Stochastic ProcessesThis comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black–Scholes formula for the pricing of derivatives in financial mathematics, the Kalman–Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature. |
Contents
| 1 | |
| 6 | |
3 Martingales | 13 |
4 Markov properties of Brownian motion | 25 |
5 The Poisson process | 32 |
6 Construction of Brownian motion | 36 |
7 Path properties of Brownian motion | 43 |
8 The continuity of paths | 49 |
26 The RayKnight theorems | 209 |
27 Brownian excursions | 214 |
28 Financial mathematics | 218 |
29 Filtering | 229 |
30 Convergence of probability measures | 237 |
31 Skorokhod representation | 244 |
32 The space C01 | 247 |
33 Gaussian processes | 251 |
9 Continuous semimartingales | 54 |
10 Stochastic integrals | 64 |
11 Itos formula | 71 |
12 Some applications of Itos formula | 77 |
13 The Girsanov theorem | 89 |
14 Local times | 94 |
15 Skorokhod embedding | 100 |
16 The general theory of processes | 111 |
17 Processes with jumps | 130 |
18 Poisson point processes | 147 |
19 Framework for Markov processes | 152 |
20 Markov properties | 160 |
21 Applications of the Markov properties | 167 |
22 Transformations of Markov processes | 177 |
23 Optimal stopping | 184 |
24 Stochastic differential equations | 192 |
25 Weak solutions of SDEs | 204 |
34 The space D01 | 259 |
35 Applications of weak convergence | 269 |
36 Semigroups | 279 |
37 Infinitesimal generators | 286 |
38 Dirichlet forms | 302 |
39 Markov processes and SDEs | 312 |
40 Solving partial differential equations | 319 |
41 Onedimensional diffusions | 326 |
42 Levy processes | 339 |
Appendix A Basic probability | 348 |
Appendix B Some results from analysis | 378 |
Appendix C Regular conditional probabilities | 380 |
Appendix D Kolmogorov extension theorem | 382 |
| 385 | |
| 387 | |
Common terms and phrases
A₁ B₁ Borel measurable Borel subset Brownian motion C₁ characteristic function continuous function continuous paths continuous with left converges weakly Corollary d-dimensional Brownian motion define dominated convergence equal Exercise exists F₁ finite finite-dimensional distributions hence independent inequality inf{t Itô's formula left limits Lemma Let F Lévy processes lim sup linear local martingale M₁ Markov process Markov property mean zero metric space minimal augmented filtration N₁ non-negative normal random variable null set o-field one-dimensional Brownian motion optional P₁ Poisson process predictable stopping probability measure Proof Let Proposition prove right continuous S₁ satisfying the usual semigroup semimartingale sequence solution square integrable martingale stochastic integral stochastic process strong Markov process strong Markov property submartingale T₁ Theorem uniformly integrable uniqueness usual conditions W₁ X₁ Y₁ Z₁ Zn(t