Continuous Martingales and Brownian MotionSpringer Science & Business Media, 29 de jun. de 2013 - 536 páginas This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersec tioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with in dependent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be success fully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of para mount importance: stochastic calculus, the use ofwhich pervades the whole book and the powerful excursion theory, both of which are introduced in a self contained fashion and with a minimum of apparatus. They have made much easier the proofs of many results found in the epoch-making book of Itö and McKean: Diffusion Processes and their Sampie Paths, Springer (1965). |
Conteúdo
1 | |
14 | |
3 Canonical Processes and Gaussian Processes | 31 |
Notes and Comments | 46 |
Optional Stopping Theorem | 73 |
12 | 89 |
39 | 127 |
Predictable Processes | 161 |
Stochastic Integrals | 316 |
3 Functionals and Transformations of Diffusion Processes | 324 |
Notes and Comments | 335 |
Existence and Uniqueness in the Case of Lipschitz Coefficients | 348 |
The Case of Hölder Coefficients in Dimension One | 358 |
Notes and Comments | 369 |
2 Representation Theorem for Additive Functionals | 379 |
3 Ergodic Theorems for Additive Functionals | 392 |
Strong Markov Property | 165 |
Representation of Martingales | 168 |
Conformal Martingales and Planar Brownian Motion | 177 |
3 Brownian Martingales | 186 |
4 Integral Representations | 195 |
Notes and Comments | 202 |
The Local Time of Brownian Motion | 221 |
3 The ThreeDimensional Bessel Process | 232 |
4 First Order Calculus | 241 |
The Skorokhod Stopping Problem | 249 |
Notes and Comments | 256 |
2 Diffusions and Itô Processes | 271 |
3 Linear Continuous Markov Processes | 278 |
Time Reversal and Applications | 290 |
Notes and Comments | 299 |
Application of Girsanovs Theorem to the Study of Wieners Space | 313 |
Asymptotic Results for the Planar Brownian Motion | 400 |
Notes and Comments | 406 |
Bessel Processes and RayKnight Theorems | 409 |
2 RayKnight Theorems | 420 |
3 Bessel Bridges | 428 |
Notes and Comments | 434 |
2 The Excursion Process of Brownian Motion | 442 |
Excursions Straddling a Given Time | 450 |
Notes and Comments | 470 |
3 Itôs Formula and First Applications | 477 |
2 Asymptotic Behavior of Additive Functionals of Brownian Motion | 478 |
Asymptotic Properties of Planar Brownian Motion | 487 |
Notes and Comments | 497 |
4 Hausdorff Measures and Dimension | 503 |
527 | |
Termos e frases comuns
A₁ additive functional B₁ BES³ Bessel processes Brownian Bridge Brownian motion Chap compact compute constant continuous local martingale continuous semimartingale converges Corollary countable D₁ defined Definition denote density distribution equal equation equivalent excursion Exercise exists Feller process filtration F follows function f Gaussian Girsanov's theorem hence Hint independent inequality inf{t invariant Itô Itô's formula L₁ Lebesgue measure Lemma local martingale locally bounded M₁ Markov process Markov property Moreover notation o-algebra o-field P₁ positive Borel function predictable process probability measure probability space Proposition prove random variables reader real number Remark resp respect result right-continuous Sect semi-group semimartingale sequence solution standard linear BM stochastic integrals stopping strong Markov property T₁ Tanaka's formula time-change uniformly integrable uniqueness X₁ Y₁ Z₁ zero