Università Roma Tre

Brownian motion (part I). Definition, properties and explicit construction of the Brownian motion. Markov property. Strong Markov property and reflection principle.
Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).

- Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf
- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf