Martingales and Lévy processes. Poisson random measures and Poisson integrals

19 Outubro 2020, 10:00 • João Guerra

MArtingales and Lévy processes. Examples. 

Poisson random measures. Examples. 

Poisson integrals: definition and properties. Examples. 

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Exercises and subordinators

14 Outubro 2020, 11:30 • João Guerra

Exercises on Poisson process and on the Cauchy distribution. 

Subordinators: examples. The alpha-stable subordinators, the Lévy subordinator, the gamma subordinator. 

Subordinators and time change of Lévy processes. Examples: time change of Brownian motion by a alpha-stable subordinator; the variance-gamma process. 

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Lévy processes and subordinators - examples

12 Outubro 2020, 10:00 • João Guerra

Lévy processes - definition and examples. 

Examples: multidimensional Brownian motion, Multidimensional Brownian motion with drift, Poisson process, Compound Poisson process, Stable Lévy processes. 

Rotationally invariant stable Lévy processes and self-similar processes. 

Discussion of exercise about the Poisson process. 

Subordinators. Definition, basic properties and simple examples. 

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Stable distributions

30 Setembro 2020, 11:30 • João Guerra

Discussion of exercises on Lévy measures, infinitely divisible distributions and stable distributions. 

Stable distributions and the central limit theorem.

Stable distributions: definition, examples and basic properties. 

The Stability index alpha. 

The Cauchy distribution (alpha=1) and the Lévy distribution (alpha=1/2). 

The "heavy tails" of stable distributions with alpha<2. 

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Infinitely divisible distributions and Lévy-Khintchine formula

28 Setembro 2020, 10:00 • João Guerra

The compound Poisson distribution: derivation of the characteristic function and proof of infinitely divisibility. 

Lévy measure: definition and some basic properties.  Exercise. 

The Lévy-Khintchine formula. Proof. 

The central limit theorem and stable distributions. 

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