Stochastic Exponential, Exponential martingales and markets

9 Novembro 2017, 10:00 • João Guerra

Stochastic exponential and ordinary exponential of Lévy-type stochastic integrals and Lévy processes. Examples. 

Exponential martingales. 

The Girsanov theorem. 

Complete and incomplete markets. Fundamental theorems of asset pricing. 

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Exercises and discussion of Group assignments

7 Novembro 2017, 10:00 • João Guerra

Exercises: Applications of Itô formula for Lévy-type stochastic integrals. 

The Geometric Lévy process. 

Discussion of the Groups assignments. 

 

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The Itô formula

2 Novembro 2017, 10:00 • João Guerra

The groups assignments: discussion. 

The Lévy-type stochastic integrals. The Lévy stochastic integrals. Example: The Ornstein-Uhlenbeck process. 

Itô formula for jump processes, for Itô processes with respect to Brownian motion and for Lévy-type stochastic integrals. Example. 

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Stochastic integrals with respect to Lévy processes

31 Outubro 2017, 10:00 • João Guerra

Stochastic integrals with respect to Lévy processes: 

stochastic integrals of simple processes. Properties and example. 

Lévy-type stochastic integrals (generalize the concept of Itô process to include jumps). 

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Poisson integration - stochastic integrals with respect to jumps

26 Outubro 2017, 10:00 • João Guerra

Cádlág functions and trajectories. 

The jumps associated to a Lévy process and the Poisson random measure (counting measure for the jumps of the process).

The Poisson integral (or stochastic integral with respect to jumps) of a deterministic function. Definition and basic properties. 

Examples and exercises. 

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