Contents......Page 8
Figures......Page 16
Tables......Page 28
Preface......Page 30
Acronyms......Page 34
General Notations......Page 36
1.1 Introduction......Page 40
1.2 Probability Space and Basic Definitions......Page 44
1.3 Probability as a Measure......Page 46
1.3.1 Carathéodory’s extension theorem......Page 49
1.3.2 Uniqueness criterion for measures......Page 55
1.4.1 Some properties of random variables......Page 57
1.5 Random Variables and Induced Probability Measures......Page 60
1.6 Probability Distribution and Density Function of a RandomVariable......Page 61
1.6.1 Probability distribution function......Page 62
1.6.2 Lebesgue−Stieltjes measure......Page 64
1.6.4 Radon−Nikodyn theorem......Page 65
1.7 Vector-valued Random Variables and Joint Probability Distributions......Page 67
1.7.2 Marginal probability distributions and density functions......Page 68
1.8 Integration of Measurable Functions and Expectation of a Random Variable......Page 69
1.8.1 Integration with respect to product measure and Fubini’s theorem......Page 70
1.8.3 Expectation of a random variable......Page 72
1.8.4 Higher order expectations of a random variable......Page 73
1.9.1 Independence of events......Page 76
1.9.4 Independence of random variables......Page 77
1.9.5 Independence in terms of CDFs......Page 78
1.9.7 Independence and expectation of random variables......Page 79
1.10 Some oft-used Probability Distributions......Page 80
1.10.2 Poisson distribution......Page 81
1.10.3 Normal distribution......Page 82
1.10.4 Uniform distribution......Page 88
1.10.5 Rayleigh distribution......Page 89
1.11 Transformation of Random Variables......Page 90
1.11.1 Transformation involving a scalar function of vector random variables......Page 91
1.11.2 Transformation involving vector functions of random variables......Page 93
1.11.3 Diagonalization of covariance matrix and transformation to uncorrelated random variables......Page 94
1.11.4 Nataf transformation......Page 96
1.12 Concluding Remarks......Page 99
Exercises......Page 100
Notations......Page 102
2.1 Introduction......Page 104
2.2 Conditional Probability......Page 107
2.2.1 Conditional expectation......Page 109
2.2.2 Change of measure......Page 112
2.2.3 Generalized Bayes’ formula and conditional probabilities......Page 114
2.2.4 Conditional expectation as the least mean square error estimator......Page 115
2.2.5 Rosenblatt transformation......Page 116
2.3.1 Convergence of a sequence of random variables......Page 120
2.3.2 Law of large numbers......Page 123
2.3.3 Central limit theorem (CLT)......Page 124
2.4 Some Useful Inequalities in Probability Theory......Page 126
2.5.1 Random number generation−−uniformly distributed random variable......Page 130
2.5.2 Simulation for other distributions......Page 132
2.5.3 Simulation of joint random variables−−uncorrelated and correlated......Page 137
2.5.4 Multidimensional integrals by MC simulation methods......Page 143
2.5.5 Rao−Blackwell theorem and a general approach to variance reduction techniques......Page 156
Exercises......Page 159
Notations......Page 162
3.1 Introduction......Page 164
3.2 Stochastic Process and its Finite Dimensional Distributions......Page 168
3.2.1 Continuity of a stochastic process......Page 169
3.2.2 Version/modification of a stochastic process......Page 170
3.3 Stochastic Processes−Measurability and Filtration......Page 171
3.3.2 Some basic stochastic processes......Page 172
3.3.3 Stationary stochastic processes......Page 174
3.3.4 Wiener process/ Brownian motion......Page 175
3.3.5 Formal definition of a Wiener process......Page 177
3.3.6 Other properties of a Wiener process......Page 178
3.4.1 Doob’s decomposition theorem......Page 189
3.4.2 Martingale transform......Page 191
3.4.3 Doob’s upcrossing inequality......Page 193
3.4.4 Martingale convergence theorem......Page 194
3.4.5 Uniform integrability......Page 196
3.5 Stopping Time and Stopped Processes......Page 201
3.5.1 Stopping time......Page 202
3.5.2 Stopped processes......Page 203
3.5.3 Doob’s optional stopping theorem......Page 204
3.5.4 A super-martingale inequality......Page 205
3.5.5 Optional stopping theorem for UI martingales......Page 206
3.6.1 Doob’s and Levy’s martingale theorem......Page 211
3.6.2 Martingale convergence theorem......Page 212
3.6.3 Optional stopping theorem......Page 213
3.7.1 Definition of a local martingale......Page 221
Exercises......Page 222
Notations......Page 225
4.1 Introduction......Page 226
4.2 Stochastic Integral......Page 228
4.2.1 Stochastic integral of a discrete stochastic process......Page 229
4.2.2 Properties of Ito integral of simple adapted processes......Page 232
4.2.3 Ito integral for continuous processes......Page 234
4.3 Ito Processes......Page 237
4.3.1 Larger class of integrands for Ito integral......Page 239
4.4 Stochastic Calculus......Page 240
4.4.1 Integral representation of an SDE......Page 241
4.4.2 Ito’s formula......Page 242
4.4.3 Ito’s formula for higher dimensions......Page 254
4.4.4 Dynamical system of higher dimension and application of Ito’s formula......Page 262
4.5 Spectral Representations of Stochastic Signals......Page 270
4.5.1 Non-stationary process and evolutionary power spectrum......Page 271
4.5.2 Some interesting aspects of evolutionary power spectrum......Page 283
4.6 Existence and Uniqueness of Solutions to SDEs......Page 286
4.6.2 Strong and weak solutions......Page 288
4.6.3 Linear SDEs......Page 289
4.6.4 Markov property of solutions to SDEs......Page 295
4.7.1 Backward Kolmogorov equation......Page 298
4.7.3 Adjoint differential operator and forward Kolmogorov PDE......Page 301
4.7.4 Generator Lt......Page 304
4.8 Solution of PDEs via Corresponding SDEs......Page 306
4.8.1 Solution to elliptic PDEs......Page 309
4.8.2 Exit time distributions from solutions of PDEs......Page 315
4.9 Recurrence and Transience of a Diffusion Process......Page 317
4.10 Girsanov’s Theorem and Change of Measure......Page 318
4.10.1 Girsanov’s theorem......Page 319
4.10.2 Girsanov’s theorem for Brownian motion......Page 320
4.10.3 Girsanov’s theorem—Version 1......Page 321
4.10.4 Girsanov’s theorem—the general version......Page 325
4.11 Martingale Representation Theorem......Page 326
4.11.1 Proof of martingale representation theorem......Page 330
4.12 A Brief Remark on the Martingale Problem......Page 331
4.13 Concluding Remarks......Page 332
Exercises......Page 333
Notations......Page 334
5.1 Introduction......Page 338
5.2 Euler−Maruyama (EM) Method for Solving SDEs......Page 340
5.2.2 Statement of the theorem for global convergence......Page 341
5.3 An Implicit EM Method......Page 354
5.4 Further Issues on Convergence of EM Methods......Page 355
5.5 An introduction to Ito−Taylor Expansion for Stochastic Processes......Page 357
5.6 Derivation of Ito−Taylor Expansion......Page 359
5.6.1 One-step approximations−−explicit integration methods......Page 362
5.7 Implementation Issues of the Numerical Integration Schemes......Page 368
5.7.1 Evaluation of MSIs......Page 369
5.8 Stochastic Implicit Methods and Ito−Taylor Expansion......Page 378
5.8.1 Stochastic Newmark method−a two-parameter implicit scheme for mechanical oscillators......Page 382
5.9 Weak One-step Approximate Solutions of SDEs......Page 390
5.9.1 Statement of the weak convergence theorem......Page 391
5.9.2 Modelling of MSIs and construction of a weak one-step approximation......Page 395
5.9.3 Stochastic Newmark scheme using weak one-step approximation......Page 403
5.10.1 LTL-based schemes......Page 409
5.11 Concluding Remarks......Page 418
Exercises......Page 419
Notations......Page 423
6.1 Introduction......Page 425
6.2 Objective of Stochastic Filtering......Page 428
6.3 Stochastic Filtering and Kushner−Stratanovitch (KS) Equation......Page 429
6.3.1 Zakai equation......Page 431
6.3.2 KS equation......Page 432
6.3.3 Circularity—the problem of moment closure in non-linear filtering problems......Page 434
6.3.4 Unnormalized conditional density and Kushner’s theorem......Page 436
6.4 Non-linear Stochastic Filtering and Solution Strategies......Page 439
6.4.1 Extended Kalman filter (EKF)......Page 440
6.4.2 EKF using locally transversal linearization (LTL)......Page 441
6.4.3 EKF applied to parameter estimation......Page 446
6.5.1 Bootstrap filter......Page 450
6.5.2 Auxiliary bootstrap filter......Page 457
6.5.3 Ensemble Kalman filter (EnKF)......Page 458
6.6 Concluding Remarks......Page 466
Exercises......Page 467
Notations......Page 468
7.2 Iterated Gain-based Stochastic Filter (IGSF)......Page 471
7.2.1 IGSF scheme......Page 472
7.3.1 Gaussian sum approximation and filter bank......Page 477
7.3.2 Filtering strategy......Page 478
7.3.3 Iterative update scheme for IGSF bank......Page 480
7.3.4 Iterative update scheme for IGSF bank with ADP......Page 481
7.4 KS Filters......Page 483
7.4.1 KS filtering scheme......Page 484
7.5 EnKS Filter−−a Variant of KS Filter......Page 490
7.5.1 EnKS filtering scheme......Page 491
7.5.2 EnKS filter−−a non-iterative form......Page 492
7.5.3 EnKS filter−−an iterative form......Page 496
7.6 Concluding Remarks......Page 503
Notations......Page 504
8.1 Introduction......Page 506
8.2 Girsanov Corrected Linearization Method (GCLM)......Page 511
8.2.1 Algorithm for GCLM......Page 516
8.3 Girsanov Corrected Euler−Maruyama (GCEM) Method......Page 530
8.3.1 Additively driven SDEs and the GCEM method......Page 531
8.3.2 Weak correction through a change of measure......Page 532
8.4 Numerical Demonstration of GCEM Method......Page 535
8.5 Concluding Remarks......Page 543
Notations......Page 544
9.1 Introduction......Page 546
9.2 Possible Ineffectiveness of Evolutionary Schemes......Page 565
9.3 Global Optimization by Change of Measure and Martingale Characterization......Page 566
9.4 Local Optimization as a Martingale Problem......Page 567
9.5 The Optimization Scheme−−Algorithmic Aspects......Page 569
9.5.1 Discretization of the extremal equation......Page 572
9.5.2 Pseudo codes......Page 580
9.6 Some Applications of the Pseudo Code 2 to Dynamical Systems......Page 582
9.7 Concluding Remarks......Page 591
Notations......Page 592
10.1 Introduction......Page 595
10.2.1 Improvements to the coalescence strategy......Page 596
10.2.2 Improvements to scrambling and introduction of a relaxation parameter......Page 598
10.2.3 Blending......Page 600
10.3 COMBEO Algorithm......Page 612
10.3.1 Some benchmark problems and solutions by COMBEO......Page 616
10.4.1 State space splitting (3S)......Page 621
10.4.2 Benchmark problems......Page 624
Notations......Page 628
Appendix A (Chapter 1)......Page 630
Appendix B (Chapter 2)......Page 646
Appendix C (Chapter 3)......Page 653
Appendix D (Chapter 4)......Page 659
Appendix E (Chapter 5)......Page 674
Appendix F (Chapter 6)......Page 681
Appendix G (Chapter 7)......Page 684
Appendix H (Chapter 8)......Page 705
Appendix I (Chapter 9)......Page 707
References......Page 712
Bibliography......Page 733
Index......Page 740