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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

General info

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

spaces.

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).

Syllabus.

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.

References

• 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.

ICMS

15 South College Street

Edinburgh

EH8 9AA

Tel : 0131 650 9816

Fax : 0131 651 4381

Mail : johanna.mcbryde@icms.org.uk