Martingales

26 Fevereiro 2018, 10:00 • João Guerra

Martingales in discrete time: examples.

A bet system example. 

A financial example: the binomial model.

Martingales in continuous time: definition and properties.

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Conditional expectation and martingales in discrete time

22 Fevereiro 2018, 10:00 • João Guerra

Conditional expectation: properties and interpretation as an orthogonal projection of a random variable in a L^2 space. 

Exercises. 

Martingales in discrete time. Definition, basic properties and examples. 

Previsible processes, martingale transform or discrete time stochastic integral. Example. 

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Stochastic processes - Review of main concepts

19 Fevereiro 2018, 10:00 • João Guerra

Finite dimensional distributions for stochastic processes. 

Gaussian processes. Example: White noise process. 

Stationary processes.

Pprocesses with independent increments and stationary increments. Example: Poisson process. 

Equivalent and undistinguishable stochastic processes. Example. 

Different concepts of continuity for stochastic processes. Example. 

The Kolmogorov continuity criterion. 

The conditional expectation: definition and main properties of conditional expectation. 

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Introduction

15 Fevereiro 2018, 10:00 • João Guerra

Presentation. 

Syllabus, bibliography and assessment. 

Introduction to stochastic calculus. Very brief history of stochastic calculus. 

Stochastic process: definition and examples. 

State of a process, state of spaces, trajectories and samples paths of processes. 

Markov process: definition and examples. 

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