corso di calcolo stocastico, prof. Mania

[Laura Sangalli <laura@dimat.unipv.it>]

UNIVERSITA' DEGLI STUDI DI PAVIA
DIPARTIMENTO DI MATEMATICA "F.CASORATI"

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Si comunica che il Professor Michael Mania (“A. Razmadze” Mathematical
Institute,
Georgian Academy of Sciences) terrà, nell'ambito delle attività del Dottorato 
di Ricerca in Matematica e Statistica dell'Università degli Studi di Pavia,
un corso dal
titolo:

A Short Course on Stochastic Calculus


Programma del corso:
*) Stochastic processes: basic notions.
*) Filtrations and stopping times.
*) 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.
*) Increasing processes and processes of finite variation. Stieltjes
integrals, change of variable formula, integration by parts.
*) Martingales and semimartingales. Optional sampling theorem. Convergence
and regularization theorems. Quadratic characteristics. Doob-Meyer
decomposition. Some fundamental inequalities.
*) 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. Brownian motion as a square
integrable martingale (Levy characterization).
*) Stochastic integrals with respect to Brownian motion.
*) The Ito rule.
*) Continuous martingales as time-changed Brownian motion.
*) Brownian martingales as stochastic integrals, integral representation
of Brownian functionals.
*) 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.


Bibliografia:
- Karatzas and S. E. Shreve (1991). Brownian motion and stochastic calculus.
2nd Edition, Springer, New York.
- R. S. Liptser and A. N. Shiryaev (2001). Statistics of random processes.
I. General Theory. 2nd Edition, Springer, Berlin.
- D. Revuz and M. Yor (1999). Continuous martingales and Brownian motion.
3rd Edition, Springer, Berlin.


Il corso ha la durata di 30 ore.
Le lezioni sarannno tenute presso il dipartimento di Matematica, dalle ore 14
alle ore 17 dei seguenti giorni:
Martedi' 19 Ottobre
Mercoledi' 20 Ottobre
Martedi' 26 Ottobre
Mercoledi' 27 Ottobre
Mercoledi' 17 Novembre
Giovedi' 18 Novembre
Mercoledi' 24 Novembre
Giovedi' 25 Novembre
Mercoledi' 1 dicembre
Giovedi' 2 dicembre


Tutti gli interessati sono cordialmente invitati a partecipare.
Per ragioni organizzative si prega di segnalare la propria presenza con una
mail all'indirizzo laura@dimat.unipv.it




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