B0319 - STOCHASTIC CALCULUS I

Learning outcomes

At the end of the course, the students know the fundamentals of the theory of stochastic processes in discrete and continuous time that naturally intervene in applications in Physics, Economics, Biology and Engineering.

Course contents

The course aims at providing the minimum basic knowledge to be able to undertake independently the study of advanced topics of stochastic analysis, such as the courses

http://www.hairer.org/Teaching.html

by Martin Hairer, Fields medalist in 2014.

The course contains an introduction to the theory of stochastic processes and stochastic differential equations that naturally intervene in applications in physics and economics, highlighting the link with the theory of elliptic-parabolic partial differential equations.

More details at the webpage of the course

Program
Introduction to stochastic processes. Law and finite-dimensional distributions. Gaussian processes. Equivalence of processes: equivalence in law, modifications, indistinguishability. Existence: Kolmogorov's extension theorem. The case of Gaussian processes. Review of conditional expectation. Filtrations and martingales. Doob's decomposition theorem: the discrete case. Transition law of a process. Markov process. Extended Markov property. Processes with independent increments, Markov processes, and martingales. Finite-dimensional distributions of a Markov process. Chapman-Kolmogorov equation. Existence. Poisson and Gaussian transition laws. Infinitesimal generator of a Markov process.

Poisson process: definition and main properties. Continuous stochastic processes. Kolmogorov continuity theorem: application to the Gaussian case. Brownian motion: definition and characterization as a Gaussian process. Markov and martingale properties.

Discrete stopping times. Optional sampling theorem. Doob's maximal inequalities. Upcrossing lemma. The continuous case: usual hypotheses. Optional sampling, maximal inequalities, and upcrossing lemma in the continuous case. Existence of cadlag modifications of martingales. BV functions.

Variation theory. Riemann-Stieltjes integral and deterministic Ito formula. Lebesgue-Stieltjes integral. Quadratic variation of Brownian motion. Continuous martingales with bounded variation. Quadratic variation process of a square-integrable continuous martingale. Covariance matrix process.

Multidimensional Brownian motion. Correlated Brownian motion. Construction of the stochastic integral with respect to Brownian motion. Progressively measurable processes. The case of simple processes: Ito isometry, martingale property of the integral, and quadratic variation of the stochastic integral. Stochastic integral in L^2. Local martingales. Stochastic integral in L^2_loc. Ito processes.

One-dimensional Ito formula for Brownian motion. Geometric Brownian motion as a solution of a linear stochastic differential equation. Brownian motion and the heat operator. Stochastic integral with respect to a (local) continuous martingale and an Ito process. One-dimensional Ito formula. Ito processes with deterministic coefficients. Multidimensional Ito processes and Ito formula. Examples and applications: Feynman-Kac formula for Brownian motion. Dirichlet problem for the Laplace operator and the heat operator.

The course is part of the University's teaching innovation project.

Readings/Bibliography

A. Pascucci, Probability Theory II, Springer (2024)

Teaching methods

Classroom lectures

Assessment methods

See Virtuale

Teaching tools

See the webpage of the course

Links to further information

https://1drv.ms/w/c/6522c228f5a947a1/ERHZvdSx5wVLkI6WH4xfHQgBe8TrvlmQm3TOX9oGvkX1qw?e=LzQUdn

Office hours

See the website of Andrea Pascucci

See the website of Stefano Pagliarani