Itô Stochastic Integrals

1 Outubro 2019, 11:00 • João Guerra

Definition of stochastic integral: the Riemann-Stieltjes approach does not work. 

Probabilistic approach for stochastic integrals of simples processes: definition and examples. 

Properties of the stochastic integral: the zero-mean, linearity and Itô isometry properties. 

Stochastic integral for adapted processes in L^2. Definition and example. 

 

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Martingales

27 Setembro 2019, 12:00 • João Guerra

Recap of conditional expectation properties. 

Filtrations and adapted processes. 

Martingales in discrete time. Definition, properties and examples. 

Martingales in continuous time. Definition, properties and examples. 

Some functions of the Brownian motion as examples of martingales in continuous time. 

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Brownian motion and Geometric Brownian motion

25 Setembro 2019, 11:30 • João Guerra

Definition sand properties of Brownian motion. 

Examples and exercises. 

The correlated Brownian motion. 

The geometric Brownian motion. Properties. 

Examples of R scripts for simulating trajectories of Brownian motion and geometric Brownian motion. 

Basic idea of the martingale concept in discrete time. 

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Introduction

20 Setembro 2019, 12:00 • João Guerra

Presentation of the course programme, bibliography and assessment.

Brief history of stochastic calculus  and Brownian motion.

Definition of stochastic process, standard Brownian motion and Brownian motion with drift.

Some basic properties of Brownian motion.

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