2. Brownian motion as a Gaussian process

Chapters in this book

1. Frontmatter I
2. Also of Interest II
3. Preface to the second edition V
4. Preface VI
5. Contents IX
6. Dependence chart XIII
7. Index of notation XV
8. 1. Robert Brown’s new thing 1
9. 2. Brownian motion as a Gaussian process 7
10. 3. Constructions of Brownian motion 20
11. 4. The canonical model 38
12. 5. Brownian motion as a martingale 46
13. 6. Brownian motion as a Markov process 59
14. 7. Brownian motion and transition semigroups 81
15. 8. The PDE connection 114
16. 9. The variation of Brownian paths 136
17. 10. Regularity of Brownian paths 150
18. 11. Brownian motion as a random fractal 160
19. 12. The growth of Brownian paths 181
20. 13. Strassen’s functional law of the iterated logarithm 191
21. 14. Skorokhod representation 211
22. 15. Stochastic integrals: L²-Theory 220
23. 16. Stochastic integrals: beyond L²_T 242
24. 17 Itô’s formula 248
25. 18. Applications of Itô’s formula 268
26. 19. Stochastic differential equations 290
27. 20. Stratonovich’s stochastic calculus 322
28. 21. On diffusions 332
29. 22. Simulation of Brownian motion by Björn Böttcher 345
30. A. Appendix 359
31. Bibliography 393
32. Index 403