Fleming W.H. Controlled Markov processes and viscosity solutions / Fleming W.H., Soner H.M. - 2nd ed. - New York: Springer-Verlag, 2006. - xvii, 428 p. - (Stochastic modelling and applied probability; 25). - ISBN 0-387-26045-5  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Preface to Second Edition ...................................... xi Preface ...................................................... xiii Notation ....................................................... xv I. Deterministic Optimal Control .............................. 1 I.1. Introduction ............................................. 1 I.2. Examples ................................................. 2 I.3. Finite time horizon problems ............................. 5 I.4. Dynamic programming principle ............................ 9 I.5. Dynamic programming equation ............................ 11 I.6. Dynamic programming and Pontryagin's principle .......... 18 I.7. Discounted cost with infinite horizon ................... 25 I.8. Calculus of variations I ................................ 33 I.9. Calculus of variations II ............................... 37 I.10. Generalized solutions to Hamilton-Jacobi equations ...... 42 I.11. Existence theorems ...................................... 49 I.12. Historical remarks ...................................... 55 II. Viscosity Solutions ....................................... 57 II.1. Introduction ............................................ 57 II.2. Examples ................................................ 60 II.3. An abstract dynamic programming principle ............... 62 II.4. Definition .............................................. 67 II.5. Dynamic programming and viscosity property .............. 72 II.6. Properties of viscosity solutions ....................... 73 II.7. Deterministic optimal control and viscosity solutions ... 78 II.8. Viscosity solutions: first order case ................... 83 II.9. Uniqueness: first order case ............................ 89 II.10. Continuity of the value function ....................... 99 II.11. Discounted cost with infinite horizon .................. 105 II.12. State constraint ....................................... 106 II.13. Discussion of boundary conditions ...................... 1ll II.14. Uniqueness: first-order case ........................... 114 II.15. Pontryagin's maximum principle (continued) ............. 115 II.16. Historical remarks ..................................... 117 III. Optimal Control of Markov Processes: Classical Solutions ................................................ 119 III.1. Introduction ........................................... 119 III.2. Markov processes and their evolution operators ......... 120 III.3. Autonomous (time-homogeneous) Markov processes ......... 123 III.4. Classes of Markov processes ............................ 124 III.5. Markov diffusion processes on Mn; stochastic differential equations ................................. 127 III.6. Controlled Markov processes ............................ 130 III.7. Dynamic programming: formal description ................ 131 III.8. A Verification Theorem; finite time horizon ............ 134 III.9. Infinite Time Horizon .................................. 139 III.10.Viscosity solutions .................................... 145 III.11.Historical remarks ..................................... 148 IV. Controlled Markov Diffusions in Mn ....................... 151 IV.1. Introduction ........................................... 151 IV.2. Finite time horizon problem ............................ 152 IV.3. Hamilton-Jacobi-Bellman PDE ............................ 155 IV.4. Uniformly parabolic case ............................... 161 IV.5. Infinite time horizon .................................. 164 IV.6. Fixed finite time horizon problem: Preliminary estimates .............................................. 171 IV.7. Dynamic programming principle .......................... 176 IV.8. Estimates for first order difference quotients ......... 182 IV.9. Estimates for second-order difference quotients ........ 186 IV.10. Generalized subsolutions and solutions ................. 190 IV.11. Historical remarks ..................................... 197 V. Viscosity Solutions: Second-Order Case ................... 199 V.l. Introduction ........................................... 199 V.2. Dynamic programming principle .......................... 200 V.3. Viscosity property ..................................... 205 V.4. An equivalent formulation .............................. 210 V.5. Semiconvex, concave approximations ..................... 214 V.6. Crandall-Ishii Lemma ................................... 216 V.7. Properties of U ........................................ 218 V.8. Comparison ............................................. 219 V.9. Viscosity solutions in Qq .............................. 222 V.10. Historical remarks ..................................... 225 VI. Logarithmic Transformations and Risk Sensitivity ......... 227 VI.l. Introduction ........................................... 227 VI.2. Risk sensitivity ....................................... 228 VI.3. Logarithmic transformations for Markov diffusions ...... 230 VI.4. Auxiliary stochastic control problem ................... 235 VI.5. Bounded region Q ....................................... 238 VI.6. Small noise limits ..................................... 239 VI.7. H-infinity norm of a nonlinear system .................. 245 VI.8. Risk sensitive control ................................. 250 VI.9. Logarithmic transformations for Markov processes ....... 255 VI.10. Historical remarks ..................................... 259 VII. Singular Perturbations ................................... 261 VII.l. Introduction ........................................... 261 VII.2. Examples ............................................... 263 VII.3. Barles and Perthame procedure .......................... 265 VII.4. Discontinuous viscosity solutions ...................... 266 VII.5. Terminal condition ..................................... 269 VII.6. Boundary condition ..................................... 271 VII.7. Convergence ............................................ 272 VII.8. Comparison ............................................. 273 VII.9. Vanishing viscosity .................................... 280 VII.10.Large deviations for exit probabilities ................ 282 VII.11.Weak comparison principle in Q0 ........................ 290 VII.12.Historical remarks ..................................... 292 VIII.Singular Stochastic Control .............................. 293 VIII.l.Introduction ........................................... 293 VIII.2.Formal discussion ...................................... 294 VIII.3.Singular stochastic control ............................ 296 VIII.4.Verification theorem ................................... 299 VIII.5.Viscosity solutions .................................... 311 VIII.6.Finite fuel problem .................................... 317 VIII.7.Historical remarks ..................................... 319 IX. Finite. Difference Numerical Approximations .............. 321 IX.1. Introduction ........................................... 321 IX.2. Controlled discrete time Markov chains ................. 322 IX.3. Finite difference approximations to HJB equations ...... 324 IX.4. Convergence of finite difference approximations. I ..... 331 IX.5. Convergence of finite difference approximations. II .... 336 IX.6. Historical remarks ..................................... 346 X. Applications to Finance .................................. 347 X.l. Introduction ........................................... 347 X.2. Financial market model ................................. 347 X.3. Merton portfolio problem ............................... 348 X.4. General utility and duality ............................ 349 X.5. Portfolio selection with transaction costs ............. 354 X.6. Derivatives and the Black-Scholes price ................ 360 X.7. Utility pricing ........................................ 362 X.8. Super-replication with portfolio constraints ........... 364 X.9. Buyer's price and the no-arbitrage interval ............ 365 X.10. Portfolio constraints and duality ...................... 366 X.ll. Merton problem with random parameters .................. 368 X.12. Historical remarks ..................................... 372 XI. Differential Games ....................................... 375 XI.l. Introduction ........................................... 375 XI.2. Static games ........................................... 376 XI.3. Differential game formulation .......................... 377 XI.4. Upper and lower value functions ........................ 381 XI.5. Dynamic programming principle .......................... 382 XI.6. Value functions as viscosity solutions ................. 384 XL 7. Risk sensitive control limit game ...................... 387 XI.8. Time discretizations ................................... 390 XI.9. Strictly progressive strategies ........................ 392 XI.10. Historical remarks ..................................... 395 A. Duality Relationships ...................................... 397 В. Dynkin's Formula for Random Evolutions with Markov Chain Parameters ........................................... 399 С. Extension of Lipschitz Continuous Functions; Smoothing ..... 401 D. Stochastic Differential Equations: Random Coefficients ..... 403 References .................................................... 409 Index ......................................................... 425