Advanced stochastic processes: Part II

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247 pages
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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A propos de l'auteur

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,...


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