Sumários

Stochastic integrals and Ito formula

17 Março 2022, 13:30 • João Guerra

Properties of the indefinite stochastic integral.

Exercise on stochastic integrals.

The on-dimensional Itô formula. Motivation and examples.

Stochastic integrals

11 Março 2022, 10:00 • João Guerra

Stochastic integrals for simple processes and for adapted and squared-integrable processes.

Examples.

Properties. The Itô isometry property. Example.

Example of the computation of a stochastic integral of an adapted process by definition.

Exercises on Brownian motion and stochastic integrals of simple processes

10 Março 2022, 13:30 • João Guerra

Discussion of exercises on Brownian motion.

Stochastic integrals: motivation and stochastic integrals of simples processes. Example.

Brownian motion: definition, continuity of trajectories, total variation and quadratic variation, the self-similarity property and processes related to Brownian motion. Martingales related to Brownian motion: examples.

Martingales

3 Março 2022, 13:30 • João Guerra

Martingales in discrete time. Example from finance: the binomial model.

Exercises.

Martingales in continuous time: definition and basic properties.

Brownian motion: definition and continuity of sample paths.