Probability and Stochastic Processes
[BOOK DESCRIPTION]
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: * Multiple examples from disciplines such as business, mathematical finance, and engineering * Chapter-by-chapter exercises and examples to allow readers to test their comprehension of the presented material * A rigorous treatment of all probability and stochastic processes concepts 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.
[TABLE OF CONTENTS]
List of Figures xvii
List of Tables xx
Preface xxi
Acknowledgments xxiii
Introduction 1 (8)
PART I PROBABILITY
1 Elements of Probability Measure 9 (36)
1.1 Probability Spaces 10 (12)
1.1.1 Null element of F. Almost sure 21 (1)
(a.s.) statements. Indicator of a set
1.2 Conditional Probability 22 (7)
1.3 Independence 29 (2)
1.4 Monotone Convergence Properties of 31 (6)
Probability
1.5 Lebesgue Measure on the Unit Interval 37 (8)
(0,1]
Problems 40 (5)
2 Random Variables 45 (42)
2.1 Discrete and Continuous Random 48 (4)
Variables
2.2 Examples of Commonly Encountered 52 (13)
Random Variables
2.3 Existence of Random Variables with 65 (3)
Prescribed Distribution
2.4 Independence 68 (4)
2.5 Functions of Random Variables. 72 (15)
Calculating Distributions
Problems 82 (5)
3 Applied Chapter: Generating Random 87 (36)
Variables
3.1 Generating One-Dimensional Random 88 (3)
Variables by Inverting the cdf
3.2 Generating One-Dimensional Normal 91 (3)
Random Variables
3.3 Generating Random Variables. 94 (15)
Rejection Sampling Method
3.4 Generating Random Variables. 109 (14)
Importance Sampling
Problems 119 (4)
4 Integration Theory 123 (34)
4.1 Integral of Measurable Functions 124 (6)
4.2 Expectations 130 (13)
4.3 Moments of a Random Variable. 143 (2)
Variance and the Correlation Coefficient
4.4 Functions of Random Variables. The 145 (3)
Transport Formula
4.5 Applications. Exercises in 148 (2)
Probability Reasoning
4.6 A Basic Central Limit Theorem: The 150 (7)
DeMoivre--Laplace Theorem
Problems 152 (5)
5 Conditional Distribution and Conditional 157 (24)
Expectation
5.1 Product Spaces 158 (4)
5.2 Conditional Distribution and 162 (3)
Expectation. Calculation in Simple Cases
5.3 Conditional Expectation. General 165 (3)
Definition
5.4 Random Vectors. Moments and 168 (13)
Distributions
Problems 177 (4)
6 Moment Generating Function. 181 (32)
Characteristic Function
6.1 Sums of Random Variables. Convolutions 181 (1)
6.2 Generating Functions and Applications 182 (6)
6.3 Moment Generating Function 188 (4)
6.4 Characteristic Function 192 (7)
6.5 Inversion and Continuity Theorems 199 (5)
6.6 Stable Distributions. Levy 204 (9)
Distribution
6.6.1 Truncated Levy flight distribution 206 (2)
Problems 208 (5)
7 Limit Theorems 213 (46)
7.1 Types of Convergence 213 (8)
7.1.1 Traditional deterministic 214 (1)
convergence types
7.1.2 Convergence in LP 215 (1)
7.1.3 Almost sure (a.s.) convergence 216 (1)
7.1.4 Convergence in probability. 217 (4)
Convergence in distribution
7.2 Relationships between Types of 221 (9)
Convergence
7.2.1 A.S. and LP 221 (2)
7.2.2 Probability, a.s., LP convergence 223 (3)
7.2.3 Uniform Integrability 226 (2)
7.2.4 Weak convergence and all the 228 (2)
others
7.3 Continuous Mapping Theorem. Joint 230 (2)
Convergence. Slutsky's Theorem
7.4 The Two Big Limit Theorems: LLN and 232 (13)
CLT
7.4.1 A note on statistics 232 (2)
7.4.2 The order statistics 234 (4)
7.4.3 Limit theorems for the mean 238 (7)
statistics
7.5 Extensions of CLT 245 (6)
7.6 Exchanging the Order of Limits and 251 (8)
Expectations
Problems 252 (7)
8 Statistical Inference 259 (34)
8.1 The Classical Problems in Statistics 259 (1)
8.2 Parameter Estimation Problem 260 (5)
8.2.1 The case of the normal 262 (2)
distribution, estimating mean when
variance is unknown
8.2.2 The case of the normal 264 (1)
distribution, comparing variances
8.3 Maximum Likelihood Estimation Method 265 (11)
8.3.1 The bisection method 267 (9)
8.4 The Method of Moments 276 (1)
8.5 Testing, the Likelihood Ratio Test 277 (7)
8.5.1 The likelihood ratio test 280 (4)
8.6 Confidence Sets 284 (9)
Problems 286 (7)
PART II STOCHASTIC PROCESSES
9 Introduction to Stochastic Processes 293 (14)
9.1 General Characteristics of Stochastic 294 (7)
Processes
9.1.1 The index set I 294 (1)
9.1.2 The state space S 294 (1)
9.1.3 Adaptiveness, nitration, standard 294 (2)
filtration
9.1.4 Pathwise realizations 296 (1)
9.1.5 The finite distribution of 296 (1)
stochastic processes
9.1.6 Independent components 297 (1)
9.1.7 Stationary process 298 (1)
9.1.8 Stationary and independent 299 (1)
increments
9.1.9 Other properties that 300 (1)
characterize specific classes of
stochastic processes
9.2 A Simple Process -- The Bernoulli 301 (6)
Process
Problems 304 (3)
10 The Poisson Process 307 (24)
10.1 Definitions 307 (3)
10.2 Inter-Arrival and Waiting Time for a 310 (7)
Poisson Process
10.2.1 Proving that the inter-arrival 311 (4)
times are independent
10.2.2 Memoryless property of the 315 (1)
exponential distribution
10.2.3 Merging two independent Poisson 316 (1)
processes
10.2.4 Splitting the events of the 316 (1)
Poisson process into types
10.3 General Poisson Processes 317 (6)
10.3.1 Nonhomogenous Poisson process 318 (1)
10.3.2 The compound Poisson process 319 (4)
10.4 Simulation techniques. Constructing 323 (8)
Poisson Processes
10.4.1 One-dimensional simple Poisson 323 (3)
process
Problems 326 (5)
11 Renewal Processes 331 (40)
11.0.2 The renewal function 333 (1)
11.1 Limit Theorems for the Renewal 334 (10)
Process
11.1.1 Auxiliary but very important 336 (4)
results. Wald's theorem. Discrete
stopping time
11.1.2 An alternative proof of the 340 (4)
elementary renewal theorem
11.2 Discrete Renewal Theory 344 (5)
11.3 The Key Renewal Theorem 349 (1)
11.4 Applications of the Renewal Theorems 350 (2)
11.5 Special cases of renewal processes 352 (7)
11.5.1 The alternating renewal process 353 (5)
11.5.2 Renewal reward process 358 (1)
11.6 The renewal Equation 359 (4)
11.7 Age-Dependent Branching processes 363 (8)
Problems 366 (5)
12 Markov Chains 371 (40)
12.1 Basic Concepts for Markov Chains 371 (12)
12.1.1 Definition 371 (1)
12.1.2 Examples of Markov chains 372 (6)
12.1.3 The Chapman--Kolmogorov equation 378 (1)
12.1.4 Communicating classes and class 379 (1)
properties
12.1.5 Periodicity 379 (1)
12.1.6 Recurrence property 380 (2)
12.1.7 Types of recurrence 382 (1)
12.2 Simple Random Walk on Integers in D 383 (3)
Dimensions
12.3 Limit Theorems 386 (1)
12.4 States in a MC. Stationary 387 (7)
Distribution
12.4.1 Examples. Calculating stationary 391 (3)
distribution
12.5 Other Issues: Graphs, First-Step 394 (2)
Analysis
12.5.1 First-step analysis 394 (1)
12.5.2 Markov chains and graphs 395 (1)
12.6 A general Treatment of the Markov 396 (15)
Chains
12.6.1 Time of absorption 399 (1)
12.6.2 An example 400 (6)
Problems 406 (5)
13 Semi-Markov and Continuous-time Markov 411 (26)
Processes
13.1 Characterization Theorems for the 413 (4)
General semi- Markov Process
13.2 Continuous-Time Markov Processes 417 (3)
13.3 The Kolmogorov Differential Equations 420 (5)
13.4 Calculating Transition Probabilities 425 (1)
for a Markov Process. General Approach
13.5 Limiting Probabilities for the 426 (3)
Continuous-Time Markov Chain
13.6 Reversible Markov Process 429 (8)
Problems 432 (5)
14 Martingales 437 (28)
14.1 Definition and Examples 438 (2)
14.1.1 Examples of martingales 439 (1)
14.2 Martingales and Markov Chains 440 (2)
14.2.1 Martingales induced by Markov 440 (2)
chains
14.3 Previsible Process. The Martingale 442 (2)
Transform
14.4 Stopping Time. Stopped Process 444 (5)
14.4.1 Properties of stopping time 446 (3)
14.5 Classical Examples of Martingale 449 (7)
Reasoning
14.5.1 The expected number of tosses 449 (2)
until a binary pattern occurs
14.5.2 Expected number of attempts 451 (1)
until a general pattern occurs
14.5.3 Gambler's ruin probability -- 452 (4)
revisited
14.6 Convergence Theorems. L1 456 (9)
Convergence. Bounded Martingales in L2
Problems 458 (7)
15 Brownian Motion 465 (20)
15.1 History 465 (2)
15.2 Definition 467 (4)
15.2.1 Brownian motion as a Gaussian 469 (2)
process
15.3 Properties of Brownian Motion 471 (9)
15.3.1 Hitting times. Reflection 474 (2)
principle. Maximum value
15.3.2 Quadratic variation 476 (4)
15.4 Simulating Brownian Motions 480 (5)
15.4.1 Generating a Brownian motion path 480 (1)
15.4.2 Estimating parameters for a 481 (1)
Brownian motion with drift
Problems 481 (4)
16 Stochastic Differential Equations 485 (42)
16.1 The Construction of the Stochastic 487 (7)
Integral
16.1.1 Ito integral construction 490 (2)
16.1.2 An illustrative example 492 (2)
16.2 Properties of the Stochastic Integral 494 (1)
16.3 Ito lemma 495 (4)
16.4 Stochastic Differential Equations 499 (3)
(SDEs)
16.4.1 A discussion of the types of 501 (1)
solution for an SDE
16.5 Examples of SDEs 502 (11)
16.5.1 An analysis of Cox--Ingersoll-- 507 (1)
Ross (CIR) type models
16.5.2 Models similar to CIR 507 (2)
16.5.3 Moments calculation for the CIR 509 (2)
model
16.5.4 Interpretation of the formulas 511 (1)
for moments
16.5.5 Parameter estimation for the CIR 511 (2)
model
16.6 Linear Systems of SDEs 513 (2)
16.7 A Simple Relationship between SDEs 515 (2)
and Partial Differential Equations (PDEs)
16.8 Monte Carlo Simulations of SDEs 517 (10)
Problems 522 (5)
A Appendix: Linear Algebra and Solving 527 (14)
Difference Equations and Systems of
Differential Equations
A.1 Solving difference equations with 528 (1)
constant coefficients
A.2 Generalized matrix inverse and 528 (1)
pseudo-determinant
A.3 Connection between systems of 529 (4)
differential equations and matrices
A.3.1 Writing a system of differential 530 (3)
equations in matrix form
A.4 Linear Algebra results 533 (2)
A.4.1 Eigenvalues, eigenvectors of a 533 (1)
square matrix
A.4.2 Matrix Exponential Function 534 (1)
A.4.3 Relationship between Exponential 534 (1)
matrix and Eigenvectors
A.5 Finding fundamental solution of the 535 (3)
homogeneous system
A.5.1 The case when all the eigenvalues 536 (1)
are distinct and real
A.5.2 The case when some of the 536 (1)
eigenvalues are complex
A.5.3 The case of repeated real 537 (1)
eigenvalues
A.6 The nonhomogeneous system 538 (2)
A.6.1 The method of undetermined 538 (1)
coefficients
A.6.2 The method of variation of 539 (1)
parameters
A.7 Solving systems when P is non-constant 540 (1)
Bibliography 541 (6)
Index 547
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