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| Shiryaev A., Peskir G. — Optimal Stopping and Free-Boundary Problems |
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| Предметный указатель |
A posteriori probability process 288 309 335 357
Adapted process 54
Admissible function 207
American option 375
Angle-bracket process 59
Appell polynomial 24
Arbitrage-free price 375 379 395
Asian option 416
Average delay 356
Average number of visits 81
Backward equation 95
Backward Kolmogorov equation 90
Bayesian problem xiii
Bellman's principle 6
Bessel inequalities 251
Blumenthal's 0-1 law 230
Blumenthal's 0-1 law for Brownian motion 97
Bouleau — Yor formula 68
Brownian motion 93 94
Burkholder — Davis — Gundy inequalities 63 284
Cadlag function 54
Cag function 55
Canonical representation 105
Canonical representation for semimartingales 69
Cauchy distribution 105
Cauchy problem 135 137
Cauchy problem, killed 136—138
Cauchy — Euler equation 376 397
Change of measure 115
Change of scale 194
Change of space 111 193
Change of time 106 109
Change of variables 195
Change-of-variable formula 74
Chapman — Kolmogorov equations 79 108 113
Chapman — Kolmogorov equations of Volterra type 221
Characteristic operator 101 128
Compensator 56
Compound Poisson process 104
Concave conjugate 248
Condition of linear growth 73
Condition of normal reflectione xix
Condition of smooth fit xix
Continuation set xvii 35
Continuous fit 49 144
Cost function 200
creation 119
Cumulant 103
Cumulant function 70
Dambis — Dubins — Schwarz theorem 110
Differential characteristics 88
Differential equation, normal form 211
Diffusion 101
Diffusion coefficient 88 199
Diffusion process 72 101
Diffusion process with jumps 72
Diffusions with angles 155
Dimension of problem 126
Dirichlet class 56
Dirichlet problem 84 130
Dirichlet problem for the Poisson equation 85
Dirichlet problem, inhomogeneous 86
Dirichlet/Poisson problem 132
Discounted problem 127
Discounting 119
Discounting rate 102 127 215
Doob convergence theorem 61
Doob inequalities 255 269
Doob inequalities in mean 62
Doob inequalities in probability 62
Doob inequalities, expected waiting time 263
Doob stopping time theorem 60
Doob type bounds 269
Doob — Meyer decomposition 56
Drift coefficient 88 199
Dynamic programming 6
Early exercise premium representation 385 403 411 420
Ellipticity condition 102
Esscher measure 119
Essential supremum 7
Euclidean velocity 187
Excessive function 83
Feynman — Kac formula 137 138
Filtered probability space 54
Filtration 53
Finite horizon 125 146
Finite horizon formulation 36
First boundary problem 84
First-passage equation 221
Fixed-point theorem for contractive mappings 237
Foellmer — Protter — Shiryaev formula 68
Forward equation 95
Forward Kolmogorov equation 90
Free-boundary equation 219 221 393
Free-boundary problem 48 143
Gain function 35 203
Generalized Markov property 78
Generating operator 82
Girsanov theorem for local martingales 117
Green function 81 200
Hardy — Littlewood inequalities 272
Harmonic function 83
Hermite polynomial 193
Hunt stopping time theorem 60
Inequality of L log L type 283
Infinite horizon 125 144
Infinite horizon formulation 36
Infinitesimal generator 129
Infinitesimal operator 101 129
information 53
Initial distribution 76
Innovation process 344
Instantaneous stopping 264
Integral process 124
Integral representation of the maximum process 447
Invariant function 83
Inverse problem 240
Iterative method 19
Iterative procedure 48
Ito formula 67
Ito — Clark representation theorem 442
Ito — Levy representation 70 106
Ito — Tanaka formula 67
Ito — Tanaka — Meyer formula 67
Khintchine inequalities 62
Killed problem 127
Killing 119
Killing coefficient 102
Killing rate 127
Kolmogorov backward equation 139
Kolmogorov backward equation, semigroup formulation 140
Kolmogorov inequalities 61
Kolmogorov test 230
Kolmogorov — Chapman equations 79 108 113
Kolmogorov — Levy — Khintchine formula 103
Kolmogorov — Levy — Khintchine formula, semimartingale analogue 72
Kummer confluent, hypergeometric function 192
Lagrange functional 132
Laplacian 86
Law of the iterated logarithm at infinity 97
Law of the iterated logarithm at zero 97
Levy characterization theorem 94
Levy convergence theorem 61
Levy distributional theorem 96
Levy measure 69
Levy process 102
Levy — Khintchine representation 104
Levy — Smirnov distribution 105
Likelihood ratio process 288 309 336
Linear problem 196
| Linear programming 49
Linear programming, dual problem 50
Linear programming, primal problem 50
Local Lipschitz condition 73
Local martingale 55
Local martingale, first decomposition 58
Local martingale, purely discontinuous 58
Local martingale, second decomposition 58
Local submartingale 55
Local supermartingale 55
Local time 67
Local time on curve 74
Local time on surfaces 75
Local time-space formula 74
Localized class 55
Localizing sequence 55
LOWER function 230
Markov chain 76
Markov chain in a wide sense 76
Markov chain, time-homogeneous 76
Markov kernel 76
Markov process 76 88
Markov property 91
Markov property in a strict sense 76
Markov property in a wide sense 76
Markov property, generalized 78
Markov property, strong 79
Markov sequence 76
Markov time 1 27
Markov time, finite 54
Markovian cost problem 217
Martingale 53 55
Martingale convergence theorem 61
Martingale maximal inequalities 61
Martingale, basic definitions 53
Martingale, fundamental theorems 60
Master equation 227 228
Maximal equality xi
Maximal inequality xii
Maximal inequality for geometric Brownian motion 271
Maximality principle 207
Maximum process 395
Mayer functional 130
Method of backward induction 3
Method of essential supremum 6
Method of measure change 197
Method of space change 193
Method of time change 165
MLS formulation 124
MLS functional 128 135
Neumann problem 134 135
Newton potential 81
Nonlinear integral equation 219
Nonlinear problem 196
Normal distribution 105
Normal reflection xix 264
Novikov condition 197
Number of visits 81
Obstacle problem 146
Occupation times formula 69
Optimal prediction problem 437
Optimal prediction problem, ultimate integral 438
Optimal prediction problem, ultimate maximum 441
Optimal prediction problem, ultimate position 437
Optimal stopping boundary 207
Optimal stopping of maximum process 199
Optimal stopping problem 2
Optimal stopping time 2
Optimal stopping, continuous time 26
Optimal stopping, discrete time 1
Optimal stopping, Markovian approach 12 34
Optimal stopping, martingale approach 1 26
Optional  -algebra 57
Optional process 57
Optional sampling theorem 60
Orthogonality of local martingales 58
Parabolic cylinder function 192
Parabolic differential equation, backward 88
Parabolic differential equation, forward 89
Perpetual option 395
Picard method 271
PIDE problem 128
Poisson disorder problem 356
Poisson equation 81 82 85
Potential measure 81
Potential of a function 81
Potential of a Markov chain 80
Potential of an operator 80
Potential theory 79
Predictable -algebra 55
Predictable process 56
Predictable quadratic covariation 59
Predictable quadratic variation 59
Principle of continuous fit 153
Principle of smooth fit 149
Probability of a false alarm 356
Probability of an error of the first kind 335
Probability of an error of the second kind 335
Probability-statistical space 287
Process of bounded variation 55
Progressive measurability 58
Quadratic characteristic 59 65
Quadratic covariation 66
Quadratic variation 65
Quickest detection of Poisson process 355
Quickest detection of Wiener process 308
Quickest detection problem for Poisson process 356
Quickest detection problem for Wiener process 308
Radon — Nikodym derivative 288
Random element 54
Reflection principle 229
Reflection principle for Brownian motion 96
Regular boundary 129
Regular diffusion process 150 156
Regular point 152 156
Russian option 395
S-concave function 157
Scale function 114 200
Scaling property 227
Self-similarity property 95 104
Semimartingale 55
Semimartingale, special 59
Sequential analysis xiii
Sequential testing of a Poisson process 334
Sequential testing of Wiener process 287
Shift operator 77
Smallest supermartingale 9 14
Smooth fit 49 144 160
Smooth fit through scale 158
Smooth-fit condition xix
Snell envelope 8 28
Solution-measure 73
Solution-process 73
Space change 193
Speed measure 107 200
Squared Bessel process 188
State space 76
Statistical experiment 287
Stefan free-boundary problem 147
Stochastic basis 53
Stochastic differential equation 72
Stochastic differential equation of “bang-bang” type 454
Stochastic exponential 72 103
Stochastic integral 63
Stochastic process with independent increments 69
Stochastic process, adapted to a filtration 54
Stochastic process, Markov in a strict sense 91
Stochastic process, Markov in a wide sense 91
Stochastic process, progressively measurable 58
Stochastic process, with stationary independent increments 69
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