Brownian Motion : : An Introduction to Stochastic Processes / / René L. Schilling, Lothar Partzsch.

• Superior document: Title is part of eBook package: De Gruyter DGBA Backlist Complete English Language 2000-2014 PART1
• VerfasserIn: Schilling, René L.,; Partzsch, Lothar,
• MitwirkendeR: Böttcher, Björn,
• TeilnehmendeR: Böttcher, Björn.
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2012]; ©2012
• Year of Publication: 2012
• Language: English
• Series: De Gruyter Textbook
• Physical Description: 1 online resource (380 p.)

Brownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance. Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors’ aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs. This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion.

Table of Contents

Frontmatter -- Preface -- Contents -- Dependence chart -- Index of notation -- Chapter 1. Robert Brown’s new thing -- Chapter 2. Brownian motion as a Gaussian process -- Chapter 3. Constructions of Brownian motion -- Chapter 4. The canonical model -- Chapter 5. Brownian motion as a martingale -- Chapter 6. Brownian motion as a Markov process -- Chapter 7. Brownian motion and transition semigroups -- Chapter 8. The PDE connection -- Chapter 9. The variation of Brownian paths -- Chapter 10. Regularity of Brownian paths -- Chapter 11. The growth of Brownian paths -- Chapter 12. Strassen’s Functional Law of the Iterated Logarithm -- Chapter 13. Skorokhod representation -- Chapter 14. Stochastic integrals: L2-Theory -- Chapter 15. Stochastic integrals: beyond L2T -- Chapter 16. Itô’s formula -- Chapter 17. Applications of Itô’s formula -- Chapter 18. Stochastic differential equations -- Chapter 19. On diffusions -- Chapter 20. Simulation of Brownian motion -- Appendix -- Index