Syllabus Stochastic calculus and applications

Corsi di Laurea Magistrali in Matematica A.A. 2024/25, Prof. L. Bertini.

• Gaussian processes. Brownian motion.
• Martingales and Doob's inequality [O]
• Convergence of probabilities on the space of continuous functions. Invariance principle. [0]
• Garsia Rademich Rumsey inequality. [O]
• Kologorov test for continuity of sample paths.
• Regularity of Brownian sample paths. Quadratic variation.
• Strong Markov property.
• Reflection principle and applications.
• Stochastic integrals and Ito's formula.
• Levy's theorem. [O]
• Stochastic equations. Ito's theory.
• Martingale problem for diffusion processes. [O]
• Brownian bridge as the solution to SDE. [O]
• Smoluchowski approximation. [O]
• McKean-Vlasov limit. [O]
• Local times of Brownian motion and Skorohood problem. [O]
• Markov processes.
• Feller semigroups and their generators (sketch).
• Dynkin's martingale.
• Solution to stochastic equations as Markov processes.
• Invariant measures for Markov processes. Case of diffusion processes. [O]
• Probabilistic proof of Liouville theorem.
• Probabilistic representation for elliptic problems.
• Regular and singular points for the Dirichelet problem. [O]
• Probabilistic representation for the Poisson equation. [O]
• Fenyman-Kac formula.
• Arcsin laws for the Brownian motion. [O]

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