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This course is run by MIGSAA. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module.
A Fourier Based Approach of (stochastic) Integration and applications by Goncalo dos Reis
In 1961, Ciesielski established a remarkable isomorphism of spaces of Hölder continuous
functions and Banach spaces of real valued sequences. The isomorphism can be established
along Fourier type expansions of Hölder continuous functions using the Haar-Schauder
wavelet. We will start with Schauder representations for a pathwise approach of the integral of
one function with respect to another one, using Ciesielski's isomorphism. We cover the
paradigm of Young integral and the rough paths integral of T. Lyons. Our approach allows
understanding this more involved theory of integration, purely from an analytical perspective
using Paley-Littlewood decompositions of distributions, and Bony paraproducts in Besov
We apply the theory within a probabilistic framework and express Brownian motion in this
language to derive several of its properties. Moreover, the 2nd part of the course focuses on
themes relating to the applications of the theory developed in the 1st part and adapts to the
audience. We cover the theory of Large Deviations Principles and some of its applications to
in the space of continuous functions, and we apply it to solve stochastic differential equations
in a pathwise manner.
Credits: 15 point course.
Prerequisites: SMSTC Probability 2 - Stochastic Processes (or “better”)
Course duration: 12 – 15 hours (8-10 lectures of 1h30)
Engagement with the course from students:
• The lectures notes is a mix of several sources and students will be asked to type some
parts of the notes.
• Students will present some material in class, for example, proofs of some results or
solve useful examples.
• Roughly 2/3 of the course is delivered through standard lectures while the remaining
1/3 is based on active student participation (reading-group style).
The lectures notes are made of a mix of several sources. A list of possible topics is as follows
(although we would like to stress that the course is designed to take into account the interest of
the audience, therefore the list below is to be intended as a guideline).
• Ciesielski’s isomorphismand Fourier type expansions of (rough) Hölder continuous functions
by means of the Haar-Schauder wavelet; isomorphism; function spaces; sequence spaces;
• General theory of Large deviations principles (LDP): concepts & basic properties;
constructions of LDP from exponential rates of elementary sets; Contraction principle.
• Large Deviations for Brownian motion: LDPs Gaussian random variables; LDP for Brownian
motion in Holder spaces; Freidling-Wentzel theory
• If time allows: we discuss further the foundations of (controlled) rough-paths theory and
discuss pathwise stochastic integration and Functional Ito Calculus.
• Notes by Goncalo dos Reis
• Herrmann, Samuel, et al. Stochastic Resonance: A Mathematical Approach in the Small
Noise Limit. Vol. 194. American Mathematical Soc., 2013.
• M. Gubinelli, P. Imkeller, and N. Perkowski, A Fourier analytic approach to pathwise
stochastic integration, EJP (2016).
• P. Mörters and Y. Peres, Brownian motion, Cambridge Series in Statistical and
Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010
• Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to
regularity structures. Springer, 2014.