Brownian Motion and Stochastic CalculusSpringer Science & Business Media, 16 de ago. de 1991 - 470 páginas This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises. |
Conteúdo
Martingales Stopping Times and Filtrations | 1 |
12 Stopping Times | 6 |
13 ContinuousTime Martingales | 11 |
A Fundamental Inequalities | 12 |
B Convergence Results | 17 |
C The Optional Sampling Theorem | 19 |
14 The DoobMeyer Decomposition | 21 |
15 Continuous SquareIntegrable Martingales | 30 |
A The MeanValue Property | 241 |
B The Dirichlet Problem | 243 |
C Conditions for Regularity | 247 |
D Integral Formulas of Poisson | 251 |
E Supplementary Exercises | 253 |
43 The OneDimensional Heat Equation | 254 |
A The Tychonoff Uniqueness Theorem | 255 |
B Nonnegative Solutions of the Heat Equation | 256 |
16 Solutions to Selected Problems | 38 |
17 Notes | 45 |
Brownian Motion | 47 |
B The KolmogorovCentsov Theorem | 53 |
23 Second Construction of Brownian Motion | 56 |
24 The Space C0 oo Weak Convergence and the Wiener Measure | 59 |
A Weak Convergence | 60 |
B Tightness | 61 |
C Convergence of FiniteDimensional Distributions | 64 |
D The Invariance Principle and the Wiener Measure | 66 |
25 The Markov Property | 71 |
A Brownian Motion in Several Dimensions | 72 |
B Markov Processes and Markov Families | 74 |
C Equivalent Formulations of the Markov Property | 75 |
26 The Strong Markov Property and the Reflection Principle | 79 |
B Strong Markov Processes and Families | 81 |
C The Strong Markov Property for Brownian Motion | 84 |
27 Brownian Filtrations | 89 |
A RightContinuity of the Augmented Filtration for a Strong Markov Process | 90 |
B A Universal Filtration | 93 |
C The Blumenthal ZeroOne Law | 94 |
A Brownian Motion and Its Running Maximum | 95 |
B Brownian Motion on a HalfLine | 97 |
D Distributions Involving Last Exit Times | 100 |
29 The Brownian Sample Paths | 103 |
B The Zero Set and the Quadratic Variation | 104 |
C Local Maxima and Points of Increase | 106 |
D Nowhere Differentiability | 109 |
E Law of the Iterated Logarithm | 111 |
F Modulus of Continuity1 | 114 |
210 Solutions to Selected Problems | 116 |
211 Notes | 126 |
Stochastic Integration | 128 |
32 Construction of the Stochastic Integral | 129 |
A Simple Processes and Approximations | 132 |
B Construction and Elementary Properties of the Integral | 137 |
C A Characterization of the Integral | 141 |
D Integration with Respect to Continuous Local Martingales | 145 |
33 The ChangeofVariable Formula | 148 |
A The Ito Rule | 149 |
B Martingale Characterization of Brownian Motion | 156 |
C Bessel Processes Questions of Recurrence | 158 |
D Martingale Moment Inequalities | 163 |
E Supplementary Exercises | 167 |
34 Representations of Continuous Martingales in Terms of Brownian Motion | 169 |
A Continuous Local Martingales as Stochastic Integrals with Respect to Brownian Motion | 170 |
B Continuous Local Martingales as TimeChanged Brownian Motions | 173 |
C A Theorem of F B Knight | 179 |
D Brownian Martingales as Stochastic Integrals | 180 |
E Brownian Functionals as Stochastic Integrals | 185 |
35 The Girsanov Theorem | 190 |
A The Basic Result | 191 |
B Proof and Ramifications | 193 |
C Brownian Motion with Drift | 196 |
D The Novikov Condition | 198 |
36 Local Time and a Generalized Ito Rule for Brownian Motion | 201 |
A Definition of Local Time and the Tanaka Formula | 203 |
B The Trotter Existence Theorem | 206 |
C Reflected Brownian Motion and the Skorohod Equation | 210 |
D A Generalized Ito Rule for Convex Functions | 212 |
E The EngelbertSchmidt ZeroOne Law | 215 |
37 Local Time for Continuous Semimartingales1 | 217 |
38 Solutions to Selected Problems | 226 |
39 Notes | 236 |
Brownian Motion and Partial Differential Equations | 239 |
42 Harmonic Functions and the Dirichlet Problem | 240 |
C BoundaryCrossing Probabilities for Brownian Motion | 262 |
D Mixed InitialBoundary Value Problems | 265 |
44 The Formulas of Feynman and Kac | 267 |
A The Multidimensional Formula | 268 |
B The OneDimensional Formula | 271 |
45 Solutions to Selected Problems | 276 |
46 Notes | 278 |
Stochastic Differential Equations | 281 |
52 Strong Solutions | 284 |
A Definitions | 285 |
B The ltd Theory | 286 |
C Comparison Results and Other Refinements | 291 |
D Approximations of Stochastic Differential Equations | 295 |
E Supplementary Exercises | 299 |
53 Weak Solutions | 300 |
A Two Notions of Uniqueness | 301 |
B Weak Solutions by Means of the Girsanov Theorem | 302 |
C A Digression on Regular Conditional Probabilities | 306 |
D Results of Yamada and Watanabe on Weak and Strong Solutions | 308 |
54 The Martingale Problem of Stroock and Varadhan | 311 |
A Some Fundamental Martingales | 312 |
B Weak Solutions and Martingale Problems | 314 |
C WellPosedness and the Strong Markov Property | 319 |
D Questions of Existence | 323 |
E Questions of Uniqueness | 325 |
F Supplementary Exercises | 328 |
55 A Study of the OneDimensional Case | 329 |
A The Method of TimeChange | 330 |
B The Method of Removal of Drift | 339 |
C Fellers Test for Explosions | 342 |
D Supplementary Exercises | 351 |
56 Linear Equations | 354 |
A GaussMarkov Processes | 355 |
B Brownian Bridge | 358 |
C The General OneDimensional Linear Equation | 360 |
D Supplementary Exercises | 361 |
57 Connections with Partial Differential Equations | 363 |
A The Dirichlet Problem | 364 |
B The Cauchy Problem and a FeynmanKac Representation | 366 |
C Supplementary Exercises | 369 |
58 Applications to Economics | 371 |
B Option Pricing | 376 |
C Optimal Consumption and Investment General Theory | 379 |
D Optimal Consumption and Investment Constant Coefficients | 381 |
59 Solutions to Selected Problems | 387 |
510 Notes | 394 |
P Levys Theory of Brownian Local Time | 399 |
62 Alternate Representations of Brownian Local Time | 400 |
B Poisson Random Measures | 403 |
C Subordinators | 405 |
D The Process of Passage Times Revisited | 411 |
E The Excursion and Downcrossing Representations of Local Time | 414 |
63 Two Independent Reflected Brownian Motions | 418 |
B The First Formula of D Williams | 421 |
C The Joint Density of Wt Lt r+t | 423 |
64 Elastic Brownian Motion | 425 |
A The FeynmanKac Formulas for Elastic Brownian Motion Our intent is to study the counterpart | 426 |
B The RayKnight Description of Local Time | 430 |
C The Second Formula of D Williams | 434 |
Transition Probabilities of Brownian Motion with TwoValued Drift | 437 |
66 Solutions to Selected Problems | 442 |
67 Notes | 445 |
Bibliography | 447 |
| 459 | |
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Termos e frases comuns
assume B₁ Borel-measurable bounded Brownian family Brownian motion Brownian path constant continuous function convergence theorem Corollary d-dimensional Brownian motion decomposition define Definition denote Dynkin system Exercise exists F-measurable filtration F finite finite-dimensional distributions fixed follows formula holds implies independent inequality inf{t Itô Itô's rule Lemma Lévy local martingale M₁ Markov family Markov process Markov property martingale problem nondecreasing nonnegative o-field obtain one-dimensional Brownian motion P}xe Poisson probability measure probability space progressively measurable proof of Theorem Proposition quadratic variation random variable Remark representation result right-continuous sample path satisfies the usual Section semimartingale sequence space Q stochastic differential equation stochastic integral stochastic process stopping strong Markov property strong solution Stroock submartingale subsection Suppose t₁ uniformly integrable vector W₁ weak solution X₁ xe Rd y₁ zero