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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 edition contains detailed solutions of selected exercises. Many readers have requested this, because it makes the book more suitable for self-study. At the same time new exercises (without solutions) have been added. They have all been placed in the end of each chapter, in order to facilitate the use of this edition together with previous ones.
Several errors have been corrected and formulations have been improved. This has been made possible by the valuable comments from (in alphabetical order) Jon Bohlin, Mark Davis, Helge Holden, Patrick Jaillet, Chen Jing, Natalia Koroleva, Mario Lefebvre, Alexander Matasov, Thilo Meyer-Brandis, Keigo Osawa, Bj0rn Thunestvedt, Jan Ub0e and Yngve Willassen. I thank them all for helping to improve the book.
My thanks also go to Dina Haraldsson, who once again has performed the typing and drawn the figures with great skill.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 6-th

Год издания: 2003

Количество страниц: 363

Добавлена в каталог: 30.10.2006

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Предметный указатель

-space      9
$\mathcal{H}_t$ -Brownian motion      72
$\sigma$ -algebra      7
$\sigma$ -algebra, finite      17
$\sigma$ -algebra, generated by a family of sets      8
$\sigma$ -algebra, generated by a random variable      8
(Multi)normal distribution      13
Absolutely continuous      161
Adapted process      25
Adjoint operator      169
Admissible control      236
Admissible portfolio      265
Almost surely (a.s.)      8
American call option      298 302
American contingent T-claim      290
American option      290—298
American option price      291
American put option      296—298
American put option, perpetual      303
Analytic functions (and Brownian motion)      78 158
Arbitrage      265
Attainable claim      274
Banach space      9
Bankruptcy time      224 235
Bayes’ rule      160 (8.6.3)
Bellman principle      254
Bequest function      224 235
Bessel process      49 148
Black — Scholes equation      203
Black — Scholes formula      4 169 204 288 289 302
Borel sets, Borel $\sigma$ -algebra      8
Borel — Cantelli lemma      17
Borrowing      247
Brownian bridge      76
Brownian motion, complex      78
Brownian motion, in $\mathbb{R}^n$       3 11—15
Brownian motion, on a Riemannian manifold      158
Brownian motion, on the ellipse      74
Brownian motion, on the unit circle      67 127
Brownian motion, on the unit sphere      157
Brownian motion, the graph of      124
Brownian motion, w.r.t.an increasing family $\mathcal{H}_t$ of $\sigma$ -algebras      72
Brownian scaling      19
Capacity      173
Carrying capacity      78
Cauchy sequence      20
Change of time      153
Change of variable in an Ito integral      156
Characteristic function      306
Characteristic operator      126
Chebychev’s Inequality      16
Claim, American      290
Claim, European      274
Closed loop control      237
Coincide in law      149
Combined Dirichlet — Poisson problem      175—178 193
Complete market      274
Complete normal linear space      20
Complete probability space      8
Complex Brownian motion      78
Conditional expectation      309—310
Conditioned Brownian motion      133
Contingent T-claim (American)      290
Contingent T-claim (European)      274
Continuation region      211 225
Continuous in mean square      40
Control, deterministic (open loop)      23
Control, feedback (closed loop)      237
Control, Markov      237
Control, optimal      236
Convolution      316
Covariance matrix      13 305
Cross-variation processes      160
Crowded environment      78
Density (of a random variable)      16
Diffusion coefficient      113
Diffusion, Dynkin      127
Diffusion, Ito      113 114
Dirichlet problem      2 177 179
Dirichlet problem (generalized)      185
Dirichlet problem (stochastic version)      181
Dirichlet — Poisson problem      175—178 193
Distribution (of a process)      11
Distribution (of a random variable)      9
Distribution function (of a random variable)      15
Doob — Dynkin lemma      8—9
Doob — Meyer decomposition      294
Drift coefficient      113
Dudley’s theorem      267
Dynkin’s formula      124 196
Eigenvalues (of the Laplacian)      198
Elementary function/process      26
Elliptic partial differential operator      175
Equivalent local martingale measure      165 268
Equivalent martingale measure      165 268 278
Estimate (linear/measurable)      89
Estimation of a parameter      101
Estimation, exact asymptotic      105 106
European call option      4 279
European contingent T-claim      274
European option      279
European option price      279 280
European put option      280
Events      8
Exact asymptotic estimation      105
Excessive function      209
Expectation      9
Explosion (of a diffusion)      68 79
Exponential martingale      55
Feedback control      237
Feller-continuity      141
Feynman — Kac formula      143 201
Filtering problem, general      2 83—85
Filtering problem, linear      85—105
Filtration      31 38
Finite-dimensional distributions (of a stochastic process)      11
First exit distribution      136
First exit time      117
Gaussian process      13
Generalized (distribution valued) process      21
Generator (of an Ito diffusion)      121 123
Geometric Brownian motion      64
Girsanov transformation      162
Girsanov’s theorem      60 159—168
Green formula      195
Green function      172 194 196
Green measure      19 194 250
Green operator      172
Gronwall inequality      70 80
h-transform (of Brownian motion)      133
Hamilton — Jacobi — Bellman (HJB) equation      238—243
Harmonic extension (w.r.t.an Ito diffusion)      128
Harmonic function (and Brownian motion)      130 158
Harmonic function (w.r.t.a diffusion)      180
Harmonic measure (of a diffusion)      120 121 135
Harmonic measure (of Brownian motion)      130
Hausdorff measure      173
Heat equation      178
Hedging portfolio      274
Hermite polynomials      38
High contact (smooth fit) principle      222 224 230
Hilbert space      9
Hitting distribution      120 121
Hunt’s condition (H)      186
Independent      9 10
Independent increments      14 22
Innovation process      86 90 91 94
Integration by parts (stochastic)      46 55
Interpolation (smoothing)      107
Irregular point      183—185 199

Iterated Ito integrals      38
Iterated logarithm (law of)      66
Ito diffusion      114
Ito integral      24—37
Ito integral; multidimensional      34 35
Ito interpretation (of a stochastic differential equation)      36 63 83
Ito isometry      26 29
Ito process      44 48
Ito representation theorem/formula      51 284
Ito’s formula      44 48
Jensen inequality      310
Kalman — Bucy filter      2 99 104
Kazarnaki condition      55
Kelly criterion      248
Kernel function      133
Killing (a diffusion)      145
Killing rate      145
Killing time      145
Kolmogorov’s backward equation      139
Kolmogorov’s continuity theorem      14
Kolmogorov’s extension theorem      11
Kolmogorov’s forward equation      168—169
Langevin equation      75
Laplace operator $\Delta$       3 57
Laplace transform      136
Laplace — Beltrami operator      158
Law of iterated logarithm      66
Least superharmonic majorant      208 210
Least supermeanvalued majorant      208 210
Levy’s characterization of Brownian motion      160
Levy’s theorem      159
Linear regulator problem      243
Lipschitz surface      225 315
Local martingale      132 268
Local time      58 59 73
Lyapunov equation      107
Malliavin derivative      53
Market      261
Market, complete      274
Market, normalized      261 262
Markov control      237
Markov process      116
Markov property      115
Martingale      31 33 312
Martingale convergence theorem      312
Martingale inequality      31
Martingale problem      146—147
Martingale representation theorem      49 53
Martingale, local      132
Maximum likelihood      102
Maximum principle      200
Mean square error      96
Mean value      13
Mean value property (for a diffusion)      120 121
Mean value property, classical      130
Measurable function (w.r.t.a cr-algebra)      8
Measurable sets (w.r.t.a cr-algebra)      8
Measurable space      7
Moan-reverting Ornstein — Uhlenbeck process      75
Moving average, exponentially weighted      101
Noise      1—4 21—22 63
Normal distribution      12 305—307
Normalization (of a market process)      261 262 263 267 298
Novikov condition      55 162
Numeraire      262
Observation process      84
Open loop control      237
Optimal control      236
Optimal performance      236
Optimal portfolio selection      4 246
Optimal stopping      3 205—227
Optimal stopping existence theorem      211
Optimal stopping time      205 211 214 224 225
Optimal stopping uniqueness theorem      214
Option pricing      4 279—298
Optional sampling theorem      208
Ornstein — Uhlenbeck equation/process      75
Orthogonal increments      86
Outer measure zero      8
Path (of a stochastic process)      10
Performance function      236
Perron — Wiener — Brelot solution      189
Poisson formula      200
Poisson kernel      200
Poisson problem      179
Poisson problem (generalized)      191
Poisson problem (stochastic version)      191
Polar set      172 186
Polish space      12
Population growth      1 63 78 136
Portfolio      4 246 262—267
Prediction      104
Probability measure      7
Probability space      8
Profit rate function      206 224 235
p’th variation process      19
Quadratic variation process      19 56
Radon — Nikodym derivative      161
Random time change      153
Random variable      9
Recurrent      126
Regular point      183—185 199
Replicating portfolio      274 289
Resolvent operator      141 192
Reward function      205
Riccati equation      97 99 100 103 245
Risky investment      246
Safe investment      246
Scaling (Brownian)      19
Self-financing portfolio      262 263—264
Semi-elliptic partial differential operator      175
Semi-polar set      186
Separation principle      237 246
Shift operator      119
Shortselling      247 290 303
Smoothing (interpolation)      107
Snell envelope      294
Solvency set      224
Stationary process      18
Stochastic control      4 235—252
Stochastic differential equation, definition      63
Stochastic differential equation, existence and uniqueness of solution      68
Stochastic differential equation, weak and strong solution      72
Stochastic Dirichlet problem      181
Stochastic integral      44
Stochastic Poisson problem      191
Stochastic process      10
Stopping time      57 116
Stratonovich integral      24 35—37 39 40
Stratonovich interpretation (of a stochastic differential equation)      36 64 65 66 83
Strong Feller process      188 190
Strong Markov property      116—121
Strong solution (of a stochastic differential equation)      72
Strong uniqueness (of a stochastic differential equation)      69 72
Submartingale      302 303 312
Superharmonic function      206 254
Superharmonic majorant      208
Supermartingale      132 208 268 280 294 312
Supermeanvalued function      206
Supermeanvalued majorant      208
Superreplicate      294
Support (of a diffusion)      109
Tanaka’s equation      73
Tanaka’s formula      58 59 73
Terminal conditions (in stochastic control)      251—252 257
Thin set      186
Time change formula Ito integrals      156
Time-homogeneous      114
Total variation process      19
Transient      126

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