The course presents the basic material for stochastic calculus. It will include the following chapters.
1. Motivations: stochastic modeling, probabilistic representations of linear PDEs, stochastic control, filtering, mathematical finance.
2. Stochastic processes in continuous time: Gaussian processes, Brownian motion, (local) martingales, semimartingales, Itˆo processes.
3. Stochastic integrals: forward and Ito integrals.
4. Ito and chain rule formulae, a first approach to stochastic differential equations.
5. Girsanov formulae. Novikov and Benˆes coondition. Predictable representation of Brownian martingales.
6. Stochastic differential equations with Lipschitz coefficients. Markov flows.