Martingale transform or discrete stochastic integral. Definition. Example. 

Martingales in continuous time. Definition and properties. 

Discussion of exercises on martingales. 

Filtration: definition.

Martingales in discrete time: definition, basic properties and example. The symmetric random walk as an example of martingale. 

Discussion of exercises about the computation of expected values and variances of random variables and about basic notions on stochastic processes. 

Informal introduction and short history of stochastic calculus.

Review of basic notions of stochastic processes: definition, finite dimensional distributions, Gaussian processes, Strongly stationary process, Processes with stationary increments, processes with independen increments, the Markov property.

Examples: the symmetric random walk and the Poisson process.  

Equivalent and undistinguishable processes. 

Concepts of continuity: Continuous processes, Continuity in probability and continuity in mean of order p. 

The Kolmogorov continuity criterion.