Properties of the Ito Stochastic integral as a process

17 Março 2017, 09:00 • João Guerra

Information about the AABCS certification of ISEG and the evaluation of the course on the learning goal of "Analytical Thinking". Discussion of the evaluation parameters and of the evaluation file for each student. 

Exercises about Stochastic integrals.

The indefinite Stochastic integral as a Stochastic process.  Definition and main properties: integral as a sum of two integrals, factorization, the martingale property, the continuity of trajectories and the maximal inequality.

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Stochastic integrals

15 Março 2017, 09:00 • João Guerra

Stochastic integrals for processes in L^2_{a,T}. 

Lemma about the density of simple processes in the space  L^2_{a,T}. 

Stochastic integral of a process L^2_{a,T} as the limit (in L^2) of a sequence of stochastic integrals of approximating simple processes. 

Example. 

The heuristic arguments for (dB_{t})^2=dt.

 

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Stochastic integrals

10 Março 2017, 09:00 • João Guerra

Exercises. Discussion of some exercises. 

Stochastic integrals. Motivation and introduction. 

Riemann-Stieltjes integrals. 

Stochastic integrals for simple processes. Definition and properties. The isometry property. 

Example. 

 

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Brownian motion

8 Março 2017, 09:00 • João Guerra

Brownian motion: definition and basic properties.

The continuity and Holder-continuity of Brownian motion trajectories.

The total variation and quadratic variation of Brownian motion.

The self-similar property of Brownian motion.

Processes related to Brownian motion.

Martingales related to Brownian motion.

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Martingales and Brownian motion

3 Março 2017, 09:00 • João Guerra

Martingales in discrete time. 

A financial example: the binomial model. 

Martingales in continuous time. 

The maximal inequality or martingale inequality.

The Brownian motion: definition and continuity of the trajectories. 

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