Brownian Motion (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> René L. Schilling and Lothar Partzsch, Dresden University of Technology, Germany.

Preface 5
Dependence chart 11
Index of notation 13
1 Robert Brown’s new thing 15
2 Brownian motion as a Gaussian process 21
2.1 The finite dimensional distributions 21
2.2 Invariance properties of Brownian motion 26
2.3 Brownian Motion in Rd 29
3 Constructions of Brownian motion 35
3.1 The Lévy-Ciesielski construction 35
3.2 Lévy’s original argument 42
3.3 Wiener’s construction 47
3.4 Donsker’s construction 50
3.5 The Bachelier-Kolmogorov point of view 51
4 The canonical model 54
4.1 Wiener measure 54
4.2 Kolmogorov’s construction 58
5 Brownian motion as a martingale 62
5.1 Some ‘Brownian’ martingales 62
5.2 Stopping and sampling 67
5.3 The exponential Wald identity 71
6 Brownian motion as a Markov process 76
6.1 The Markov property 76
6.2 The strong Markov property 79
6.3 Desiré André’s reflection principle 82
6.4 Transience and recurrence 87
6.5 Lévy’s triple law 90
6.6 An arc-sine law 93
6.7 Some measurability issues 94
7 Brownian motion and transition semigroups 100
7.1 The semigroup 100
7.2 The generator 106
7.3 The resolvent 110
7.4 The Hille-Yosida theorem and positivity 114
7.5 Dynkin’s characteristic operator 117
8 The PDE connection 127
8.1 The heat equation 128
8.2 The inhomogeneous initial value problem 131
8.3 The Feynman-Kac formula 133
8.4 The Dirichlet problem 137
9 The variation of Brownian paths 151
9.1 The quadratic variation 152
9.2 Almost sure convergence of the variation sums 154
9.3 Almost sure divergence of the variation sums 157
9.4 Levy’s characterization of Brownian motion 160
10 Regularity of Brownian paths 166
10.1 Hölder continuity 166
10.2 Non-differentiability 169
10.3 Lévy’s modulus of continuity 171
11 The growth of Brownian paths 178
11.1 Khintchine’s Law of the Iterated Logarithm 178
11.2 Chung’s ‘other’ Law of the Iterated Logarithm 182
12 Strassen’s Functional Law of the Iterated Logarithm 187
12.1 The Cameron-Martin formula 188
12.2 Large deviations (Schilder’s theorem) 195
12.3 The proof of Strassen’s theorem 200
13 Skorokhod representation 207
14 Stochastic integrals: L2-Theory 217
14.1 Discrete stochastic integrals 217
14.2 Simple integrands 221
14.3 Extension of the stochastic integral to L2T 225
14.4 Evaluating Itô integrals 229
14.5 What is the closure of ET? 233
14.6 The stochastic integral for martingales 236
15 Stochastic integrals: beyond L2T 241
16 Itô’s formula 247
16.1 Itô processes and stochastic differentials 247
16.2 The heuristics behind Itô’s formula 249
16.3 Proof of Itô’s formula (Theorem 16.1) 250
16.4 Itô’s formula for stochastic differentials 253
16.5 Itô’s formula for Brownian motion in Rd 256
16.6 Tanaka’s formula and local time 257
17 Applications of Itô’s formula 262
17.1 Doléans-Dade exponentials 262
17.2 Lévy’s characterization of Brownian motion 267
17.3 Girsanov’s theorem 269
17.4 Martingale representation – 1 272
17.5 Martingale representation – 2 275
17.6 Martingales as time-changed Brownian motion 277
17.7 Burkholder-Davis-Gundy inequalities 280
18 Stochastic differential equations 286
18.1 The heuristics of SDEs 287
18.2 Some examples 288
18.3 Existence and uniqueness of solutions 294
18.4 Solutions as Markov processes 299
18.5 Localization procedures 300
18.6 Dependence on the initial values 303
19 On diffusions 312
19.1 Kolmogorov’s theory 314
19.2 Itô’s theory 320
20 Simulation of Brownian motion 326
20.1 Introduction 326
20.2 Normal distribution 331
20.3 Brownian motion 333
20.4 Multivariate Brownian motion 335
20.5 Stochastic differential equations 337
20.6 Monte Carlo method 342
Appendix 343
A.1 Kolmogorov’s existence theorem 343
A.2 A property of conditional expectations 347
A.3 From discrete to continuous time martingales 349
A.4 Stopping and sampling 355
A.4.1 Stopping times 355
A.4.2 Optional sampling 358
A.5 Remarks on Feller processes 362
A.6 The Doob-Meyer decomposition 364
A.7 BV functions and Riemann-Stieltjes integrals 370
A.7.1 Functions of bounded variation 370
A.7.2 The Riemann-Stieltjes Integral 371
A.8 Some tools from analysis 374
A.8.1 Gronwall’s lemma 374
A.8.2 Completeness of the Haar functions 375
A.8.3 A multinomial identity 376
Index 389