Stochastics Books

Book SynopsisThis book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems.The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects.In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution. Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems. Table of ContentsPreface.- Introduction.- Preliminaries.- Bounds of the Potential Function.- Some Explicit Asymptotic Forms of a(x).- Applications Under m+/m → 0.- The Two-Sided Exit Problem – General Case.- The Two-Sided Exit Problem for Relatively Stable Walks.- Absorption Problems for Asymptotically Stable Random Walks.- Asymptotically Stable RandomWalks Killed Upon Hitting a Finite Set.- Appendix.- References.- Notation Index.- Subject Index.

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Book SynopsisThe series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

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Book SynopsisThis book is a translation of the third edition of the well accepted German textbook 'Stochastik', which presents the fundamental ideas and results of both probability theory and statistics, and comprises the material of a one-year course. The stochastic concepts, models and methods are motivated by examples and problems and then developed and analysed systematically.Trade Review"The book can be used by undergraduate mathematics majors but also by science and engeneering students who wish not only to apply probability and statistics but also to understand how the methods work."Vladimir P. Kurenok in: Mathematical Reviews 2009b "The book is well-written and mathematically oriented students and researchers will certainly find that it provides a high level introduction to probability theory and mathematical statistics."In: EMS Newsletter 9/2008

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Book SynopsisThis book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms. Moreover this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on additive functional. Since the publication of the first edition in 1994, this book has attracted constant interests from readers and is by now regarded as a standard reference for the theory of Dirichlet forms. For the present second edition, the authors not only revised the existing text, but also added sections on capacities and Sobolev type inequalities, irreducible recurrence and ergodicity, recurrence and Poincaré type inequalities, the Donsker-Varadhan type large deviation principle, as well as several new exercises with solutions. The book addresses to researchers and graduate students who wish to comprehend the area of Dirichlet forms and symmetric Markov processes.

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Book SynopsisMarkov processes represent a universal model for a large variety of real life random evolutions. The wide flow of new ideas, tools, methods and applications constantly pours into the ever-growing stream of research on Markov processes that rapidly spreads over new fields of natural and social sciences, creating new streamlined logical paths to its turbulent boundary. Even if a given process is not Markov, it can be often inserted into a larger Markov one (Markovianization procedure) by including the key historic parameters into the state space. This monograph gives a concise, but systematic and self-contained, exposition of the essentials of Markov processes, together with recent achievements, working from the "physical picture" - a formal pre-generator, and stressing the interplay between probabilistic (stochastic differential equations) and analytic (semigroups) tools. The book will be useful to students and researchers. Part I can be used for a one-semester course on Brownian motion, Lévy and Markov processes, or on probabilistic methods for PDE. Part II mainly contains the author's research on Markov processes. From the contents: Tools from Probability and Analysis Brownian motion Markov processes and martingales SDE, ψDE and martingale problems Processes in Euclidean spaces Processes in domains with a boundary Heat kernels for stable-like processes Continuous-time random walks and fractional dynamics Complex chains and Feynman integralTable of ContentsPart I Brownian Motion, Markov Processes, Martingales. 1 Preliminaries in Probability and Analysis. 2 Browninan Motion I: Constructions. 3 Martingales and Markov Processes. 4 Browninan Motion II: Elements of Analysis. Part II Basic Constructions of Markov Semigroups. 1 Analytic Constructions. 2 Probabilistic Constructions. 3 Heat Kernel Estimates. 4 Process in Cones and Bounded Domains. Part III Extensions, Developments, Applications. 1 CTRW and Fractional Dynamics. 2 Complex Markov Chains and Feynman integral. 3 Controlled Processes. 4 Semiclassical Asymptotic. 5 Miscellany. 6 Bibliographical Comments.

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Book SynopsisThe book gives a systematical presentation of stochastic approximation methods for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The volume presents results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

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Book SynopsisThe aim of this book is to provide methods and algorithms for the optimization of input signals so as to estimate parameters in systems described by PDE’s as accurate as possible under given constraints. The optimality conditions have their background in the optimal experiment design theory for regression functions and in simple but useful results on the dependence of eigenvalues of partial differential operators on their parameters. Examples are provided that reveal sometimes intriguing geometry of spatiotemporal input signals and responses to them. An introduction to optimal experimental design for parameter estimation of regression functions is provided. The emphasis is on functions having a tensor product (Kronecker) structure that is compatible with eigenfunctions of many partial differential operators. New optimality conditions in the time domain and computational algorithms are derived for D-optimal input signals when parameters of ordinary differential equations are estimated. They are used as building blocks for constructing D-optimal spatio-temporal inputs for systems described by linear partial differential equations of the parabolic and hyperbolic types with constant parameters. Optimality conditions for spatially distributed signals are also obtained for equations of elliptic type in those cases where their eigenfunctions do not depend on unknown constant parameters. These conditions and the resulting algorithms are interesting in their own right and, moreover, they are second building blocks for optimality of spatio-temporal signals. A discussion of the generalizability and possible applications of the results obtained is presented.

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Book SynopsisOne of the first books to provide in-depth and systematic application of finite element methods to the field, Stochastic Structural Dynamics presents and illustrates direct integration methods for analyzing the statistics of the response of structures to stochastic loads.Table of ContentsDedication xi Preface xiii Acknowledgements xv 1. Introduction 1 1.1 Displacement Formulation Based Finite Element Method 2 1.2 Element Equations of Motion for Temporally and Spatially Stochastic Systems 13 1.3 Hybrid Stress Based Element Equations of Motion 14 1.4 Incremental Variational Principle and Mixed Formulation Based Nonlinear Element Matrices 18 1.5 Constitutive Relations and Updating of Configurations and Stresses 36 1.6 Concluding Remarks 48 References 49 2. Spectral Analysis and Response Statistics of Linear Structural Systems 53 2.1 Spectral Analysis 53 2.2 Evolutionary Spectral Analysis 56 2.3 Evolutionary Spectra of Engineering Structures 60 2.4 Modal Analysis and Time-Dependent Response Statistics 76 2.5 Response Statistics of Engineering Structures 79 References 94 3. Direct Integration Methods for Linear Structural Systems 97 3.1 Stochastic Central Difference Method 97 3.2 Stochastic Central Difference Method with Time Co-ordinate Transformation 100 3.3 Applications 102 3.4 Extended Stochastic Central Difference Method and Narrow-band Force Vector 114 3.5 Stochastic Newmark Family of Algorithms 122 References 128 4. Modal Analysis and Response Statistics of Quasi-linear Structural Systems 131 4.1 Modal Analysis of Temporally Stochastic Quasi-linear Systems 131 4.2 Response Analysis Based on Melosh-Zienkiewicz-Cheung Bending Plate Finite Element 141 4.3 Response Analysis Based on High Precision Triangular Plate Finite Element 156 4.4 Concluding Remarks 166 References 166 5. Direct Integration Methods for Response Statistics of Quasi-linear Structural Systems 169 5.1 Stochastic Central Difference Method for Quasi-linear Structural Systems 169 5.2 Recursive Covariance Matrix of Displacements of Cantilever Pipe Containing Turbulent Fluid 174 5.3 Quasi-linear Systems under Narrow-band Random Excitations 184 5.4 Concluding Remarks 188 References 190 6. Direct Integration Methods for Temporally Stochastic Nonlinear Structural Systems 191 6.1 Statistical Linearization Techniques 191 6.2 Symplectic Algorithms of Newmark Family of Integration Schemes 194 6.3 Stochastic Central Difference Method with Time Co-ordinate Transformation and Adaptive Time Schemes 199 6.4 Outline of steps in computer program 211 6.5 Large Deformations of Plate and Shell Structures 213 6.6 Concluding Remarks 224 References 226 7. Direct Integration Methods for Temporally and Spatially Stochastic Nonlinear Structural Systems 231 7.1 Perturbation Approximation Techniques and Stochastic Finite Element Methods 232 7.2 Stochastic Central Difference Methods for Temporally and Spatially Stochastic Nonlinear Systems 241 7.3 Finite Deformations of Spherical Shells with Large Spatially Stochastic Parameters 251 7.4 Closing Remarks 255 References 257 Appendices 1A Mass and Stiffness Matrices of Higher Order Tapered Beam Element 261 1B Consistent Stiffness Matrix of Lower Order Triangular Shell Element 267 1B.1 Inverse of Element Generalized Stiffness Matrix 267 1B.2 Element Leverage Matrices 268 1B.3 Element Component Stiffness Matrix Associated with Torsion 271 References 276 1C Consistent Mass Matrix of Lower Order Triangular Shell Element 277 Reference 280 2A Eigenvalue Solution 281 References 282 2B Derivation of Evolutionary Spectral Densities and Variances of Displacements 283 2B.1 Evolutionary Spectral Densities Due to Exponentially Decaying Random Excitations 283 2B.2 Evolutionary Spectral Densities Due to Uniformly Modulated Random Excitations 286 2B.3 Variances of Displacements 288 References 297 2C Time-dependent Covariances of Displacements 299 2D Covariances of Displacements and Velocities 311 2E Time-dependent Covariances of Velocities 317 2F Cylindrical Shell Element Matrices 323 3A Deterministic Newmark Family of Algorithms 327 Reference 331 Index 333

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Book SynopsisA beginner's guide to stochastic growth modeling The chief advantage of stochastic growth models over deterministic models is that they combine both deterministic and stochastic elements of dynamic behaviors, such as weather, natural disasters, market fluctuations, and epidemics. This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of outcomes. However, the background requirements for studying SDEs can be daunting for those who lack the rigorous course of study received by math majors. Designed to be accessible to readers who have had only a few courses in calculus and statistics, this book offers a comprehensive review of the mathematical essentials needed to understand and apply stochastic growth models. In addition, the book describes deterministic and stochastic applications of population growth models including logistic, generalized logistic, Gompertz, negative exponentiTrade Review"An indispensable resource for students and practitioners with limited exposure tomathematics and statistics, Stochastic Differential Equations: An Introduction withApplications in Population Dynamics Modeling is an excellent fit for advanced under-graduates and beginning graduate students, as well as practitioners who need a gentleintroduction to SDEs" Mathematical Reviews, October 2017Table of ContentsDedication x Preface xi Symbols and Abbreviations xiii 1 Mathematical Foundations 1: Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration 1 1.1 Set Notation and Operations 1 1.1.1 Sets and Set Inclusion 1 1.1.2 Set Algebra 2 1.2 Single-Valued Functions 4 1.3 Real and Extended Real Numbers 6 1.4 Metric Spaces 7 1.5 Limits of Sequences 8 1.6 Point-Set Theory 10 1.7 Continuous Functions 12 1.8 Operations on Sequences of Sets 13 1.9 Classes of Subsets of Ω 15 1.9.1 Topological Space 15 1.9.2 σ-Algebra of Sets and the Borel σ-Algebra 15 1.10 Set and Measure Functions 17 1.10.1 Set Functions 17 1.10.2 Measure Functions 18 1.10.3 Outer Measure Functions 19 1.10.4 Complete Measure Functions 21 1.10.5 Lebesgue Measure 21 1.10.6 Measurable Functions 23 1.10.7 Lebesgue Measurable Functions 26 1.11 Normed Linear Spaces 27 1.11.1 Space of Bounded Real-Valued Functions 27 1.11.2 Space of Bounded Continuous Real-Valued Functions 28 1.11.3 Some Classical Banach Spaces 29 1.12 Integration 31 1.12.1 Integral of a Non-negative Simple Function 32 1.12.2 Integral of a Non-negative Measurable Function Using Simple Functions 33 1.12.3 Integral of a Measurable Function 33 1.12.4 Integral of a Measurable Function on a Measurable Set 34 1.12.5 Convergence of Sequences of Functions 35 2 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 37 2.1 Probability Spaces 37 2.2 Probability Distributions 42 2.3 The Expectation of a Random Variable 49 2.3.1 Theoretical Underpinnings 49 2.3.2 Computational Considerations 50 2.4 Moments of a Random Variable 52 2.5 Multiple Random Variables 54 2.5.1 The Discrete Case 54 2.5.2 The Continuous Case 59 2.5.3 Expectations and Moments 63 2.5.4 The Multivariate Discrete and Continuous Cases 69 2.6 Convergence of Sequences of Random Variables 72 2.6.1 Almost Sure Convergence 73 2.6.2 Convergence in Lp,p>0 73 2.6.3 Convergence in Probability 75 2.6.4 Convergence in Distribution 75 2.6.5 Convergence of Expectations 76 2.6.6 Convergence of Sequences of Events 78 2.6.7 Applications of Convergence of Random Variables 79 2.7 A Couple of Important Inequalities 80 Appendix 2.A The Conditional Expectation E(X|Y) 81 3 Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 85 3.1 Stochastic Processes 85 3.1.1 Finite-Dimensional Distributions of a Stochastic Process 86 3.1.2 Selected Characteristics of Stochastic Processes 88 3.1.3 Filtrations of A 89 3.2 Martingales 91 3.2.1 Discrete-Time Martingales 91 3.2.1.1 Discrete-Time Martingale Convergence 93 3.2.2 Continuous-Time Martingales 96 3.2.2.1 Continuous-Time Martingale Convergence 97 3.2.3 Martingale Inequalities 97 3.3 Path Regularity of Stochastic Processes 98 3.4 Symmetric Random Walk 99 3.5 Brownian Motion 100 3.5.1 Standard Brownian Motion 100 3.5.2 BM as a Markov Process 104 3.5.3 Constructing BM 106 3.5.3.1 BM Constructed from N(0, 1) Random Variables 106 3.5.3.2 BM as the Limit of Symmetric Random Walks 108 3.5.4 White Noise Process 109 Appendix 3.A Kolmogorov Existence Theorem: Another Look 109 Appendix 3.B Nondifferentiability of BM 110 4 Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 113 4.1 Introduction 113 4.2 Stochastic Integration: The Itô Integral 114 4.3 One-Dimensional Itô Formula 120 4.4 Martingale Representation Theorem 126 4.5 Multidimensional Itô Formula 127 Appendix 4.A Itô’s Formula 129 Appendix 4.B Multidimensional Itô Formula 130 5 Stochastic Differential Equations 133 5.1 Introduction 133 5.2 Existence and Uniqueness of Solutions 134 5.3 Linear SDEs 136 5.3.1 Strong Solutions to Linear SDEs 137 5.3.2 Properties of Solutions 147 5.3.3 Solutions to SDEs as Markov Processes 152 5.4 SDEs and Stability 154 Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 159 5.A.1 Linear Homogeneous Variety 159 5.A.2 Linear Variety 161 Appendix 5.B Integrating Factors and Variation of Parameters 162 5.B.1 Integrating Factors 163 5.B.2 Variation of Parameters 164 6 Stochastic Population Growth Models 167 6.1 Introduction 167 6.2 A Deterministic Population Growth Model 168 6.3 A Stochastic Population Growth Model 169 6.4 Deterministic and Stochastic Logistic Growth Models 170 6.5 Deterministic and Stochastic Generalized Logistic Growth Models 174 6.6 Deterministic and Stochastic Gompertz Growth Models 177 6.7 Deterministic and Stochastic Negative Exponential Growth Models 179 6.8 Deterministic and Stochastic Linear Growth Models 181 6.9 Stochastic Square-Root Growth Model with Mean Reversion 182 Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184 Appendix 6.B Reducible SDEs 189 7 Approximation and Estimation of Solutions to Stochastic Differential Equations 193 7.1 Introduction 193 7.2 Iterative Schemes for Approximating SDEs 194 7.2.1 The EM Approximation 194 7.2.2 Strong and Weak Convergence of the EM Scheme 196 7.2.3 The Milstein (Second-Order) Approximation 196 7.3 The Lamperti Transformation 199 7.4 Variations on the EM and Milstein Schemes 203 7.5 Local Linearization Techniques 205 7.5.1 The Ozaki Method 205 7.5.2 The Shoji–Ozaki Method 207 7.5.3 The Rate of Convergence of the Local Linearization Method 211 Appendix 7.A Stochastic Taylor Expansions 212 Appendix 7.B The EM and Milstein Discretizations 217 7.B.1 The EM Scheme 217 7.B.2 The Milstein Scheme 218 Appendix 7.C The Lamperti Transformation 219 8 Estimation of Parameters of Stochastic Differential Equations 221 8.1 Introduction 221 8.2 The Transition Probability Density Function Is Known 222 8.3 The Transition Probability Density Function Is Unknown 227 8.3.1 Parameter Estimation via Approximation Methods 228 8.3.1.1 The EM Routine 228 8.3.1.2 The Ozaki Routine 230 8.3.1.3 The SO Routine 233 Appendix 8.A The ML Technique 235 Appendix 8.B The Log-Normal Probability Distribution 238 Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239 Appendix 8.D Change of Variable 241 Appendix A: A Review of Some Fundamental Calculus Concepts 245 Appendix B: The Lebesgue Integral 259 Appendix C: Lebesgue–Stieltjes Integral 261 Appendix D: A Brief Review of Ordinary Differential Equations 263 References 275 Index 279

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Book SynopsisStochastic Modeling for Medical Image Analysis provides a brief introduction to medical imaging, stochastic modeling, and model-guided image analysis.Today, image-guided computer-assisted diagnostics (CAD) faces two basic challenging problems. The first is the computationally feasible and accurate modeling of images from different modalities to obtain clinically useful information. The second is the accurate and fast inferring of meaningful and clinically valid CAD decisions and/or predictions on the basis of model-guided image analysis.To help address this, this book details original stochastic appearance and shape models with computationally feasible and efficient learning techniques for improving the performance of object detection, segmentation, alignment, and analysis in a number of important CAD applications.The book demonstrates accurate descriptions of visual appearances and shapes of the goal objects and their background to help solve aTable of ContentsMedical Imaging Modalities. From Images to Graphical Models. IRF Models: Estimating Marginals. Markov-Gibbs Random Field Models: Estimating Signal Interactions. Applications: Image Alignment. Segmenting Multimodal Images. Segmenting with Deformable Models. Segmenting with Shape and Appearance Priors. Cine Cardiac MRI Analysis. Sizing Cardiac Pathologies.

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Book SynopsisThe papers collected here focus on probabilistic causality, addressing topics such as the search for causal mechanisms, epistemic and metaphysical views of causality, Bayesian nets and causal dependence, and causation in the special sciences. Some papers stress the statistical analysis of probabilistic data; others address causal issues in physics, with an emphasis on physical processes that are also probabilistic—i.e., stochastic processes.

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Book SynopsisSwitching processes, invented by the author in 1977, is the main tool used in the investigation of traffic problems from automotive to telecommunications. The title provides a new approach to low traffic problems based on the analysis of flows of rare events and queuing models. In the case of fast switching, averaging principle and diffusion approximation results are proved and applied to the investigation of transient phenomena for wide classes of overloading queuing networks. The book is devoted to developing the asymptotic theory for the class of switching queuing models which covers models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc.Table of ContentsPreface 13 Definitions 17 Chapter 1. Switching Stochastic Models 19 1.1. Random processes with discrete component 19 1.1.1.Markov and semi-Markov processes 21 1.1.2. Processes with independent increments and Markov switching 21 1.1.3. Processes with independent increments and semi-Markov switching 23 1.2. Switching processes 24 1.2.1. Definition of switching processes 24 1.2.2. Recurrent processes of semi-Markov type (simple case) 26 1.2.3.RPSMwithMarkov switching 26 1.2.4. General case of RPSM 27 1.2.5. Processes with Markov or semi-Markov switching 27 1.3. Switching stochastic models 28 1.3.1. Sums of random variables 29 1.3.2. Random movements 29 1.3.3. Dynamic systems in a random environment 30 1.3.4. Stochastic differential equations in a random environment 30 1.3.5. Branching processes 31 1.3.6. State-dependent flows 32 1.3.7. Two-level Markov systems with feedback 32 1.4. Bibliography 33 Chapter 2. Switching Queueing Models 37 2.1. Introduction 37 2.2. Queueing systems 38 2.2.1. Markov queueing models 38 2.2.1.1. A state-dependent system MQ/MQ/1/∞ 39 2.2.1.2. Queueing system MM,Q/MM,Q/1/m 40 2.2.1.3. System MQ,B/MQ,B/1/∞ 41 2.2.2.Non-Markov systems 42 2.2.2.1. Semi-Markov system SM/MSM,Q/1 42 2.2.2.2. System MSM,Q/MSM,Q/1/∞ 43 2.2.2.3. System MSM,Q/MSM,Q/1/V 44 2.2.3. Models with dependent arrival flows 45 2.2.4. Polling systems 46 2.2.5. Retrial queueing systems 47 2.3. Queueing networks 48 2.3.1. Markov state-dependent networks 49 2.3.1.1. Markov network (MQ/MQ/m/∞)r 49 2.3.1.2. Markov networks (MQ,B/MQ,B/m/∞)r with batches 50 2.3.2.Non-Markov networks 50 2.3.2.1. State-dependent semi-Markov networks 50 2.3.2.2. Semi-Markov networks with random batches 52 2.3.2.3. Networks with state-dependent input 53 2.4.Bibliography 54 Chapter 3. Processes of Sums of Weakly-dependent Variables 57 3.1. Limit theorems for processes of sums of conditionally independent random variables 57 3.2. Limit theorems for sums with Markov switching 65 3.2.1. Flows of rare events 67 3.2.1.1. Discrete time 67 3.2.1.2. Continuous time 69 3.3. Quasi-ergodic Markov processes 70 3.4. Limit theorems for non-homogenous Markov processes 73 3.4.1. Convergence to Gaussian processes 74 3.4.2. Convergence to processes with independent increments 78 3.5. Bibliography 81 Chapter 4. Averaging Principle and Diffusion Approximation for Switching Processes 83 4.1. Introduction 83 4.2. Averaging principle for switching recurrent sequences 84 4.3. Averaging principle and diffusion approximation for RPSMs 88 4.4. Averaging principle and diffusion approximation for recurrent processes of semi-Markov type (Markov case) 95 4.4.1. Averaging principle and diffusion approximation for SMP 105 4.5. Averaging principle for RPSM with feedback 106 4.6. Averaging principle and diffusion approximation for switching processes 108 4.6.1. Averaging principle and diffusion approximation for processes with semi-Markov switching 112 4.7. Bibliography 113 Chapter 5. Averaging and Diffusion Approximation in Overloaded Switching Queueing Systems and Networks 117 5.1. Introduction 117 5.2. Markov queueing models 120 5.2.1. System MQ,B/MQ,B/1/∞ 121 5.2.2. System MQ/MQ/1/∞ 124 5.2.3. Analysis of the waiting time 129 5.2.4. An output process 131 5.2.5. Time-dependent system MQ,t/MQ,t/1/∞ 132 5.2.6. Asystemwith impatient calls 134 5.3. Non-Markov queueing models 135 5.3.1. System GI/MQ/1/∞ 135 5.3.2. Semi-Markov system SM/MSM,Q/1/∞ 136 5.3.3. System MSM,Q/MSM,Q/1/∞ 138 5.3.4. System SMQ/MSM,Q/1/∞ 139 5.3.5. System GQ/MQ/1/∞ 142 5.3.6. A system with unreliable servers 143 5.3.7. Polling systems 145 5.4. Retrial queueing systems 146 5.4.1. Retrial system MQ/G/1/w.r 147 5.4.2. System M¯ /G¯/1/w.r 150 5.4.3. Retrial system M/M/m/w.r 154 5.5. Queueing networks 159 5.5.1. State-dependent Markov network (MQ/MQ/1/∞)r 159 5.5.2. Markov state-dependent networks with batches 161 5.6. Non-Markov queueing networks 164 5.6.1. A network (MSM,Q/MSM,Q/1/∞)r with semi-Markov switching 164 5.6.2. State-dependent network with recurrent input 169 5.7. Bibliography 172 Chapter 6. Systems in Low Traffic Conditions 175 6.1. Introduction 175 6.2. Analysis of the first exit time from the subset of states 176 6.2.1. Definition of S-set 176 6.2.2. An asymptotic behavior of the first exit time 177 6.2.3. State space forming a monotone structure 180 6.2.4. Exit time as the time of first jump of the process of sums with Markov switching 182 6.3. Markov queueing systems with fast service 183 6.3.1. M/M/s/m systems 183 6.3.1.1. System MM/M/l/m in a Markov environment 185 6.3.2. Semi-Markov queueing systems with fast service 188 6.4. Single-server retrial queueing model 190 6.4.1. Case 1: fast service 191 6.4.1.1. State-dependent case 194 6.4.2. Case 2: fast service and large retrial rate 195 6.4.3. State-dependent model in a Markov environment 197 6.5. Multiserver retrial queueing models 201 6.6. Bibliography 204 Chapter 7. Flows of Rare Events in Low and Heavy Traffic Conditions 207 7.1. Introduction 207 7.2. Flows of rare events in systems with mixing 208 7.3. Asymptotically connected sets (Vn-S-sets) 211 7.3.1. Homogenous case 211 7.3.2. Non-homogenous case 214 7.4. Heavy traffic conditions 215 7.5. Flows of rare events in queueing models 216 7.5.1. Light traffic analysis in models with finite capacity 216 7.5.2. Heavy traffic analysis 218 7.6. Bibliography 219 Chapter 8. Asymptotic Aggregation of State Space 221 8.1. Introduction 221 8.2. Aggregation of finite Markov processes (stationary behavior) 223 8.2.1. Discrete time 223 8.2.2. Hierarchic asymptotic aggregation 225 8.2.3. Continuous time 227 8.3. Convergence of switching processes 228 8.4. Aggregation of states in Markov models 231 8.4.1. Convergence of the aggregated process to a Markov process (finite state space) 232 8.4.2. Convergence of the aggregated process with a general state space 236 8.4.3. Accumulating processes in aggregation scheme 237 8.4.4. MP aggregation in continuous time 238 8.5. Asymptotic behavior of the first exit time from the subset of states (non-homogenous in time case) 240 8.6. Aggregation of states of non-homogenous Markov processes 243 8.7. Averaging principle for RPSM in the asymptotically aggregated Markov environment 246 8.7.1. Switching MP with a finite state space 247 8.7.2. Switching MP with a general state space 250 8.7.3. Averaging principle for accumulating processes in the asymptotically aggregated semi-Markov environment 251 8.8. Diffusion approximation for RPSM in the asymptotically aggregated Markov environment 252 8.9. Aggregation of states in Markov queueing models 255 8.9.1. System MQ/MQ/r/∞ with unreliable servers in heavy traffic 255 8.9.2. System MM,Q/MM,Q/1/∞ in heavy traffic 256 8.10. Aggregation of states in semi-Markov queueing models 258 8.10.1. System SM/MSM,Q/1/∞ 258 8.10.2. System MSM,Q/MSM,Q/1/∞ 259 8.11. Analysis of flows of lost calls 260 8.12. Bibliography 263 Chapter 9. Aggregation in Markov Models with Fast Markov Switching 267 9.1. Introduction 267 9.2. Markov models with fast Markov switching 269 9.2.1.Markov processes with Markov switching 269 9.2.2. Markov queueing systems with Markov type switching 271 9.2.3. Averaging in the fast Markov type environment 272 9.2.4. Approximation of a stationary distribution 274 9.3. Proofs of theorems 275 9.3.1. Proof of Theorem 9.1 275 9.3.2. Proof of Theorem 9.2 277 9.3.3. Proof of Theorem 9.3 279 9.4. Queueing systems with fast Markov type switching 279 9.4.1. System MM,Q/MM,Q/1/N 279 9.4.1.1. Averaging of states of the environment 279 9.4.1.2. The approximation of a stationary distribution 280 9.4.2. Batch system BMM,Q/BMM,Q/1/N 281 9.4.3. System M/M/s/mwith unreliable servers 282 9.4.4. Priority model MQ/MQ/m/s,N 283 9.5. Non-homogenous in time queueing models 285 9.5.1. SystemMM,Q,t/MM,Q,t/s/m with fast switching – averaging of states 286 9.5.2. System MM,Q/MM,Q/s/m with fast switching – aggregation of states 287 9.6. Numerical examples 288 9.7. Bibliography 289 Chapter 10. Aggregation in Markov Models with Fast Semi-Markov Switching 291 10.1. Markov processes with fast semi-Markov switches 292 10.1.1.Averaging of a semi-Markov environment 292 10.1.2. Asymptotic aggregation of a semi-Markov environment 300 10.1.3. Approximation of a stationary distribution 305 10.2. Averaging and aggregation in Markov queueing systems with semi-Markov switching 309 10.2.1.Averaging of states of the environment 309 10.2.2. Asymptotic aggregation of states of the environment 310 10.2.3. The approximation of a stationary distribution 311 10.3. Bibliography 313 Chapter 11. Other Applications of Switching Processes 315 11.1. Self-organization in multicomponent interacting Markov systems 315 11.2. Averaging principle and diffusion approximation for dynamic systems with stochastic perturbations 319 11.2.1. Recurrent perturbations 319 11.2.2. Semi-Markov perturbations 321 11.3. Random movements 324 11.3.1. Ergodic case 324 11.3.2. Case of the asymptotic aggregation of state space 325 11.4. Bibliography 326 Chapter 12. Simulation Examples 329 12.1. Simulation of recurrent sequences 329 12.2. Simulation of recurrent point processes 331 12.3. Simulation ofRPSM 332 12.4. Simulation of state-dependent queueing models 334 12.5. Simulation of the exit time from a subset of states of a Markov chain 337 12.6. Aggregation of states in Markov models 340 Index 343

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Book SynopsisThis 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.Table of ContentsPreface ix Chapter 1. Introduction to Stochastic Processes 1 1.1. Sequences of random variables 1 1.2. The notion of stochastic process 10 1.3. Martingales 13 1.4. Markov chains 17 1.5. State classification 24 1.6. Continuous-time Markov processes 27 1.7. Semi-Markov processes 33 Chapter 2. Simple Stochastic Models 37 2.1. Urn models 37 2.2. Random walks 39 2.3. Brownian motion 44 2.4. Poisson processes 50 2.5. Birth and death processes 59 Chapter 3. Elements of Markov Modeling 61 3.1. Markov models: ideas, history, applications 61 3.2. The discrete-time Ehrenfest model 63 3.3. Markov models in genetics 79 3.4. Markov storage models 110 3.5. Reliability of Markov models 124 Chapter 4. Renewal Models 149 4.1. Fundamental concepts and examples 149 4.2. Waiting times 155 4.3. Modified renewal processes 159 4.4. Replacement models 161 4.5. Renewal reward processes 165 4.6. The risk problem of an insurance company 168 4.7. Counter models 171 4.8. Alternating renewal processes 180 4.9. Superposition of renewal processes 182 4.10. Regenerative processes 186 Chapter 5. Semi-Markov Models 189 5.1. Introduction 189 5.2. Markov renewal processes 190 5.3. First-passage times and state classification 196 5.4. Reliability 200 5.5. Reservoir models 207 5.6. Queues 218 5.7. Digital communication channels 222 Chapter 6. Branching Models 227 6.1. The Bienaymé-Galton-Watson model 227 6.2. Generalizations of the B-G-W model 271 6.3. Continuous-time models 302 Chapter 7. Optimal Stopping Models 315 7.1. The classic optimal stopping problem 315 7.2. Renewal with binary decision 333 Bibliography 343 Notation 367 Index 369

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Book SynopsisOptimal 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.Table of ContentsPreface ix Introduction xi Chapter 1. Random Vectors 1 1.1. Definitions and general properties. 1 1.2. Spaces L1 (dP) and L2 (dP) 20 1.3. Mathematical expectation and applications 23 1.4. Second order random variables and vectors. 39 1.5. Linear independence of vectors of L2 (dP) 46 1.6. Conditional expectation (concerning random vectors with density function) 51 1.7. Exercises for Chapter 1 56 Chapter 2. Gaussian Vectors 63 2.1. Some reminders regarding random Gaussian vectors 63 2.2. Definition and characterization of Gaussian vectors 66 2.3. Results relative to independence 68 2.4. Affine transformation of a Gaussian vector 72 2.5. The existence of Gaussian vectors. 74 2.6. Exercises for Chapter 2 84 Chapter 3. Introduction to Discrete Time Processes 93 3.1. Definition 93 3.2. WSS processes and spectral measure 105 3.3. Spectral representation of a WSS process 109 3.4. Introduction to digital filtering 114 3.5. Important example: autoregressive process 127 3.6. Exercises for Chapter 3 132 Chapter 4. Estimation 139 4.1. Position of the problem 139 4.2. Linear estimation 142 4.3. Best estimate – conditional expectation 154 4.4. Example: prediction of an autoregressive process AR (1) 162 4.5. Multivariate processes 163 4.6. Exercises for Chapter 4 172 Chapter 5. The Wiener Filter 177 5.1. Introduction 177 5.2. Resolution and calculation of the FIR filter 179 5.3. Evaluation of the least error 181 5.4. Resolution and calculation of the IIR filter 183 5.5. Evaluation of least mean square error 187 5.6. Exercises for Chapter 5 188 Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 195 6.1. Introduction 195 6.2. Position of problem 198 6.3. Data representation 200 6.4. Minimization of the cost function 202 6.5. Gradient algorithm 209 6.6. Geometric interpretation 212 6.7. Stability and convergence 216 6.8. Estimation of gradient and LMS algorithm 221 6.9. Example of the application of the LMS algorithm 224 6.10. Exercises for Chapter 6 233 Chapter 7. The Kalman Filter 235 7.1. Position of problem 235 7.2. Approach to estimation 239 7.3. Kalman filtering 243 7.4. Exercises for Chapter 7 261 7.5. Appendices 267 7.6. Examples treated using Matlab software 273 Table of Symbols and Notations 281 Bibliography 283 Index 285

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Book SynopsisAufbauend auf dem ersten Band, werden in diesem Buch weiterführende Konzepte der Wahrscheinlichkeitstheorie ausführlich und verständlich diskutiert. Mit vielen exemplarisch durchgerechneten Aufgaben, einer Vielzahl weiterer Problemstellungen und ausführlichen Lösungen bietet es dem Leser die Möglichkeit, die eigenen Fähigkeiten ständig zu erweitern und kritisch zu überprüfen und ein tieferes Verständnis der Materie zu erlangen. Realitätsnahe Anwendungen ermöglichen 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 ständige Verbesserung von Forschungsinstrumenten für die Untersuchung unserer Welt erleben kann. Gerichtet ist das Buch an Gymnasiasten, Studienanfänger an Hochschulen, Lehrer und Interessierte, die sich mit diesem Gebiet vertraut machen möchten. Table of ContentsEinleitung.- Standardabweichung.- Wahrscheinlichkeiten von Wertebereichen.- Das Gesetz der grossen Zahlen.- Stetige Zufallsvariablen.- Erwartungswert und Standardabweichung von stetigen Zufallsvariablen.- Modellieren von Umfragen.- Hypothesentests.- Lineare Regression.

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Book SynopsisThis volume contains papers which were presented at the XV Latin American Congress of Probability and Mathematical Statistics (CLAPEM) in December 2019 in Mérida-Yucatán, México. They represent well the wide set of topics on probability and statistics that was covered at this congress, and their high quality and variety illustrates the rich academic program of the conference.

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Book SynopsisThis book is the first volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume presents necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes defined on regularly perturbed finite Markov chains and semi-Markov processes. The text introduces new asymptotic recurrent algorithms of phase space reduction. It also addresses both effective conditions of weak convergence for distributions of hitting times as well as convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications. Trade Review“The book is concluded with the most up-to-date references, which complement these topics for the interested reader. This book is well-organized and expansive. … The results are properly motivated with precise, detailed proofs provided. This book is a must-have item for researchers interested in limit theorems as well as for other probability theorists.” (Steve Drekic, Mathematical Reviews, February, 2023)Table of ContentsPreface.- List of symbols.- Introduction.- Part I: First-Rare-Event Times for Regularly Perturbed Semi-Markov Processes.- Flows of Rare Events for Regularly Perturbed Semi-Markov Processes.- Generalizations of Limit Theorems for First-Rare-Event Times.- First-Rare-Event Times for Perturbed Risk Processes.- First-Rare-Event Times for Perturbed Closed Queuing Systems.- First-Rare-Event Times for Perturbed M/M-Type Queuing Systems.- Part II: Hitting Times and Phase Space Reduction for Perturbed Semi-Markov Processes.- Asymptotically Comparable Functions.- Perturbed Semi-Markov Processes and Reduction of Phase Space.- Asymptotics of Hitting Times for Perturbed Semi-Markov Processes.- Asymptotics for Expectations of Hitting Times for Perturbed Semi-Markov Processes.- Generalizations and Examples of Limit Theorems for Hitting Times.- Limit Theorems for Randomly Stopped Stochastic Processes.- Methodological and Bibliographical Notes.- References.- Index.

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Book SynopsisThis book describes extensions of Sudakov's classical result on the concentration of measure phenomenon for weighted sums of dependent random variables. The central topics of the book are weighted sums of random variables and the concentration of their distributions around Gaussian laws. The analysis takes place within the broader context of concentration of measure for functions on high-dimensional spheres. Starting from the usual concentration of Lipschitz functions around their limiting mean, the authors proceed to derive concentration around limiting affine or polynomial functions, aiming towards a theory of higher order concentration based on functional inequalities of log-Sobolev and Poincaré type. These results make it possible to derive concentration of higher order for weighted sums of classes of dependent variables.While the first part of the book discusses the basic notions and results from probability and analysis which are needed for the remainder of the book, the latter parts provide a thorough exposition of concentration, analysis on the sphere, higher order normal approximation and classes of weighted sums of dependent random variables with and without symmetries. Table of ContentsPart I. Generalities.- 1. Moments and correlation conditions.- 2. Some classes of probability distributions.- 3. Characteristic functions.- 4. Sums of independent random variables.- Part II. Selected topics on concentration.- 5. Standard analytic conditions.- 6. Poincaré-type inequalities.- 7. Logarithmic Sobolev inequalities.- 8. Supremum and infimum convolutions.- Part IV. Analysis on the sphere.- 9. Sobolev-type inequalities.- 10. Second order spherical concentration.- 11. Linear functionals on the sphere.- Part V. First applications to randomized sums.- 12. Typical distributions.- 13. Characteristic functions of weighted sums.- 14. Fluctuations of distributions.- Part VI. Refined bounds and rates.- 15. L^2 expansions and estimates.- 16. Refinements for the Kolmogorov distance.- 17. Applications of the second order correlation condition.- Part VII. Distributions and coefficients of special types.- 18. Special systems and examples.- 19. Distributions with symmetries.- 20. Product measures.- 21. Coefficients of Special type.- Glossary.

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Book SynopsisThis book extends the theory and applications of random evolutions to semi-Markov random media in discrete time, essentially focusing on semi-Markov chains as switching or driving processes. After giving the definitions of discrete-time semi-Markov chains and random evolutions, it presents the asymptotic theory in a functional setting, including weak convergence results in the series scheme, and their extensions in some additional directions, including reduced random media, controlled processes, and optimal stopping. Finally, applications of discrete-time semi-Markov random evolutions in epidemiology and financial mathematics are discussed. This book will be of interest to researchers and graduate students in applied mathematics and statistics, and other disciplines, including engineering, epidemiology, finance and economics, who are concerned with stochastic models of systems.Table of Contents- 1. Discrete-Time Stochastic Calculus in Banach Space. - 2. Discrete-Time Semi-Markov Chains. - 3. Discrete-Time Semi-Markov Random Evolutions. - 4. Weak Convergence of DTSMRE in Series Scheme. - 5. DTSMRE in Reduced Random Media. - 6. Controlled Discrete-Time Semi-Markov Random Evolutions. - 7. Epidemic Models in Random Media. - 8. Optimal Stopping of Geometric Markov Renewal Chains and Pricing.

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Book SynopsisThis book constitutes the referred proceedings of the 11th International Workshop on Practical Applications of Stochastic Modelling, PASM 2022, was held in Alicante, Spain, in September 2022.The 7 full papers presented in this volume were carefully reviewed and selected from 9 submissions. The papers demonstrate a diverse set of applications and approaches of stochastic modelling.Table of ContentsPerformance modelling of attack graphs.- Towards Calculating the Resilience of a Urban Transport Network under Attack.- Analysis of the Battery Level in Complex Wireless Sensor Networks using a Two Time Scales Second Order Fluid Model.- To Confine or not to Confine: A Mean Field Game Analysis of the End of an Epidemic.- Data Center Organization and Optimization Strategy as a k-ary n-cube Topology.- Towards energy-aware management of shared printers.- Modelling Performance and Fairness of Frame Bursting in IEEE 802.11n using PEPA.

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Book SynopsisThis proceedings book covers a wide range of topics related to uncertainty analysis and its application in various fields of engineering and science. It explores uncertainties in numerical simulations for soil liquefaction potential, the toughness properties of construction materials, experimental tests on cyclic liquefaction potential, and the estimation of geotechnical engineering properties for aerogenerator foundation design. Additionally, the book delves into uncertainties in concrete compressive strength, bio-inspired shape optimization using isogeometric analysis, stochastic damping in rotordynamics, and the hygro-thermal properties of raw earth building materials. It also addresses dynamic analysis with uncertainties in structural parameters, reliability-based design optimization of steel frames, and calibration methods for models with dependent parameters. The book further explores mechanical property characterization in 3D printing, stochastic analysis in computational simulations, probability distribution in branching processes, data assimilation in ocean circulation modeling, uncertainty quantification in climate prediction, and applications of uncertainty quantification in decision problems and disaster management. This comprehensive collection provides insights into the challenges and solutions related to uncertainty in various scientific and engineering contexts.Table of ContentsUncertainties of numerical simulation for static liquefaction potential of saturated soils.- Uncertainties about the Toughness Property of raw earth construction materials.- Uncertainties of Experimental Tests on Cyclic Liquefaction Potential of Unsaturated Soils.- Analysis of the Impact of Uncertainties on the Estimation of Geotechnical Engineering Properties of Soil from SPT on the Design of Aerogenerators Foundation.- Uncertainties on the unconfined compressive strength of raw and textured concrete.- Bio-inspired shape optimization for structural robust design using isogeometric analysis.

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Book SynopsisNow in its second edition, this textbook serves as an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. More classical examples such as Montmort's problem, the ballot problem, and Bertrand’s paradox are now included, along with applications such as the Maxwell-Boltzmann and Bose-Einstein distributions in physics.Key features in new edition:* 35 new exercises* Expanded section on the algebra of sets * Expanded chapters on probabilities to include more classical examples* New section on regression* Online instructors' manual containing solutions to all exercises<Advanced undergraduate and graduate students in computer science, engineering, and other natural and social sciences with only a basic background in calculus will benefit from this introductory text balancing theory with applications.Review of the first edition: This textbook is a classical and well-written introduction to probability theory and statistics. … the book is written ‘for an audience such as computer science students, whose mathematical background is not very strong and who do not need the detail and mathematical depth of similar books written for mathematics or statistics majors.’ … Each new concept is clearly explained and is followed by many detailed examples. … numerous examples of calculations are given and proofs are well-detailed." (Sophie Lemaire, Mathematical Reviews, Issue 2008 m)Trade Review“Schay (emer., Univ. of Massachusetts) has created a text for a two semester, calculus-based course in mathematical statistics. … The prose reads well. Physical production is good. … Summing Up: Recommended. Upper-division undergraduates and graduate students.” (W. R. Lee, Choice, Vol. 54 (6), February, 2017)“I believe that students concentrating in mathematics and related subjects will find this book readable and interesting. … I think that students learning the probability for the first time will get real value out of working through the examples and exercises of the text. … Introduction to Probability with Statistical Applications is very clearly written and reading the book is enjoyable. I would certainly recommend Schay’s book as a primary textbook for an undergraduate course in calculus-based probability.” (Jason M. Graham, MAA Reviews, September, 2016)Table of ContentsIntroduction.- The Algebra of Events.- Combinatorial Problems.- Probabilities.- Random Variables.- Expectation, Variance, Moments.- Some Special Distributions.- The Elements of Mathematical Statistics.

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Book SynopsisThis updated and revised first-course textbook in applied probability provides a contemporary and lively post-calculus introduction to the subject of probability. The exposition reflects a desirable balance between fundamental theory and many applications involving a broad range of real problem scenarios. It is intended to appeal to a wide audience, including mathematics and statistics majors, prospective engineers and scientists, and those business and social science majors interested in the quantitative aspects of their disciplines.The textbook contains enough material for a year-long course, though many instructors will use it for a single term (one semester or one quarter). As such, three course syllabi with expanded course outlines are now available for download on the book’s page on the Springer website.A one-term course would cover material in the core chapters (1-4), supplemented by selections from one or more of the remaining chapters on statistical inference (Ch. 5), Markov chains (Ch. 6), stochastic processes (Ch. 7), and signal processing (Ch. 8—available exclusively online and specifically designed for electrical and computer engineers, making the book suitable for a one-term class on random signals and noise).For a year-long course, core chapters (1-4) are accessible to those who have taken a year of univariate differential and integral calculus; matrix algebra, multivariate calculus, and engineering mathematics are needed for the latter, more advanced chapters. At the heart of the textbook’s pedagogy are 1,100 applied exercises, ranging from straightforward to reasonably challenging, roughly 700 exercises in the first four “core” chapters alone—a self-contained textbook of problems introducing basic theoretical knowledge necessary for solving problems and illustrating how to solve the problems at hand – in R and MATLAB, including code so that students can create simulations. New to this edition• Updated and re-worked Recommended Coverage for instructors, detailing which courses should use the textbook and how to utilize different sections for various objectives and time constraints• Extended and revised instructions and solutions to problem sets• Overhaul of Section 7.7 on continuous-time Markov chains• Supplementary materials include three sample syllabi and updated solutions manuals for both instructors and studentsTable of ContentsProbability.- Discrete Random Variables and Probability Distributions.- Continuous Random Variables and Probability Distributions.- Joint probability distributions and their applications.- The Basics of Statistical Inference.- Markov chains.- Random processes.- Introduction to signal processing.

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Book SynopsisLong-memory processes are known to play an important part in many areas of science and technology, including physics, geophysics, hydrology, telecommunications, economics, finance, climatology, and network engineering. In the last 20 years enormous progress has been made in understanding the probabilistic foundations and statistical principles of such processes. This book provides a timely and comprehensive review, including a thorough discussion of mathematical and probabilistic foundations and statistical methods, emphasizing their practical motivation and mathematical justification. Proofs of the main theorems are provided and data examples illustrate practical aspects. This book will be a valuable resource for researchers and graduate students in statistics, mathematics, econometrics and other quantitative areas, as well as for practitioners and applied researchers who need to analyze data in which long memory, power laws, self-similar scaling or fractal properties are relevant.Trade ReviewFrom the book reviews:“This encyclopaedic book covers almost the whole literature on univariate and multivariate long-range dependent (LRD) processes, or long-memory processes or strongly dependent processes. … This volume is then of strong interest for both researchers and teachers familiar with the topic, as it gives an overall, structured and balanced picture of the current state of the art. Readers less familiar with the topic will easily find their way in the vast literature on this issue, and will have their curiosity satisfied.” (Gilles Teyssière, Mathematical Reviews, October, 2014)“This book aims to cover probabilistic and statistical aspects of long-memory processes in as much detail as possible, including a broad range of topics. The authors did an excellent job to reach their goals, and the book would be a must for researchers interested in long-memory processes and practioners on time series and data analysis. … the book is an excellent choice for anyone who is working in fields related to long-memory processes with many update information and research topics.” (Weiping Li, zbMATH, Vol. 1282, 2014)Table of ContentsDefinition of Long Memory.- Origins and Generation of Long Memory.- Mathematical Concepts.- Limit Theorems.- Statistical Inference for Stationary Processes.- Statistical Inference for Nonlinear Processes.- Statistical Inference for Nonstationary Processes.- Forecasting.- Spatial and Space-Time Processes.- Resampling.- Function Spaces.- Regularly Varying Functions.- Vague Convergence.- Some Useful Integrals.- Notation and Abbreviations.

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Book SynopsisThe lectures concentrate on some old and new relations between quasiderivatives of solutions to Ito stochastic equations and interior smoothness of harmonic functions associated with degenerate elliptic equations. Recent progress in the case of constant coefficients is discussed in full detail.

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Book SynopsisThis volume explores, from a mathematical and a philosophical perspective, the virtuous circle connecting logic and rationality. While logic lends its methods, techniques and ideas to the investigation of rationality, the practical problems which arise in modelling rational behaviour, especially in the social sciences, motivate logicians to develop more refined logical formalisms. This is why non classical logics - a unifying theme of this volume - play a fundamental role in the construction of formal models of rationality.Table of ContentsIntroduction.- 1. Foundations.- Ordered algebras and logic. George Metcalfe, Francesco Paoli and Constantine Tsinakis.- 2. Probability. The social entropy process: Axiomatising the aggregation of probabilistic beliefs. George Wilmers.- Conditional probability in the light of qualitative belief change. David Makinson.- Is there a probability theory of many-valued events? Vincenzo Marra.- 3. Uncertainty.- On Giles style dialogue games and hypersequent systems. Christian G. Fermüller.- Poset representation for free RDP-algebras. Diego Valota.- Uncertainty, indeterminacy and fuzziness: A probabilistic approach. Martina Fedel.- 4. Rationality.- Tractable depth-bounded logics and the problem of logical omniscience. Marcello D’Agostino.- Rational behaviour at trust nodes. Hykel Hosni and Silvia Milano.

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Book SynopsisThis treatise delves into the latest advancements in stochastic volatility models, highlighting the utilization of Markov chain Monte Carlo simulations for estimating model parameters and forecasting the volatility and quantiles of financial asset returns. The modeling of financial time series volatility constitutes a crucial aspect of finance, as it plays a vital role in predicting return distributions and managing risks. Among the various econometric models available, the stochastic volatility model has been a popular choice, particularly in comparison to other models, such as GARCH models, as it has demonstrated superior performance in previous empirical studies in terms of fit, forecasting volatility, and evaluating tail risk measures such as Value-at-Risk and Expected Shortfall. The book also explores an extension of the basic stochastic volatility model, incorporating a skewed return error distribution and a realized volatility measurement equation. The concept of realized volatility, a newly established estimator of volatility using intraday returns data, is introduced, and a comprehensive description of the resulting realized stochastic volatility model is provided. The text contains a thorough explanation of several efficient sampling algorithms for latent log volatilities, as well as an illustration of parameter estimation and volatility prediction through empirical studies utilizing various asset return data, including the yen/US dollar exchange rate, the Dow Jones Industrial Average, and the Nikkei 225 stock index. This publication is highly recommended for readers with an interest in the latest developments in stochastic volatility models and realized stochastic volatility models, particularly in regards to financial risk management.Table of Contents1 Introduction.- 2 Stochastic Volatility Model.- 3 Asymmetric Stochastic Volatility Model.- 4 Stochastic Volatility Model with Generalized Hyperbolic Skew Student’s t Error.- 5 Realized Stochastic Volatility Model.

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Book Synopsis Unifies existing and emerging concepts concerning multi-objective control and stochastic control with engineering-oriented phenomena Establishes a unified theoretical framework for control and filtering problems for a class of discrete-time nonlinear stochastic systems with consideration to performance Includes case studies of several nonlinear stochastic systems Investigates the phenomena of incomplete information, including missing/degraded measurements, actuator failures and sensor saturations Considers both time-invariant systems and time-varying systems Exploits newly developed techniques to handle the emerging mathematical and computational challenges Table of ContentsPreface vii Acknowledgements ix List of Abbreviations xi 1 Introduction 1 1.1 Analysis and Synthesis of Nonlinear Stochastic Systems 2 1.1.1 Nonlinear Systems 3 1.1.2 Stochastic Systems 4 1.2 Multi-Objective Control and Filtering with Variance Constraints 5 1.2.1 Covariance Control Theory 5 1.2.2 Multiple Performance Requirements 7 1.2.3 Design Techniques for Nonlinear Stochastic Systems with Variance Constraints 9 1.2.4 A Special Case of Multi-Objective Design: Mixed H2/H1 Control/Filtering 11 1.3 Outline 12 2 Robust H1 Control with Variance Constraints 17 2.1 Problem Formulation 18 2.2 Stability, H1 Performance and Variance Analysis 20 2.2.1 Stability, H1 Performance Analysis 21 2.2.2 Variance Analysis 23 2.3 Robust Controller Design 27 2.4 Numerical Example 30 2.5 Summary 33 3 Robust Mixed H2=H1 Filtering 41 3.1 System Description and Problem Formulation 42 3.2 Algebraic Characterizations for Robust H2=H1 Filtering 44 3.2.1 Robust H2 Filtering 44 3.2.2 Robust H1 Filtering 50 3.3 Robust H2=H1 Filter Design Techniques 51 3.4 An Illustrative Example 60 3.5 Summary 62 4 Filtering with Missing Measurements 63 4.1 Problem Formulation 64 4.2 Stability and Variance Analysis 67 4.3 Robust Filter Design 71 4.4 Numerical Example 75 4.5 Summary 78 5 Robust Fault-Tolerant Control 87 5.1 Problem Formulation 88 5.2 Stability and Variance Analysis 90 5.3 Robust Controller Design 92 5.4 Numerical Example 98 5.5 Summary 103 6 Robust H2 SMC 105 6.1 The System Model 106 6.2 Robust H2 Sliding Mode Control 107 6.2.1 Switching Surface 107 6.2.2 Performances of the Sliding Motion 108 6.2.3 Computational Algorithm 114 6.3 Sliding Mode Controller 115 6.4 Numerical Example 116 6.5 Summary 118 7 Dissipative Control with Degraded Measurements 125 7.1 Problem Formulation 126 7.2 Stability, Dissipativity and Variance Analysis 129 7.3 Observer-Based Controller Design 134 7.3.1 Solvability of Multi-Objective Control Problem 134 7.3.2 Computational Algorithm 139 7.4 Numerical Example 140 7.5 Summary 142 8 Variance-Constrained H1 Control with Multiplicative Noises 145 8.1 Problem Formulation 146 8.2 Stability, H1 Performance, Variance Analysis 147 8.2.1 Stability 148 8.2.2 H1 performance 150 8.2.3 Variance analysis 152 8.3 Robust State Feedback Controller Design 153 8.4 A Numerical Example 156 8.5 Summary 157 9 Robust Finite-Horizon H1 Control 159 9.1 Problem Formulation 160 9.2 Performance Analysis 162 9.2.1 H1 Performance 162 9.2.2 Variance Analysis 164 9.3 Robust Finite Horizon Controller Design 167 9.4 Numerical Example 171 9.5 Summary 173 10 Finite-Horizon Filtering with Degraded Measurements 177 10.1 Problem Formulation 178 10.2 Performance Analysis 181 10.2.1 H1 Performance Analysis 181 10.2.2 System Covariance Analysis 186 10.3 Robust Filter Design 187 10.4 Numerical Example 190 10.5 Summary 191 11 Mixed H2=H1 Control with Randomly Occurring Nonlinearities: the Finite-Horizon Case 197 11.1 Problem Formulation 199 11.2 H1 Performance 200 11.3 Mixed H2=H1 Controller Design 204 11.3.1 State-Feedback Controller Design 204 11.3.2 Computational Algorithm 207 11.4 Numerical Example 207 11.5 Summary 211 12 Finite-Horizon H2=H1 Control of MJSs with Sensor Failures 213 12.1 Problem Formulation 214 12.2 H1 Performance 216 12.3 Mixed H2=H1 Controller Design 220 12.3.1 Controller Design 220 12.3.2 Computational Algorithm 224 12.4 Numerical Example 224 12.5 Summary 227 13 Finite-Horizon Control with ROSF 229 13.1 Problem Formulation 230 13.2 H1 And Covariance Performances Analysis 234 13.2.1 H1 Performance 234 13.2.2 Covariance Analysis 238 13.3 Robust Finite-Horizon Controller Design 240 13.3.1 Controller Design 240 13.3.2 Computational Algorithm 243 13.4 Numerical Example 243 13.5 Summary 244 14 Finite-Horizon H2=H1 Control with Actuator Failures 247 14.1 Problem Formulation 248 14.2 H1 Performance 251 14.3 Multi-Objective Controller Design 253 14.3.1 Controller Design 253 14.3.2 Computational Algorithm 256 14.4 Numerical Example 257 14.5 Summary 259 15 Conclusions and Future Topics 261 References 263

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Book SynopsisThis title considers the special 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. The class of semi-Markov processes includes strong Markov processes, Lévy and Smith stepped semi-Markov processes, and some other subclasses. Extensive coverage is devoted to non-Markovian semi-Markov processes with continuous trajectories and, in particular, to semi-Markov diffusion processes. Readers looking to enrich their knowledge on Markov processes will find this book a valuable resource.Table of ContentsIntroduction 9 Chapter 1. Stepped Semi-Markov Processes 17 1.1. Random sequence 17 1.2. Markov chain 20 1.3. Two-dimensional Markov chain 25 1.4. Semi-Markov process 29 1.5. Stationary distributions 32 Chapter 2. Sequences of First Exit Times and Regeneration Times 37 2.1. Basic maps 38 2.2. Markov times 42 2.3. Deducing sequences 46 2.4. Correct exit and continuity 55 2.5. Time of regeneration 63 Chapter 3. General Semi-Markov Processes 71 3.1. Definition of a semi-Markov process 72 3.2. Transition function of a SM process 79 3.3. Operators and SM walk 82 3.4. Operators and SM process 91 3.5. Criterion of Markov property for SM processes 103 3.6. Intervals of constancy 110 Chapter 4. Construction of Semi-Markov Processes using Semi-Markov Transition Functions 115 4.1. Realization of an innite systemof pairs 116 4.2. Extension of a measure 121 4.3. Construction of a measure 124 4.4. Construction of a projective system of measures 127 4.5. Semi-Markov processes 133 Chapter 5. Semi-Markov Processes of Diffusion Type 137 5.1. One-dimensional semi-Markov processes of diffusion type 138 5.1.1. Differential equation 138 5.1.2. Construction SM process 143 5.1.3. Some properties of the process 159 5.2. Multi-dimensional processes of diffusion type 168 5.2.1. Differential equations of elliptic type 168 5.2.2. Neighborhood of arbitrary form 171 5.2.3. Neighborhood of spherical form 178 5.2.4. Characteristic operator 188 Chapter 6. Time Change and Semi-Markov Processes 197 6.1. Time change and trajectories 198 6.2. Intrinsic time and traces 206 6.3. Canonical time change 211 6.4. Coordination of function and time change 223 6.5. Random time changes 228 6.6. Additive functionals 233 6.7. Distribution of a time run along the trace 242 6.8. Random curvilinear integrals 252 6.9. Characteristic operator and integral 264 6.10. Stochastic integral 268 6.10.1. Semi-martingale and martingale 268 6.10.2. Stochastic integral 275 6.10.3. Ito-Dynkin’s formula 277 Chapter 7. Limit Theorems for Semi-Markov Processes 281 7.1. Weak compactness and weak convergence 281 7.2. Weak convergence of semi-Markov processes 289 Chapter 8. Representation of a Semi-Markov Process as a Transformed Markov Process 299 8.1. Construction by operator 300 8.2. Comparison of processes 302 8.3. Construction by parameters of Lévy formula 307 8.4. Stationary distribution 311 Chapter 9. Semi-Markov Model of Chromatography 325 9.1. Chromatography 326 9.2. Model of liquid column chromatography 328 9.3. Some monotone Semi-Markov processes 332 9.4. Transfer with diffusion 337 9.5. Transfer with nal absorption 346 Bibliography 361 Index 369

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Book SynopsisStochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due 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 this kind of risk. These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance. Volume 1 starts with the introduction of the basic financial instruments and the fundamental principles of financial modeling and arbitrage valuation of derivatives. Next, we use the discrete-time binomial model to introduce all relevant concepts. The mathematical simplicity of the binomial model also provides us with the opportunity to introduce and discuss in depth concepts such as conditional expectations and martingales in discrete time. However, we do not expand beyond the needs of the stochastic finance framework. Numerous examples, each highlighted and isolated from the text for easy reference and identification, are included. The book concludes with the use of the binomial model to introduce interest rate models and the use of the Markov chain model to introduce credit risk. This volume is designed in such a way that, among other uses, makes it useful as an undergraduate course.Table of ContentsPreface xi Chapter 1. Probability and Random Variables 1 1.1. Introductory notes 1 1.2. Probability space 2 1.3. Conditional probability and independence 8 1.4. Random variables 12 1.5. Expectation and variance of a random variable 24 1.6. Jointly distributed random variables 28 1.7. Moment generating functions 32 1.8. Probability inequalities and limit theorems 37 1.9.Multivariate normal distribution 44 Chapter 2. An Introduction to Financial Instruments and Derivatives 49 2.1. Introduction 49 2.2. Bonds and basic interest rates 50 2.3. Forward contracts 58 2.4. Futures contracts 60 2.5.Swaps 60 2.6.Options 62 2.7. Types of market participants 67 2.8.Arbitrage relationships between call and put options 67 2.9.Exercises 69 Chapter 3. Conditional Expectation and Markov Chains 71 3.1. Introduction 71 3.2. Conditional expectation: the discrete case 72 3.3. Applications of conditional expectations 75 3.4. Properties of the conditional expectation 81 3.5. Markov chains 85 3.6. Exercises 131 4.1. Introductory notes 137 Chapter 4. The No-Arbitrage Binomial Pricing Model 137 4.2. Binomial model 138 4.3. Stochastic evolution of the asset prices 141 4.4. Binomial approximation to the lognormal distribution 143 4.5. One-period European call option 145 4.6. Two-period European call option 150 4.7. Multiperiod binomial model 153 4.8. The evolution of the asset prices as a Markov chain 154 4.9.Exercises 158 Chapter 5. Martingales 163 5.1. Introductory notes 163 5.2.Martingales 164 5.3. Optional sampling theorem 169 5.4. Submartingales, supermartingales and martingales convergence theorem 178 5.5.Martingale transforms 182 5.6. Uniform integrability and Doob’s decomposition 184 5.7.The snell envelope 187 5.8.Exercises 190 Chapter 6. Equivalent Martingale Measures, No-Arbitrage and Complete Markets 195 6.1. Introductory notes 195 6.2. Equivalent martingale measure and the Randon-Nikodým derivative process 196 6.3. Finite general markets 204 6.4. Fundamental theorem of asset pricing 215 6.5.Completemarkets andmartingale representation 222 6.6. Finding the equivalent martingale measure 228 6.7.Exercises 238 Chapter 7. American Derivative Securities 241 7.1. Introductory notes 241 7.2.A three-periodAmerican put option 242 7.3. Hedging strategy for an American put option 249 7.4.The algorithm of the American put option 254 7.5.Optimal time for the holder to exercise 255 7.6. American derivatives in general markets 262 7.7. Extending the concept of self-financing strategies 266 7.8.Exercises 269 Chapter 8. Fixed-Income Markets and Interest Rates 273 8.1. Introductory notes 273 8.2. The zero coupon bonds of all maturities 274 8.3. Arbitrage-free family of bond prices 278 8.4. Interest rate process and the term structure of bond prices 282 8.5. The evolution of the interest rate process 290 8.6. Binomial model with normally distributed spread of interest rates 293 8.7. Binomial model with lognormally distributed spread of interest rates 296 8.8. Option arbitrage pricing on zero coupon bonds 298 8.9. Fixed income derivatives 302 8.10. T-period equivalent forward measure 308 8.11. Futures contracts 317 8.12.Exercises 319 Chapter 9. Credit Risk 323 9.1. Introductory notes 323 9.2. Credit ratings and corporate bonds 324 9.3. Credit risk methodologies 326 9.4. Arbitrage pricing of defaultable bonds 327 9.5. Migration process as a Markov chain 330 9.6. Estimation of the real world transition probabilities 334 9.7. Term structure of credit spread and model calibration 337 9.8. Migration process under the real-world probability measure 341 9.9.Exercises 352 Chapter 10. The Heath-Jarrow-Morton Model 355 10.1. Introductory notes 355 10.2. Heath-Jarrow-Morton model 356 10.3. Hedging strategies for zero coupon bonds 362 10.4.Exercises 364 References 365 Appendices 374 Index 395

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Book SynopsisQuantitative finance has become these last years a extraordinary field of research and interest as well from an academic point of view as for practical applications. At the same time, pension issue is clearly a major economical and financial topic for the next decades in the context of the well-known longevity risk. Surprisingly few books are devoted to application of modern stochastic calculus to pension analysis. The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. In these various problems, financial as well as demographic risks will be addressed and modelled.Table of ContentsPreface xiii Chapter 1. Introduction: Pensions in Perspective 1 1.1. Pension issues 1 1.2. Pension scheme 7 1.3. Pension and risks 11 1.4. The multi-pillar philosophy 14 Chapter 2. Classical Actuarial Theory of Pension Funding 15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes 21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control 31 3.1. Introduction 31 3.2. Deterministic optimal control 31 3.3. Necessary conditions for optimality 33 3.4. The maximum principle 42 3.5. Extension to the one-dimensional stochastic optimal control 45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit Pension Plans 55 4.1. Introduction 55 4.2. The defined benefit method 56 4.3. The defined contribution method 57 4.4. The notional defined contribution (NDC) method 58 4.5. Conclusions 93 Chapter 5. Fair and Market Values and Interest Rate Stochastic Models 95 5.1. Fair value 95 5.2. Market value of financial flows 96 5.3. Yield curve 97 5.4. Yield to maturity for a financial investment and for a bond 99 5.5. Dynamic deterministic continuous time model for an instantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for an instantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage 114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values 132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds 149 6.1. Introduction 149 6.2. Risks in defined contribution 149 6.3. Solvency modeling for a DC pension scheme 150 6.4. Risks in defined benefit 170 6.5. Solvency modeling for a DB pension scheme 171 Chapter 7. Optimal Control of a Defined Benefit Pension Scheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortization strategy 181 7.3. Optimal control of a pension fund in continuous time 194 Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates 223 Chapter 9. Simulation Models 231 9.1. Introduction231 9.2. The direct method 233 9.3. The Monte Carlo models 250 9.4. Salary lines construction 252 Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 284 10.4. Discrete time reward processes 294 10.5. General algorithms for DTSMRWP 304 Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 307 11.1. Application to pension funds evolution 307 11.2. Generalized non-homogeneous semi-Markov model for manpower management 338 11.3. Algorithms 347 APPENDICES 359 Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361 Appendix 2. Itô Calculus and Diffusion Processes 397 Bibliography 437 Index 449

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Book SynopsisThis 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 practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.Trade Review“Thus, the book is a welcome addition in the effort to make stochastic integration and SDE as accessible as possible to the greater public interested in or in need of using them.” (Mathematical Reviews, 1 February 2013) “If I have a chance to teach (again) a course in stochastic financial modelling, I will definitely choose this to be among two or three sources to use. I have all the reasons to strongly recommend it to anybody in the area of modern stochastic modelling.” (Zentralblatt MATH, 1 December 2012)Table of ContentsPreface 9 Notation 13 Chapter 1. Introduction: Basic Notions of Probability Theory 17 1.1. Probability space 17 1.2. Random variables 21 1.3. Characteristics of a random variable 21 1.4. Types of random variables 23 1.5. Conditional probabilities and distributions 26 1.6. Conditional expectations as random variables 27 1.7. Independent events and random variables 29 1.8. Convergence of random variables 29 1.9. Cauchy criterion 31 1.10. Series of random variables 31 1.11. Lebesgue theorem 32 1.12. Fubini theorem 32 1.13. Random processes 33 1.14. Kolmogorov theorem 34 Chapter 2. Brownian Motion 35 2.1. Definition and properties 35 2.2. White noise and Brownian motion 45 2.3. Exercises 49 Chapter 3. Stochastic Models with Brownian Motion and White Noise 51 3.1. Discrete time 51 3.2. Continuous time 55 Chapter 4. Stochastic Integral with Respect to Brownian Motion 59 4.1. Preliminaries. Stochastic integral with respect to a step process 59 4.2. Definition and properties 69 4.3. Extensions 81 4.4. Exercises 85 Chapter 5. Itô’s Formula 87 5.1. Exercises 94 Chapter 6. Stochastic Differential Equations 97 6.1. Exercises 105 Chapter 7. Itô Processes 107 7.1. Exercises 121 Chapter 8. Stratonovich Integral and Equations 125 8.1. Exercises 136 Chapter 9. Linear Stochastic Differential Equations 137 9.1. Explicit solution of a linear SDE 137 9.2. Expectation and variance of a solution of an LSDE 141 9.3. Other explicitly solvable equations 145 9.4. Stochastic exponential equation 147 9.5. Exercises 153 Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155 10.1. Introduction 155 10.2. Backward and forward Kolmogorov equations 161 10.3. Stationary density of a diffusion process 172 10.4. Exercises 176 Chapter 11. Examples 179 11.1. Additive noise: Langevin equation 180 11.2. Additive noise: general case 180 11.3. Multiplicative noise: general remarks 184 11.4. Multiplicative noise: Verhulst equation 186 11.5. Multiplicative noise: genetic model 189 Chapter 12. Example in Finance: Black–Scholes Model 195 12.1. Introduction: what is an option? 195 12.2. Self-financing strategies 197 12.3. Option pricing problem: the Black–Scholes model 204 12.4. Black–Scholes formula 206 12.5. Risk-neutral probabilities: alternative derivation of Black–Scholes formula 210 12.6. Exercises 214 Chapter 13. Numerical Solution of Stochastic Differential Equations 217 13.1. Memories of approximations of ordinary differential equations 218 13.2. Euler approximation 221 13.3. Higher-order strong approximations 224 13.4. First-order weak approximations 231 13.5. Higher-order weak approximations 238 13.6. Example: Milstein-type approximations 241 13.7. Example: Runge–Kutta approximations 244 13.8. Exercises 249 Chapter 14. Elements of Multidimensional Stochastic Analysis 251 14.1. Multidimensional Brownian motion 251 14.2. Itô’s formula for a multidimensional Brownian motion 252 14.3. Stochastic differential equations 253 14.4. Itô processes 254 14.5. Itô’s formula for multidimensional Itô processes 256 14.6. Linear stochastic differential equations 256 14.7. Diffusion processes 257 14.8. Approximations of stochastic differential equations 259 Solutions, Hints, and Answers 261 Bibliography 271 Index 273

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