20402101 CP430 - STOCHASTIC CALCULUS in Mathematics LM-40 N0 CAPUTO PIETRO
(syllabus)
Brownian motion I
Gaussian multivariate distribution. Processes with stationary and independent increments.
Definition and continuity properties of Brownian motion. Non differentiability of trajectories.
Markov and Strong Markov property.
Brownian motion II
Brownian motion in higher dimensions. Harmonic functions and the Dirichlet problem in regular domains.
The Poisson problem. Iterated logarithm law. Skorohod embedding.
Donsker's invariance principle. Applications: arcsine law and the maximum of random walks.
Stochastic integrals
The Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. It\^o formula and applications. Local time and the Tanaka formula.
Stochastic differential equations
The linear case. Existence and uniqueness theorem for stochastic differential equations.
Diffusion process in the limit of zero noise. Infinitesimal generator and partial differential equations.
Feynman-Kac formula and applications.
(reference books)
P. M\"orters and Y. Peres
Brownian Motion
Cambridge University Press
2010
L.C. Evans
An Introduction to Stochastic Differential Equations
AMS bookstore
2013
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Università Roma Tre