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Index

Lq(n, :F, P) space, 34 a-field, 6, 116

Borel, 7 generated by random process, 107 generated by random variable,

22

Backward Kolmogorov equation, 490, 494

Bochner's theorem, 132 Borel-Cantelli Lemma, 21 Brownian motion, 107, 256

in the first orthant of JR2 , 387 properties, 186 reflected at two thresholds, 383 reflected at zero, 372

local time, 372 Tanaka formula, 373

Cadlag, caglad, 113 Cauchy distribution, 46 Chapman-Kolmogorov equation, 121 Compound Poisson process, 111, 182 Conditional expectation, 82

change of fields, 89 conditional probability with re­

spect to a a-field, 91 defining relation, 84 properties, 89

Conditional probability, 16 Convergence of random variables, 70

Lp, 70 a.s., 70 distribution, 70 mean square, 70 probability, 70

771

Decomposition method, 563 Differential equations for

characteristic function, 455, 478, 510

density, 481 backward Kolmogorov, 490 Fokker-Planck equation, 481

moments, 433, 437, 452, 463, 475,494,508

Diffusion coefficient, 254 Diffusion process, 254, 258, 262, 267,

432,475 Doob decomposition, 95 Doob-Meyer decomposition, 175 Drift coefficient, 254 Dynkin formula, 350

Earthquake engineering and seismol-ogy, 741

cellular automata model, 743 fragility surface, 528 soil liquefaction, 663

Eigenvalue problem, 370, 413, 421, 566,599

Equivalent linearization method, 564 Expectation operator, 28, 33, 143 Extremes of stochastic processes, 165

first passage time, 168, 527 mean crossing rate, 165

Fatou's lemma, 21 Feynman-Kac functional, 364, 365 Filtration, 78, 107 Finance, 534

Black-Scholes formula, 536 stock price model, 535

772

Finite difference method, 411, 488, 531,533,600,700

Finite dimensional distribution, 117 Finite element method, 600, 652, 668,

701 First passage time, 168, 527 Fokk:er-Planckequation, 269,481,494,

531 Fubini theorem, 39,99

Girsanov's theorem, 337, 338

Independence a-fields, 36 events, 36 random variables, 38

Independentincrements, 122, 182, 186, 189

Inequalities Cauchy-Schwarz, 69 Chebyshev, 68 Doob, 98 Holder, 69 Jensen, 68 Minkowski, 69

Infinitely divisible characteristic func-tion, 52

a-stable, 57 canonical representation, 55 construction, 54 Levy-Khinchine representation,

56 properties, 53

Integrals of random variables Fatou lemma, 32 Lebesgue theorem, 33 properties,30

Ito formula, 237 multi-dimensional case, 247 one-dimensional case, 238

Iteration method, 565, 574, 604

Karhunen-Loeve representation, 161 Kolmogorov criterion, 110

Levy measure, 56 Levy process, 189

Index

Levy decomposition, 193, 197 Levy-Khinchine formula, 197

Levy-Khinchine representation, 56 Liouville equation, 687, 714, 730 Lipschitz condition, 258 Local solution

algebraic and integral equations homogeneous,413,421 inhomogeneous,407,418

differential equation boundary walk method, 403 Feynman-Kac functional, 364 random walk method, 345, 550 Schrodingerequation,367,370 spherical process method, 394

Lyapunov exponent modelocalization,666 noise induced transitions, 720 stochastic stability, 655

Markov property, 81 Martingale, 94

Doob decomposition, 95 Doob inequality, 98 Doob-Meyerdecomposition, 175 Jensen inequality, 176 stopped, 96, 175 submartingale, 92, 94, 169 supermartingale, 92, 94, 169 variation and covariation, 179

Martingales, 92, 169 Materials science, 518

effective properties, 605 evolution, 633 Reuss average, 623 Voigt average, 621

Measurable functions, 21 Measurable space, 6 Mixture of translation process, 126 Modeling

non-Gaussian input, 515 partially known input, 745

Index

unknown input and system, 7 48 Monte Carlo simulation

applications, 507,527,528,531, 533, 552, 554, 567, 581, 584,614,677,736

improved measure change, 334 time change, 330

non-Gaussian process and field memory less transformations,

316 transformation with memory,

320 non-stationary Gaussian process

and field Fourier series, 312,315 linear differential equations,

310 point processes, 325 random variable, 288 stationary Gaussian process and

field, 293 sampling theorem, 304, 309 spectral representation, 293,

299

Neumann series method, 561, 591, 630,693

Omstein-Uhlenbeckprocess, 257,263, 314,340

Path integral solution, 494 Perturbation method, 499, 558, 572,

587,603,629,692 Poisson process, 182, 326 Probability measure, 8 Probability space, 5 Product probability space, 13

Quadratic variation and covariation, 179,228

integration by parts, 229 Kunita-Watanabe inequality, 235 polarization identity, 229

Radon-Nikodym derivative, 41 Random field, 104, 158 Random variable, 22

?-integrable, 29 arbitrary, 29 characteristic function, 47 density function, 45 distribution function, 43 finite-valued simple, 28 positive, 29

Random variables, 42 Random vector, 22

characteristic function, 62 Gaussian vector, 65 independence,61

773

joint density function, 59 joint distribution function, 59 moments, 64 second moment properties, 65

Random vectors, 58 Random walk, 7 5 Reliability, 522 Reuss average, 623 Riemann-Stieltjes integral, 206

Sample space, 5 SchrOdingerequation, 346,367,370 Second moment calculus for processes

in L2, 139 expectation and mean square in­

tegrals, 152 Karhunen-Loeve representation,

161 mean square continuity, 111, 141 mean square differentiation, 14 2 mean square integration, 145 spectral representation, 153 variation functions, 146

Second moment properties, 65, 138, 159

Semimartingale, 324 Sequence of events, 20 State augmentation, 460, 512, 682

774

Stationary increments, 111, 122, 182, 186, 189

Stationary process, 119 in the strict sense, 119 in the weak sense, 128 spectral density, 132

Stochastic differential equation, 253 numerical solution, 275

Euler, 277 Milstein, 280

semimartingale input, 271 Brownian motion input, 256

diffusion process, 262 equations for characteristic func-

tions, 267 equations for densities, 267 equations for moments, 267 semimartingale, 263 Wong-Zakai theorem, 267

Stochastic integral, 208 associativity, 225 Ito integral, 208, 250 preservation, 225 semimartingale, 217 simple predictable integrand, 221 Stratonovich integral, 210, 249,

250 Stochastic process, 104

adapted, 107 dtdlag, caglact, 113 classes of, 119 correlation, 127, 130 covariance, 127 finite dimensional densities, 117 finite dimensional distribution,

117 measurable, 106 progressively measurable, 108 sample properties, 110 second moment properties, 127

Stopping time, 78, 114 Stratonovich integral, 249

Index

Taylor series method, 554, 570, 584, 691

Translation process, 125

Voigt average, 621

White noise process, 144, 184, 186, 254,322,323

Wong-Zakai theorem, 267