Stochastic processes with independent increments and their applications

Objectifs
Processes with independent increments frequently appear in various fields, including insurance,
financial mathematics, ecology, physics, etc. This advanced course will cover the construction of
processes with independent increments, their properties, computer simulations, and applications to
finance and insurance. While prior knowledge of probability is desirable, the course will include
necessary refreshers.

Contenu
1. Examples: Brownian motion and Poisson process
2. Compound Poisson process
3. The Lévy-Khintchine representation for characteristic function of a Lévy process
4. Poisson point measures. The Lévy-Itô decomposition.
5. Important classes of processes with independent increments: subordinators, stable processes,
etc.
6. Ruin theory and the classical Cramér-Lundberg model
7. Simulation of processes with independent increments.
8. Stochastic integration and a generalization of the Black-Scholes equation