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Oksendal B. — Stochastic differential equations : an introduction with applications
Oksendal B. — Stochastic differential equations : an introduction with applications

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Название: Stochastic differential equations : an introduction with applications

Автор: Oksendal B.

Аннотация:

This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided.


Язык: en

Рубрика: Математика/Вероятность/Стохастические процессы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: fifth edition

Год издания: 2000

Количество страниц: 332

Добавлена в каталог: 05.06.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$H_t$-Brownian motion      70
$\sigma$-algebra      7
$\sigma$-algebra, generated by a family of sets      8
$\sigma$-algebra, generated by a random variable      8
Adapted process      25
Admissible portfolio      251
American call option      284 289
American contingent T-claim      276
American options      275—284
American put option      283—284
American put option, perpetual      289
Analytic functions (and Brownian motion)      76 150
Arbitrage      251
Attainable claim      260
Bayes' rule      152
Bellman principle      241
Bequest function      223
Bessel process      49 140
Black and Scholes formula      4 160 274 288
Borel sets, Borel $\sigma$-algebra      8
Borel — Cantelli lemma      16
Brownian bridge      75
Brownian motion, complex      76
Brownian motion, in $R^n$      3 11—14
Brownian motion, on a Riemannian manifold      150
Brownian motion, on the ellipse      73
Brownian motion, on the unit circle      65 121
Brownian motion, on the unit sphere      149
Brownian motion, the graph of      118
Brownian motion, w.r.t. an increasing family $H_t$ of $\sigma$-algebras      70
Capacity      163
Carrying capacity      77
Change of time      145
Change of variable in an Ito integral      148
Characteristic function      292
Characteristic operator      120
Chebychev's inequality      16
Coincide in law      140 141
Combined Dirichlet — Poisson problem      165—167
Complete market      260
Complete probability space      8
Complex Brownian motion      76
Conditional expectation      295
Conditioned Brownian motion      127
Contingent T-claim (American)      276
Contingent T-claim (European)      259
Continuation region      201
Continuous in mean square      40
Control, deterministic (open loop)      225
Control, feedback (closed loop)      225
Control, Markov      225
Control, optimal      224
Convolution      302
Covariance matrix      12 291
Cross-variation processes      152
Crowded environment      77
Density (of a random variable)      15
Diffusion coeffcient      107
Diffusion, Dynkin      121
Diffusion, Ito      107
Dirichlet problem      2 167
Dirichlet problem (generalized)      174
Dirichlet problem (stochastic version)      170
Dirichlet — Poisson problem      165—167 182
Distribution (of a process)      10
Distribution (of a random variable)      9
Distribution function (of a random variable)      15
Doob — Dynkin lemma      8—9
Doob —Meyer decomposition      279
Drift coeffcient      107
Dudley's theorem      253
Dynkin's formula      118
Eigenvalues (of the Laplacian)      187
Elementary function/process      26
Elliptic partial differential operator      165 176
Equivalent martingale measure      254 264
Estimate (linear/measurable)      85
Estimation of a parameter      97
Estimation, exact asymptotic      101 102
European call option      4 265 288—289
European contingent T-claim      259
European option      265
European put option      266
Events      8
Excessive function      197
Expectation      9
Explosion (of a diffusion)      66 78
Exponential martingale      55
Feller-continuity      133
Feynman — Kac formula      135 190
Filtering problem, general      2 79—81
Filtering problem, linear      81—101
Filtration      31 38
Finite-dimensional distributions (of a stochastic process)      10
First exit distribution      130 192
First exit time      111
Gaussian process      12
Generalized (distribution valued) process      21
Generator (of an Ito diffusion)      115 117
Geometric Brownian motion      62
Girsanov transformation      153
Girsanov's theorem      60 153—158
Green formula      184
Green function      163 183 191
Green function (classical)      183 185
Green measure      18 183 238
Green operator      164
Gronwall inequality      68 78
h-transform (of Brownian motion)      127
Hamilton — Jacobi — Bellman (HJB) equation      226—230
Harmonic extension (w.r.t. an Ito diffusion)      122
Harmonic function (and Brownian motion)      150
Harmonic function (w.r.t. a diffusion)      169
Harmonic measure (of a diffusion)      114 115 129
Harmonic measure (of Brownian motion)      124
Hedging portfolio      260
Hermite polynomials      38
High contact (smooth fit) principle      210 212 218
Hitting distribution      114 115
Hunt's condition (H)      175
Independent      9
Independent increments      13 22
Innovation process      82 86 87 90
Integration by parts (stochastic)      46 55
Interpolation (smoothing)      103
Irregular point      172 188
Iterated Ito integrals      38
Iterated logarithm (law of)      64
Ito diffsion      107
Ito integral      24—37
Ito integral; multidimensional      34 35
Ito interpretation (of a stochastic differential equation)      36 61 79
Ito isometry      26 29
Ito process      44 48
Ito representation theorem      51
Ito's formula      44 48
Jensen inequality      296
Kalman — Bucy filter      2 95 100
Kazamaki condition      55
Kernel function      127
Killing (a diffusion)      137
Killing rate      138 164
Kolmogorov's backward equation      131
Kolmogorov's continuity theorem      14
Kolmogorov's extension theorem      11
Kolmogorov's forward equation      159
Langevin equation      74
Laplace operator $\Delta$      3 57
Laplace — Beltrami operator      150
Law of iterated logarithm      64
Least superharmonic majorant      196
Least supermeanvalued majorant      196
Levy's characterization of Brownian motion      152
Levy's theorem      151
Linear regulator problem      231
Lipschitz surface      213 301
Local martingale      126
Local time      58 59 72
Lyapunov equation      103
Malliavin derivative      53
Market      247
Market, complete      260
Market, normalized      247 248
Markov control      225
Markov process      110
Markov property      109
Martingale      31 33 298
Martingale convergence theorem      298
Martingale inequality      31
Martingale problem      138
Martingale representation theorem      49 53
Martingale, local      126
Maximum likelihood      98
Maximum principle      189
Mean square error      92
Mean value property (for a diffusion)      114 115
Mean value property, classical      124
Mean-reverting Ornstein — Uhlenbeck process      74
Measurable sets (w.r.t. a $\sigma$-algebra)      8
Measurable space      7
Moving average, exponentially weighted      97
Noise      1—4 21—22 61
Normal distribution      12 291
Normalization (of a market process)      248
Novikov condition      55
Numeraire      248
Observation process      80
Optimal control      224
Optimal performance      224
Optimal portfolio selection      4 234
Optimal stopping      3 193—215
Optimal stopping existence theorem      199
Optimal stopping time      193 199 202 213
Optimal stopping uniqueness theorem      202
Option pricing      4 265—284
Ornstein — Uhlenbeck equation/process      74
Orthogonal increments      82
p'th variation process      19
Path (of a stochastic process)      10
Performance function      224
Perron — Wiener — Brelot solution      178
Poisson formula      189
Poisson kernel      189
Poisson problem      168
Poisson problem (generalized)      180
Poisson problem (stochastic version)      180
Polar set      162 175
Population growth      1 61 77
Portfolio      4 236 248—251
Prediction      103
Probability measure      7
Probability space      8
Quadratic variation process      19 56
Random time change      145
Random variable      9
Recurrent      120
Regular point      172—174 188
Replicating portfolio      260
Resolvent operator      133
Reward function      193
Reward rate function      194
Riccati equation      93 95 101 233
Scaling (Brownian)      19
Self-financing portfolio      248
Semi-elliptic partial differential operator      165
Semi-polar set      175
Separation principle      225 233
Shift operator      113
Smoothing (interpolation)      103
Snell envelope      279
Stationary process      21 22
Stochastic control      4 223—240
Stochastic differential equation; definition      61
Stochastic differential equation; existence and uniqueness of solution      66
Stochastic differential equation; weak and strong solution      70
Stochastic Dirichlet problem      170
Stochastic integral      44
Stochastic Poisson problem      180
Stochastic process      9
Stopping time      57 110
Stratonovich integral      24 35—37 39 40
Stratonovich interpretation (of a stochastic differential equation)      36 62 63 64 79
Strong Feller process      177
Strong Markov property      110—113
Strong solution (of a stochastic differential equation)      70
Strong uniqueness (of a stochastic differential equation)      67 71
Submartingale      298
Superharmonic function      194
Superharmonic majorant      196
Supermartingale      126 196 253 266 279 298
Supermeanvalued function      194
Supermeanvalued majorant      196
Superreplicate      279
Support (of a diffusion)      105
Tanaka's equation      71
Tanaka's formula      58 59 72
Terminal conditions (in stochastic control)      239—240 245
Thin set      175
Time change formula Ito integrals      148
Time-homogeneous      108
Total variation process      19
Transient      120
Transition measure      184
Transition operator      164
Trap      121
Uniformly elliptic partial differential operator      176 269
Uniformly integrable      297—298
Utility function      4 234
Value function      224
Value process      248
Value process, normalized      249
Variational inequalities (and optimal stopping)      3 212—215
Version (of a process)      12 14 32
Volterra equation, deterministic      89
Volterra equation, stochastic      75
Weak solution (of a stochastic differential equation)      70
Weak uniqueness      71
Well posed (martingale problem)      139
White noise      21 61
Wiener criterion      174
X-harmonic      169
Zero-one law      171
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