Mercoledí 2 marzo, alle ore 10.30, in aula Buzano, avrà inizio il corso di dottorato di "Calcolo Stocastico". Docenti: M. Mania, G. Pistone, E. Riccomagno e M. Santacroce. Tutti gli interessati sono invitati a partecipare.

Calendario delle lezioni. (1): 8 marzo ore 14:30 aula 5B; (2): 15 marzo ore 14:30 aula 7D; (3): 22 marzo ore 14:30 aula 5B; (4): 5 aprile ore 14:30 aula 5B; (5): 12 aprile ore 14:30 aula 5B; (6): 19 aprile ore 14:30 aula Buzano (DIMAT); (7): 26 aprile ore 14:30 aula 5B; (8): 3 maggio ore 14:30 aula 5B; (9): 10 maggio ore 14:30 aula 5B; (10): 17 maggio ore 14:30 aula 5B.

Programma del corso (30 ore)

Settimana (1) *) Stochastic processes: basic notions. *) Filtrations and stopping times.

Settimana (2) *) Brownian motion. Construction of Brownian motion. Coordinate process and the Wiener measure. Properties of Brownian paths (continuity of paths, quadratic variation of Brownian motion, non-differentiability of Brownian paths). Continuity of Brownian filtration.

Settimana (3) *) Martingales and semimartingales. Optional sampling theorem. Convergence and regularization theorems. Quadratic characteristics. Doob-Meyer decomposition. Some fundamental inequalities.

Settimane (4),(5),(6) *) Markov processes and Markov families. Equivalent formulations of the Markov property. The strong Markov property. Processes with independent increments. Examples: Brownian motion, Poisson process, processes in discrete time. *) Brownian motion as a square integrable martingale (Levy characterization). *) Stochastic integrals with respect to Brownian motion and semimartingales. *) The Ito rule. *) Continuous martingales as time-changed Brownian motion. *) Brownian martingales as stochastic integrals, integral representation of Brownian functionals.

Settimane (7)-(10) *) Girsanov's theorem. Exponential martingales. Novikov's and Kazamaki's conditions. *) Stochastic differential equations. Strong solutions, strong uniqueness. Ito's theorem on existence and uniqueness of a strong solution. Comparison theorem. Weak solution, weak uniqueness. Weak solution by means of Girsanov's theorem. Jamada-Watanabe theorems on weak and strong solutions. Weak solutions and martingale problems. Well-posedness and the strong Markov property. *) Connection with partial differential equations. The Dirichlet problem. Harmonic functions and the mean-value property. The Cauchy problem and the Feynman-Kac representation. *) Backward stochastic differential equations. *) Some applications to optimal stopping, stochastic control and mathematical finance.

Riferimenti bibliografici: 1) Probabilities and Potential, C. Dellacherie and P. Meyer, 1978; 2) Capacities et Processus stochastiques, C. Dellacherie, 1972; 3) Stochastic Integration and Differential Equations, P. Protter, 1990; 4) Brownian Motion and Stochastic Calculus, I. Karatzas and S. E. Shreve, 1991.