Teacher: Dario Gasbarra 

Scope: 5 cr

Type: Advanced studies

Teaching:

Topics:Kolmogorov extension theorem and stochastic processes, Kolmogorov continuity criterium, Paul Levy Construction of Brownian motion. Processes with  jumps: Poisson process and counting processes, random measures.  General theory in  continuous time, filtration, predictable σ-algebra, stopping times and predictable times. Stochastic integrals with respect to processes with locally finite variation. Continuous time martingales, existence of  right continuous modification,  predictable and dual predictable projections,  compensator and Doob-Meyer decomposition, locally square integrable martingales, quadratic and predictable variation. Stochastic integration with respect to continuous martingales,  Kunita-Watanabe inequality, Ito isometry and Ito integral, Ito formula and generalizations.  Burkholder Davis Gundy inequalities. Ito-Tanaka formula and local times. Change of measure and Girsanov theorem. Stochastic differential equations, weak and strong solutions. Partial differential equations and Feynman-Kac formula. Applications: stochastic filtering, stochastic control, option pricing in mathematical finance.

Prerequisites: Probability theory, stochastic analysis I