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Advanced stochastic processes: Part II

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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.

  • Chapter 4. Stochastic differential equations

  1. Solutions to stochastic differential equations
  2. A martingale representation theorem
  3. Girsanov transformation

  • Chapter 5. Some related results

  1. Fourier transforms
  2. Convergence of positive measures
  3. A taste of ergodic theory
  4. Projective limits of probability distributions
  5. Uniform integrability
  6. Stochastic processes
  7. Markov processes
  8. The Doob-Meyer decomposition via Komlos theorem
  9. Subjects for further research and presentations

  • Chapter 6. Advanced stochastic processes: a summary of the lectures
  1. Introduction
  2. Brownian motion as a Gaussian process
  3. Brownian motion as a Markov process
  4. Brownian motion as a martingale
  5. Some relevant martingales
  6. Conditional expectation
  • 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.