical models are either parametric or non-parametric – this distinction is based on whether or not the model can be described in terms of a finite-dimensional paramete...

# Non-parametric Tests for Censored Data

Statistical analysis of data sets usually involves construction of a statistical model of the distribution of data within the available sample – and by extension the distribution of all data of the same category in the world. Statistical models are either parametric or non-parametric – this distinction is based on whether or not the model can be described in terms of a finite-dimensional paramete...

# Introduction to Stochastic Models

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, reliab...

# Discrete-time Asset Pricing Models in Applied Stochastic Finance

Stochastic finance and financial engineering are fields of science that have expanded rapidly over the past four decades, due mainly to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging these types of risk, and o...

# Continuous-time Asset Pricing Models in Applied Stochastic Finance

These two volumes provide a foundation course on applied stochastic finance for students, financial analysts and practitioners, and any professional interested in learning advanced mathematical and stochastic methods through finance.

The books are illustrated with numerous examples, each highlighted and isolated from the text for easy reference and identification.

This volume studies continuou...

# Mathematical Finance

## Deterministic and Stochastic Models

This textbook covers the uniform laws of financial practice and classical methods for evaluating amortization, annuities, capital markets, interest rates, liability, sinking funds and term structures. An overview of stochastic processes and the Itô calculus, the main tool of stochastic finance, is provided. It then goes on to discuss the following classical fields: the evaluation of equity option...

# Mathematics for Engineers

This book offers comprehensive coverage of the basic mathematical

tools for the engineer in the field of communication technologies – generation, processing and transport of all forms of information, data and images. It provides the essential results in probability theory and statistics, which constitute the basis for the presentation of signal processing, information theory, traffic and queuein...

# Switching Processes in Queueing Models

The book is devoted to developing the asymptotic theory for the class of switching queueing models which covers state-dependent models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc. Switching processes, invented by the author in 1977, are the main tools used in the investigation. Asymptotic...

# Mathematical Methods in Survival Analysis, Reliability and Quality of Life

Reliability and survival analysis are important applications of stochastic mathematics (probability, statistics and stochastic processes) that are usually covered separately in spite of the similarity of the mathematical theory involved.

This book aims to redress this situation: it includes 21 chapters divided into four parts: survival analysis, reliability, quality of life and related topics. M...

# Continuous Semi-Markov Processes

This book considers the special class of random processes known as semi-Markov processes. These possess the Markov property with respect to any intrinsic Markov time such as the first exit time from an open set or a finite iteration of these times.

This class of semi-Markov processes includes strong Markov processes, Lévy and Smith stepped semi-Markov processes, as well as some other subclasses....

# Particle Swarm Optimization

This is the first book devoted entirely to Particle Swarm Optimization (PSO), which is a non-specific algorithm, similar to evolutionary algorithms, such as taboo search and ant colonies. Since its original development in 1995, PSO has mainly been applied to continuous-discrete heterogeneous strongly non-linear numerical optimization and it is thus used almost everywhere in the world. Its converge...

# Branching Random Walks in Non-homogeneous Environments

The book is devoted to a modern section of the probability theory, the so-called theory of branching random walks.

Chapter 1 describes the random walk model in the finite branching one-source environment.

Chapter 2 is devoted to a model of homogeneous, symmetrical, irreducible random walk (without branching) with finite variance of the jumps on the multidimensional integer continuous-time lattic...

# Statistical Modeling in Clinical Trials

Clinical trials at the confirmatory stage of drug development typically require a large number of patients recruited by many clinical centers. The design of multi-center clinical trials consists of several interconnected stages including patient recruitment planning, choosing a randomization scheme and a statistical method for analyzing patient responses on different treatments, and planning the d...

# Distributions of Goodness-of-Fit Statistics

This book is devoted to the theory of goodness-of-fit tests based on weighted empirical processes. Much attention has been given to the limit distributions of statistics of these tests as well as to convergence problems. Cramér-von Mises statistics are studied throughout most of the book, but attention is also given to the Kolmogorov-Smirnov test, the chi-square goodness-offit test as well as some...