Stochastic integral of L^2 processes

18 Março 2019, 10:00 • João Guerra

Stochastic integral for L^2 processes: definition and approximating sequence of simple processes stochastic integrals. 

Example of calculation of a stochastic integral by definition. 

(dB)^2=dt. Motivation for Itô formula

Properties of the indefinite stochastic integral: additivity, martingale property, continuity, maximal inequality. 

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Stochastic integral for simple processes

14 Março 2019, 10:00 • João Guerra

An exercise about martingales. 

Simple processes. Examples. 

Stochastic integral for simple processes. Definition and properties. Example.

 

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Exercises, Brownian motion properties

11 Março 2019, 10:00 • João Guerra

Exercises and examples. 

Brownian motion martingales. Examples. 

The total variation and the quadratic variation of Brownian motion.

The stochastic integral definition motivation: the Riemann-Stiletjes approach does not work. 

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Martingales in continuous time and Brownian motion

7 Março 2019, 10:00 • João Guerra

Martingales in continuous time: definition and properties.

The Martingale Doob inequality.

The Brownian motion: definition, the covariance function, the continuity of trajectories, the degree of Holder continuity of trajectories, infinite total variation and finite quadratic variation, the non differentiability of trajectories at any point and the self-similar property.

Processes related with Brownian motion: Brownian motion with drift, geometric Brownian motion and Brownian bridge. 

 

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Martingale transform or discrete stochastic integral

28 Fevereiro 2019, 10:00 • João Guerra

Martingale transform or discrete stochastic integral: definition, properties, examples. 

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