Brownian Motion and Stochastic Calculus

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface......Page 6
Contents......Page 9
Suggestions for the Reader......Page 16
Interdependence of the Chapters......Page 18
Frequently Used Notation......Page 20
Chapter 1 Martingales, Stopping Times, and Filtrations......Page 24
1.1. Stochastic Processes and a-Fields......Page 24
1.2. Stopping Times......Page 29
1.3. Continuous-Time Martingales......Page 34
A. Fundamental inequalities......Page 35
B. Convergence results......Page 40
C. The optional sampling theorem......Page 42
1.4. The Doob-Meyer Decomposition......Page 44
1.5. Continuous, Square-Integrable Martingales......Page 53
1.6. Solutions to Selected Problems......Page 61
1.7. Notes......Page 68
Chapter 2 Brownian Motion......Page 70
2.1. Introduction......Page 70
2.2. First Construction of Brownian Motion......Page 72
A. The consistency theorem......Page 72
B. The Kolmogorov-entsov theorem......Page 76
2.3. Second Construction of Brownian Motion......Page 79
2.4. The Space C[0, co), Weak Convergence, and Wiener Measure......Page 82
A. Weak convergence......Page 83
B. Tightness......Page 84
C. Convergence of finite-dimensional distributions......Page 87
D. The invariance principle and the Wiener measure......Page 89
2.5. The Markov Property......Page 94
A. Brownian motion in several dimensions......Page 95
B. Markov processes and Markov families......Page 97
C. Equivalent formulations of the Markov property......Page 98
2.6. The Strong Markov Property and the Reflection Principle......Page 102
A. The reflection principle......Page 102
B. Strong Markov processes and families......Page 104
C. The strong Markov property for Brownian motion......Page 107
2.7. Brownian Filtrations......Page 112
A. Right-continuity of the augmented filtration for a strong Markov process......Page 113
B. A \"niversal\\\" filtration......Page 116
C. The Blumenthal zero-one law......Page 117
2.8. Computations Based on Passage Times......Page 117
A. Brownian motion and its running maximum......Page 118
B. Brownian motion on a half-line......Page 120
C. Brownian motion on a finite interval......Page 120
D. Distributions involving last exit times......Page 123
2.9. The Brownian Sample Paths......Page 126
A. Elementary properties......Page 126
B. The zero set and the quadratic variation......Page 127
C. Local maxima and points of increase......Page 129
D. Nowhere differentiability......Page 132
E. Law of the iterated logarithm......Page 134
F. Modulus of continuity......Page 137
2.10. Solutions to Selected Problems......Page 139
2.11. Notes......Page 149
Chapter 3 Stochastic Integration......Page 151
3.1. Introduction......Page 151
3.2. Construction of the Stochastic Integral......Page 152
A. Simple processes and approximations......Page 155
B. Construction and elementary properties of the integral......Page 160
C. A characterization of the integral......Page 164
D. Integration with respect to continuous, local martingales......Page 168
3.3. The Change-of-Variable Formula......Page 171
A. The Ito rule......Page 172
B. Martingale characterization of Brownian motion......Page 179
C. Bessel processes, questions of recurrence......Page 181
D. Martingale moment inequalities......Page 186
E. Supplementary exercises......Page 190
3.4. Representations of Continuous Martingales in Terms of Brownian Motion......Page 192
A. Continuous local martingales as stochastic integrals with respect to Brownian motion......Page 193
B. Continuous local martingales as time-changed Brownian motions......Page 196
C. A theorem of F. B. Knight......Page 202
D. Brownian martingales as stochastic integrals......Page 203
E. Brownian functionals as stochastic integrals......Page 208
3.5. The Girsanov Theorem......Page 213
A. The basic result......Page 214
B. Proof and ramifications......Page 216
C. Brownian motion with drift......Page 219
D. The Novikov condition......Page 221
3.6. Local Time and a Generalized It8 Rule for Brownian Motion......Page 224
A. Definition of local time and the Tanaka formula......Page 226
B. The Trotter existence theorem......Page 229
C. Reflected Brownian motion and the Skorohod equation......Page 233
D. A generalized Ito rule for convex functions......Page 235
E. The Engelbert-Schmidt zero-one law......Page 238
3.7. Local Time for Continuous Semimartingales......Page 240
3.8. Solutions to Selected Problems......Page 249
3.9. Notes......Page 259
Chapter 4 Brownian Motion and Partial Differential Equations......Page 262
4.1. Introduction......Page 262
4.2. Harmonic Functions and the Dirichlet Problem......Page 263
A. The mean-value property......Page 264
B. The Dirichlet problem......Page 266
C. Conditions for regularity......Page 270
D. Integral formulas of Poisson......Page 274
E. Supplementary exercises......Page 276
4.3. The One-Dimensional Heat Equation......Page 277
A. The Tychonoff uniqueness theorem......Page 278
B. Nonnegative solutions of the heat equation......Page 279
C. Boundary crossing probabilities for Brownian motion......Page 285
D. Mixed initial/boundary value problems......Page 288
4.4. The Formulas of Feynman and Kac......Page 290
A. The multidimensional formula......Page 291
B. The one-dimensional formula......Page 294
4.5. Solutions to selected problems......Page 298
4.6. Notes......Page 301
Chapter 5 Stochastic Differential Equations......Page 304
5.1. Introduction......Page 304
5.2. Strong Solutions......Page 307
A. Definitions......Page 308
B. The Ito theory......Page 309
C. Comparison results and other refinements......Page 314
D. Approximations of stochastic differential equations......Page 318
E. Supplementary exercises......Page 322
5.3. Weak Solutions......Page 323
A. Two notions of uniqueness......Page 324
B. Weak solutions by means of the Girsanov theorem......Page 325
C. A digression on regular conditional probabilities......Page 329
D. Results of Yamada and Watanabe on weak and strong solutions......Page 331
5.4. The Martingale Problem of Stroock and Varadhan......Page 334
A. Some fundamental martingales......Page 335
B. Weak solutions and martingale problems......Page 337
C. Well-posedness and the strong Markov property......Page 342
D. Questions of existence......Page 346
E. Questions of uniqueness......Page 348
F. Supplementary exercises......Page 351
5.5. A Study of the One-Dimensional Case......Page 352
A. The method of time change......Page 353
B. The method of removal of drift......Page 362
C. Feller\'s test for explosions......Page 365
D. Supplementary exercises......Page 374
5.6. Linear Equations......Page 377
A. Gauss-Markov processes......Page 378
B. Brownian bridge......Page 381
C. The general, one-dimensional, linear equation......Page 383
D. Supplementary exercises......Page 384
5.7. Connections with Partial Differential Equations......Page 386
A. The Dirichlet problem......Page 387
B. The Cauchy problem and a Feynman-Kac representation......Page 389
C. Supplementary exercises......Page 392
5.8. Applications to Economics......Page 394
A. Portfolio and consumption processes......Page 394
B. Option pricing......Page 399
C. Optimal consumption and investment (general theory)......Page 402
D. Optimal consumption and investment (constant coefficients)......Page 404
5.9. Solutions to Selected Problems......Page 410
5.10. Notes......Page 417
Chapter 6 P. Levy\'s Theory of Brownian Local Time......Page 422
6.1. Introduction......Page 422
6.2. Alternate Representations of Brownian Local Time......Page 423
A. The process of passage times......Page 423
B. Poisson random measures......Page 426
C. Subordinators......Page 428
D. The process of passage times revisited......Page 434
E. The excursion and downcrossing representations of local time......Page 437
6.3. Two Independent Reflected Brownian Motions......Page 441
A. The positive and negative parts of a Brownian motion......Page 441
B. The first formula of D. Williams......Page 444
C. The joint density of (W(t), L(t), f (t))......Page 446
6.4. Elastic Brownian Motion......Page 448
A. The Feynman-Kac formulas for elastic Brownian motion......Page 449
B. The Ray-Knight description of local time......Page 453
C. The second formula of D. Williams......Page 457
6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift......Page 460
6.6. Solutions to Selected Problems......Page 465
6.7. Notes......Page 468
Bibliography......Page 470
Index......Page 482