Probability and Stochastic Processes
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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 It¿ integral construction 490
16.1.2 An illustrative example 492
16.2 Properties of the Stochastic Integral 494
16.3 It¿ 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
