Details

Probability and Stochastic Processes

1. Aufl.

von: Ionut Florescu
100,99 €
Verlag:	Wiley
Format:	PDF
Veröffentl.:	07.11.2014
ISBN/EAN:	9781118593202
Sprache:	englisch
Anzahl Seiten:	576

In den Warenkorb

Als Gutschein

DRM-geschütztes eBook, Sie benötigen z.B. Adobe Digital Editions und eine Adobe ID zum Lesen.

Beschreibungen

Titelbeschreibung

A comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and real-world applications With a sophisticated approach, Probability and Stochastic Processes successfully balances theory and applications in a pedagogical and accessible format. The book’s primary focus is on key theoretical notions in probability to provide a foundation for understanding concepts and examples related to stochastic processes. Organized into two main sections, the book begins by developing probability theory with topical coverage on probability measure; random variables; integration theory; product spaces, conditional distribution, and conditional expectations; and limit theorems. The second part explores stochastic processes and related concepts including the Poisson process, renewal processes, Markov chains, semi-Markov processes, martingales, and Brownian motion. Featuring a logical combination of traditional and complex theories as well as practices, Probability and Stochastic Processes also includes: <ul> <li>Multiple examples from disciplines such as business, mathematical finance, and engineering</li> <li>Chapter-by-chapter exercises and examples to allow readers to test their comprehension of the presented material</li> <li>A rigorous treatment of all probability and stochastic processes concepts</li> </ul> An appropriate textbook for probability and stochastic processes courses at the upper-undergraduate and graduate level in mathematics, business, and electrical engineering, Probability and Stochastic Processes is also an ideal reference for researchers and practitioners in the fields of mathematics, engineering, and finance.

Inhaltsverzeichnis

List of Figures xvii List of Tables xx Preface xxi Acknowledgments xxiii Introduction 1 Part I Probability 1 Elements of Probability Measure 9 1.1 Probability Spaces 10 1.1.1 Null element of ℱ. Almost sure (a.s.) statements. Indicator of a set 21 1.2 Conditional Probability 22 1.3 Independence 29 1.4 Monotone Convergence Properties of Probability 31 1.5 Lebesgue Measure on the Unit Interval (0,1] 37 Problems 40 2 Random Variables 45 2.1 Discrete and Continuous Random Variables 48 2.2 Examples of Commonly Encountered Random Variables 52 2.3 Existence of Random Variables with Prescribed Distribution 65 2.4 Independence 68 2.5 Functions of Random Variables. Calculating Distributions 72 Problems 82 3 Applied Chapter: Generating Random Variables 87 3.1 Generating One-Dimensional Random Variables by Inverting the cdf 88 3.2 Generating One-Dimensional Normal Random Variables 91 3.3 Generating Random Variables. Rejection Sampling Method 94 3.4 Generating Random Variables. Importance Sampling 109 Problems 119 4 Integration Theory 123 4.1 Integral of Measurable Functions 124 4.2 Expectations 130 4.3 Moments of a Random Variable. Variance and the Correlation Coefficient 143 4.4 Functions of Random Variables. The Transport Formula 145 4.5 Applications. Exercises in Probability Reasoning 148 4.6 A Basic Central Limit Theorem: The DeMoivre–LaplaceTheorem: 150 Problems 152 5 Conditional Distribution and Conditional Expectation 157 5.1 Product Spaces 158 5.2 Conditional Distribution and Expectation. Calculation in Simple Cases 162 5.3 Conditional Expectation. General Definition 165 5.4 Random Vectors. Moments and Distributions 168 Problems 177 6 Moment Generating Function. Characteristic Function 181 6.1 Sums of Random Variables. Convolutions 181 6.2 Generating Functions and Applications 182 6.3 Moment Generating Function 188 6.4 Characteristic Function 192 6.5 Inversion and Continuity Theorems 199 6.6 Stable Distributions. Lvy Distribution 204 6.6.1 Truncated Lévy flight distribution 206 Problems 208 7 Limit Theorems 213 7.1 Types of Convergence 213 7.1.1 Traditional deterministic convergence types 214 7.1.2 Convergence in Lp 215 7.1.3 Almost sure (a.s.) convergence 216 7.1.4 Convergence in probability. Convergence in distribution 217 7.2 Relationships between Types of Convergence 221 7.2.1 A.S. and Lp 221 7.2.2 Probability, a.s., Lp convergence 223 7.2.3 Uniform Integrability 226 7.2.4 Weak convergence and all the others 228 7.3 Continuous Mapping Theorem. Joint Convergence. Slutsky’s Theorem 230 7.4 The Two Big Limit Theorems: LLN and CLT 232 7.4.1 A note on statistics 232 7.4.2 The order statistics 234 7.4.3 Limit theorems for the mean statistics 238 7.5 Extensions of CLT 245 7.6 Exchanging the Order of Limits and Expectations 251 Problems 252 8 Statistical Inference 259 8.1 The Classical Problems in Statistics 259 8.2 Parameter Estimation Problem 260 8.2.1 The case of the normal distribution, estimating mean when variance is unknown 262 8.2.2 The case of the normal distribution, comparing variances 264 8.3 Maximum Likelihood Estimation Method 265 8.3.1 The bisection method 267 8.4 The Method of Moments 276 8.5 Testing, the Likelihood Ratio Test 277 8.5.1 The likelihood ratio test 280 8.6 Confidence Sets 284 Problems 286 Part II Stochastic Processes 9 Introduction to Stochastic Processes 293 9.1 General Characteristics of Stochastic Processes 294 9.1.1 The index set I 294 9.1.2 The state space S 294 9.1.3 Adaptiveness, filtration, standard filtration 294 9.1.4 Pathwise realizations 296 9.1.5 The finite distribution of stochastic processes 296 9.1.6 Independent components 297 9.1.7 Stationary process 298 9.1.8 Stationary and independent increments 299 9.1.9 Other properties that characterize specific classes of stochastic processes 300 9.2 A Simple Process – The Bernoulli Process 301 Problems 304 10 The Poisson Process 307 10.1 Definitions 307 10.2 Inter-Arrival and Waiting Time for a Poisson Process 310 10.2.1 Proving that the inter-arrival times are independent 311 10.2.2 Memoryless property of the exponential distribution 315 10.2.3 Merging two independent Poisson processes 316 10.2.4 Splitting the events of the Poisson process into types 316 10.3 General Poisson Processes 317 10.3.1 Nonhomogenous Poisson process 318 10.3.2 The compound Poisson process 319 10.4 Simulation techniques. Constructing Poisson Processes 323 10.4.1 One-dimensional simple Poisson process 323 Problems 326 11 Renewal Processes 331 11.0.2 The renewal function 333 11.1 Limit Theorems for the Renewal Process 334 11.1.1 Auxiliary but very important results. Wald’s theorem. Discrete stopping time 336 11.1.2 An alternative proof of the elementary renewal theorem 340 11.2 Discrete Renewal Theory 344 11.3 The Key Renewal Theorem 349 11.4 Applications of the Renewal Theorems 350 11.5 Special cases of renewal processes 352 11.5.1 The alternating renewal process 353 11.5.2 Renewal reward process 358 11.6 The renewal Equation 359 11.7 Age-Dependent Branching processes 363 Problems 366 12 Markov Chains 371 12.1 Basic Concepts for Markov Chains 371 12.1.1 Definition 371 12.1.2 Examples of Markov chains 372 12.1.3 The Chapman– Kolmogorov equation 378 12.1.4 Communicating classes and class properties 379 12.1.5 Periodicity 379 12.1.6 Recurrence property 380 12.1.7 Types of recurrence 382 12.2 Simple Random Walk on Integers in d Dimensions 383 12.3 Limit Theorems 386 12.4 States in a MC. Stationary Distribution 387 12.4.1 Examples. Calculating stationary distribution 391 12.5 Other Issues: Graphs, First-Step Analysis 394 12.5.1 First-step analysis 394 12.5.2 Markov chains and graphs 395 12.6 A general Treatment of the Markov Chains 396 12.6.1 Time of absorption 399 12.6.2 An example 400 Problems 406 13 Semi-Markov and Continuous-time Markov Processes 411 13.1 Characterization Theorems for the General semi- Markov Process 413 13.2 Continuous-Time Markov Processes 417 13.3 The Kolmogorov Differential Equations 420 13.4 Calculating Transition Probabilities for a Markov Process General Approach 425 13.5 Limiting Probabilities for the Continuous-Time Markov Chain 426 13.6 Reversible Markov Process 429 Problems 432 14 Martingales 437 14.1 Definition and Examples 438 14.1.1 Examples of martingales 439 14.2 Martingales and Markov Chains 440 14.2.1 Martingales induced by Markov chains 440 14.3 Previsible Process. The Martingale Transform 442 14.4 Stopping Time. Stopped Process 444 14.4.1 Properties of stopping time 446 14.5 Classical Examples of Martingale Reasoning 449 14.5.1 The expected number of tosses until a binary pattern occurs 449 14.5.2 Expected number of attempts until a general pattern occurs 451 14.5.3 Gambler’s ruin probability – revisited 452 14.6 Convergence Theorems. L1 Convergence. Bounded Martingales in L2 456 Problems 458 15 Brownian Motion 465 15.1 History 465 15.2 Definition 467 15.2.1 Brownian motion as a Gaussian process 469 15.3 Properties of Brownian Motion 471 15.3.1 Hitting times. Reflection principle. Maximum value 474 15.3.2 Quadratic variation 476 15.4 Simulating Brownian Motions 480 15.4.1 Generating a Brownian motion path 480 15.4.2 Estimating parameters for a Brownian motion with drift 481 Problems 481 16 Stochastic Differential Equations 485 16.1 The Construction of the Stochastic Integral 487 16.1.1 Itȏ integral construction 490 16.1.2 An illustrative example 492 16.2 Properties of the Stochastic Integral 494 16.3 Itȏ lemma 495 16.4 Stochastic Differential Equations (SDEs) 499 16.4.1 A discussion of the types of solution for an SDE 501 16.5 Examples of SDEs 502 16.5.1 An analysis of Cox– Ingersoll– Ross (CIR) type models 507 16.5.2 Models similar to CIR 507 16.5.3 Moments calculation for the CIR model 509 16.5.4 Interpretation of the formulas for moments 511 16.5.5 Parameter estimation for the CIR model 511 16.6 Linear Systems of SDEs 513 16.7 A Simple Relationship between SDEs and Partial Differential Equations (PDEs) 515 16.8 Monte Carlo Simulations of SDEs 517 Problems 522 A Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations 527 A.1 Solving difference equations with constant coefficients 528 A.2 Generalized matrix inverse and pseudo-determinant 528 A.3 Connection between systems of differential equations and matrices 529 A.3.1 Writing a system of differential equations in matrix form 530 A.4 Linear Algebra results 533 A.4.1 Eigenvalues, eigenvectors of a square matrix 533 A.4.2 Matrix Exponential Function 534 A.4.3 Relationship between Exponential matrix and Eigenvectors 534 A.5 Finding fundamental solution of the homogeneous system 535 A.5.1 The case when all the eigenvalues are distinct and real 536 A.5.2 The case when some of the eigenvalues are complex 536 A.5.3 The case of repeated real eigenvalues 537 A.6 The nonhomogeneous system 538 A.6.1 The method of undetermined coefficients 538 A.6.2 The method of variation of parameters 539 A.7 Solving systems when P is non-constant 540 Bibliography 541 Index 547

Autorenportrait

Ionut Florescu, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. His areas of research interest include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. He is also the coauthor of Handbook of Probability and coeditor of Handbook of Modeling High-Frequency Data in Finance, both published by Wiley.

Back cover copy

A comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and real-world applications With a sophisticated approach, Probability and Stochastic Processes successfully balances theory and applications in a pedagogical and accessible format. The book’s primary focus is on key theoretical notions in probability to provide a foundation for understanding concepts and examples related to stochastic processes. Organized into two main sections, the book begins by developing probability theory with topical coverage on probability measure; random variables; integration theory; product spaces, conditional distribution, and conditional expectations; and limit theorems. The second part explores stochastic processes and related concepts including the Poisson process, renewal processes, Markov chains, semi-Markov processes, martingales, and Brownian motion. Featuring a logical combination of traditional and complex theories as well as practices, Probability and Stochastic Processes also includes: <ul> <li>Multiple examples from disciplines such as business, mathematical finance, and engineering</li> </ul> <ul> <li>Chapter-by-chapter exercises and examples to allow readers to test their comprehension of the presented material</li> </ul> <ul> <li>A rigorous treatment of all probability and stochastic processes concepts</li> </ul> An appropriate textbook for probability and stochastic processes courses at the upper-undergraduate and graduate level in mathematics, business, and electrical engineering, Probability and Stochastic Processes is also an ideal reference for researchers and practitioners in the fields of mathematics, engineering, and finance. Ionut Florescu, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. His areas of research interest include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. He is also the coauthor of Handbook of Probability and coeditor of Handbook of Modeling High-Frequency Data in Finance, both published by Wiley.