Learning objectives
Give a good theoretical understanding of stochastic differential equations, such that the student may then deal with them in a qualitative and quantitative way, both in the field of pure research and in industrial applications (for example in finance and in the modeling of noisy systems).
Prerequisites
Measure spaces, probability spaces, Borel-Cantelli lemmas, random variables, mathematical expectation, modes of convergence for random variables, L^p spaces
Course unit content
Stochastic processes, Brownian motion, stochastic integral, martingales, stochastic differential equations
Full programme
"Stochastic processes, Gaussian vectors, law of a process, Gaussian processes, modifications, equivalent processes, Kolmogorov's extension theorem, Doob's lemma, independence;
Brownian motion, Kolmogorov's regularity theorem, existence and uniqueness of BM, elementary properties and transformations, quadratic variation, BM is not BV, Hölder property, Stieljes integral and extensions, filtrations and adapted processes;
conditional expectation, existence and uniqueness, elementary properties;
progressively measurable processes, simple processes and their density in M^2, stochastic integral for M^2 processes, elementary properties, Itô isometry;
discrete-time and continuous-time martingales, stopping times, Doob's optional stopping theorem, Doob's optional sampling theorem, continuity of the stochastic integral process;
stochastic integral for M^2_loc processes;
stochastic differential equations."
Bibliography
Francesco Caravenna - Moto browniano e analisi stocastica
Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion
Ioannis Karatzas, Steven E. Shreve - Brownian Motion and Stochastic Calculus
David Williams - Probability with Martingales
Paolo Baldi - Equazioni differenziali stocastiche e applicazioni
Bernt Øksendal - Stochastic Differential Equations: An Introduction with Applications"
Teaching methods
Traditional classes (42 hours)
Assessment methods and criteria
Oral exam
Other information
On the website lea.unipr.it the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC
