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Advanced stochastic processes: Part I

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Language:  English
In this book the following topics are treated thoroughly: Brownian motion as a Gaussian process, Brownian motion as a Markov process...
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In this book, which is basically self-contained, the following topics are treated thoroughly: Brownian motion as a Gaussian process, Brownian motion as a Markov process, and Brownian motion as a martingale. Brownian motion can also be considered as a functional limit of symmetric random walks, which is, to some extent, also discussed. Related topics which are treated include Markov chains, renewal theory, the martingale problem, Itô calculus, cylindrical measures, and ergodic theory. Convergence of measures, stochastic differential equations, Feynman-Kac semigroups, and the Doob-Meyer decomposition theorem theorem are discussed in the second part of the book.

  • Preface
  • Chapter 1. Stochastic processes: prerequisites
  1. Conditional expectation
  2. Lemma of Borel-Cantelli
  3. Stochastic processes and projective systems of measures
  4. A definition of Brownian motion
  5. Martingales and related processes
  • Chapter 2. Renewal theory and Markov chains
  1. Renewal theory
  2. Some additional comments on Markov processes
  3. More on Brownian motion
  4. Gaussian vectors
  5. Radon-Nikodym Theorem
  6. Some martingales
  • Chapter 3. An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales
  1. Gaussian processes
  2. Brownian motion and related processes
  3. Some results on Markov processes, on Feller semigroups and on the martingale problem
  4. Martingales, submartingales, supermartingales and semimartingales
  5. Regularity properties of stochastic processes
  6. Stochastic integrals, Itˆo’s formula
  7. Black-Scholes model
  8. An Ornstein-Uhlenbeck process in higher dimensions
  9. A version of Fernique’s theorem
  10. Miscellaneous
  • Bibliography
  • Index

About the Author

Jan A. Van Casteren

Since 2009 the author is retired from the University of Antwerp. Until the present day his teaching duties include a course on ``Partial Differential Equations and Operators’’ and one on ``Advanced Stochastic Processes’’. In the sixties the author was a student at the Catholic University of Nijmegen, Netherlands (nowadays Radboud University), and he earned his Ph.D. from the University of Hawaii, USA, (1971). Since 1972 he has been a member of the academic staff of the University of Antwerp, Department of Mathematics and Computer Science, Belgium. Most of his professional life he has been teaching courses in analysis and stochastic processes. His research lies in the area of stochastic analysis. A recent book authored by him is Markov Processes, Feller Semigroups and Evolution Equations, published by WSPC, Singapore, 2011, of about 800 pages.