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Oksendal B. — Stochastic Differential Equations: An Introduction With Applications
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Название: Stochastic Differential Equations: An Introduction With ApplicationsАвтор: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 6-thГод издания: 2003Количество страниц: 363Добавлена в каталог: 30.10.2006Операции: Положить на полку |
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9 72 7 17 8 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 8 Borel — Cantelli lemma 17 Borrowing 247 Brownian bridge 76 Brownian motion, complex 78 Brownian motion, in 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 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 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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