Shiryaev A., Peskir G. — Optimal Stopping and Free-Boundary Problems :: Электронная библиотека попечительского совета мехмата МГУ

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Shiryaev A., Peskir G. — Optimal Stopping and Free-Boundary Problems

Shiryaev A., Peskir G. — Optimal Stopping and Free-Boundary Problems

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Название: Optimal Stopping and Free-Boundary Problems

Авторы: Shiryaev A., Peskir G.

Аннотация:

The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples.

The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations).

A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples.

Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 500

Добавлена в каталог: 19.06.2008

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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 $\sigma$ -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 $\sigma$ -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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