Rigorous definition Ito integral. Properties: martingale, Markov and strong Markov properties, Doob and Burkholder inequalities. Stochastic differential equations: local and global solutions under local Lipschitz assumptions. Applications to PDEs (Feynman-Kac formula) and Mathematical Finance (Black-Scholes formula)
Prerequisiti
Probability and measure theory, basic stochastic processes including Wiener and Poission processes
Metodi didattici
Oral lectures with Exercises (4+2 per week)
Verifica Apprendimento
Oral examination
Testi
Z. Brzeźniak, T. Zastawniak, Basic stochastic processes. A course through exercises. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 1999. x+225 pp. ISBN: 3-540-76175-6 60-01 P. Baldi, Stochastic calculus. An introduction through theory and exercises. Universitext. Springer, Cham, 2017. xiv+627 pp. ISBN: 978-3-319-62225-5
Contenuti
1. Rigorous definition Ito integral via step processes. 2. Ito isometry 3. Processes defined via Ito integral 4. Ito Lemma 5. Martingale properties of processes in 3. 6. Markov and strong Markov properties of processes in 3. 7. Doob and Burkholder inequalities for processes in 3. 8. Stochastic differential equations with globally Lipschitz coefficients 9. Stochastic differential equations with locally Lipschitz coefficient: local, local maximal and global soltions 10. Applications to PDEs (Feynman-Kac formula) 11. Applications to Mathematical Finance (Black-Scholes formula)
Lingua Insegnamento
English
Altre informazioni
Period: 9th of June 2025 - 4th of July 2025
In the "Universities in Colleges" guide - on page 35 - you can find a description of the course: https://www.calameo.com/read/005970760798e48dd160c?authid=GNJVwmi62eZX
