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.
Description
Content
- Chapter 4. Stochastic differential equations
- Solutions to stochastic differential equations
- A martingale representation theorem
- Girsanov transformation
- Chapter 5. Some related results
- Fourier transforms
- Convergence of positive measures
- A taste of ergodic theory
- Projective limits of probability distributions
- Uniform integrability
- Stochastic processes
- Markov processes
- The Doob-Meyer decomposition via Komlos theorem
- Subjects for further research and presentations
- Chapter 6. Advanced stochastic processes: a summary of the lectures
- Introduction
- Brownian motion as a Gaussian process
- Brownian motion as a Markov process
- Brownian motion as a martingale
- Some relevant martingales
- Conditional expectation
- Bibliography
- Index