Exercises. Stable processes

18 Outubro 2018, 10:00 • João Guerra

Discussion of some exercises on Lévy measures.

Stable processes: examples and characteristic exponents. 

The self-similarity property for strictly stable Lévy processes. 

Subordinators: definition, characteristic exponent and simple examples.

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Exercises and the characteristic exponents of Lévy processes

16 Outubro 2018, 10:00 • João Guerra

Discussion of some exercises. 

Lévy processes have associated infinitely divisible distributions.

The characteristic exponent and the characteristic triple associated to a Lévy process.

Examples of characteristic triples and exponents for: the Brownian motion, contibuous Gaussian processes, Poisson PRocess, Compound Poisson process, Sum of a  Gaussian process and a Compound Poisson (Jump-diffusion process). 

The interlacing-representation for a jump-diffusion process (sum of a Gaussian and a Compound Poisson). 

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Stable Distributions

11 Outubro 2018, 10:00 • João Guerra

Stable distributions and stable random variables: definition, connection with the general central limit theorem and examples. 

The Lévy measure and the characteristic functions of Stable distributions. 

The fat-tails of stable distributions with alpha<2.  

 

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Infinitely divisible distributions

9 Outubro 2018, 10:00 • João Guerra

The characteristic function associated to a probability distribution: definition and basic properties.

Infinitely divisible distribution: definition, properties and examples. 

The Lévy-Khintchine formula. Proof and examples for the following cases: Gaussian, Poisson, Compound Poisson distribution. 

Lévy measure: examples of Lévy measures. 

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General introduction to Lévy processes.

4 Outubro 2018, 10:00 • João Guerra

The characteristic function of a jump-diffusion process.

The infinitely divisible distributions and the Lévy-Khintchine formula. 

The Lévy measure: examples of Lévy measures. 

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