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