Francesco Caravenna

Pavia-Bicocca-Indam Ph.D. program in Mathematics

PhD course, Spring 2017

Rough Paths and Stochastic Differential Equations

Francesco Caravenna and Gianmario Tessitore [15h + 15h]

Presentation

This course is an introduction to rough paths, a modern theory which provides an analytic counterpart to the probabilistic theory of stochastic integration.

Rough paths allow to define a notion of integral with respect to the derivative of a Holder function, generalizing the usual Riemann-Steltjes integral. A key motivation of rough paths is to shed new light on the stochastic integration with respect to Brownian motion and on the corresponding stochastic differential equations.

The course is devoted to students interested in analysis and/or probability. The only prerequisite is a basic knowledge of Hölder functions and Brownian motion.

Interested students are invited to contact us via email.

References

• Peter Friz and Martin Hairer
  A Course on Rough Paths
  Springer (2014)
  (downloadable here)
• Massimiliano Gubinelli
  Lecture sheets on "Rough paths and controlled paths"
  (downloadable from this page)

Lectures Log

• Lecture 1 (2 Feb). 0 - Introduction and motivation.
  I - Hölder spaces and their properties. Extension of linear operators.
• Lecture 2 (8 Feb). II - The Young integral. Extension by continuity from smooth functions. Characterizing property: I(t) - I(s) = f(s) (g(t)- g(s)) + o(|t-s|). Properties of the integral (chain rule, integration by parts, associativity).
  III - The Sewing Map. General constructon of the integral: the sewing map.
• Lecture 3 (9 Feb). Abstract characterization of the sewing map.
  IV - Young Differential Equations (YDE). A priori estimate. Uniform convergence with bounded norm in Hölder spaces. Global existence for YDE using Ascoli-Arzelà. (Local existence and) Uniqueness for YDE using contractions.
• Lecture 4 (13 Feb). Lipschitz property of the composition operator. Global existence and uniqueness for YDE using contractions.
  V - Beyond Young. Counter-examples to a naive extension of the Young integral when α + β < 1. Generalized notion of integral. The paraintegral: preparatory lemma; beginning of the proof.
• Lecture 5 (28 Feb). The paraintegral: completion of the proof.
  VI - Stochastic Integrals. A generalization of the Kolmogorov continuity criterion. The Ito integral with respect to Brownian motion, when the integrand is α-Hölder, is a generalized integral, i.e. the remainder is O(|t-s|^α+β) for every β < 1/2.
• Lecture 6 (6 Mar). VII - Rough Paths. Generalized integrals and rough paths. Basic properties. (Ito and Stratonovich) Brownian motion as a rough path. Convergence of rough paths. Geometric rough paths.
• Lecture 7 (7 Mar). General (non geometric) rough paths. Pure area rough paths. Stochastic integration by parts and quadratic covariation of Brownian motion.
  VIII - Controlled Paths. Canonical choice of the generalized integral, when the integrand is "controlled". Controlled paths: definition and convergence.
• Lecture 8 (14 Mar). Rough integral: integral of a controlled path with respect to a rough path. Relation with the stochastic integral.
  IX - Rough Differential Equations. Introduction to RDEs. A priori estimate.
• Lecture 9 (21 Mar). Formulation of RDEs. Composition of controlled paths with smooth functions. Existence and uniqueness of solutions for RDEs via contraction.
• Lecture 10 (11 Apr). [Seminar by Simone Dovetta: Ito formula for rough paths.]
  X - The Ito-Lyons Map. Pseudo-distance between controlled paths (with possibly different rough paths). The composition and rough integral operators are locally Lipschitz. The Ito-Lyons map: continuity of solutions of RDEs with respect to initial data and rough path. Example: comparison between Brownian SDEs and RDEs.

Where and when

Lectures will take place in room 3014, 3rd floor of the U5 building (map)

Department of Mathematics
University of Milano-Bicocca
via Cozzi 55, Milano

Schedule of the lectures:

1. Thursday 2 February 14:00-17:00
2. Wednesday 8 February 11:00-12:30 + 14:00-15:30
3. Thursday 9 February 14:00-17:00
4. Monday 13 February 11:00-12:30 + 14:00-15:30
5. Tuesday 28 February 11:00-12:30 + 14:00-15:30
6. Thursday 2 March 11:00-12:30 + 14:00-15:30
   Monday 6 March 11:00-12:30 + 13:30-15:00
7. Tuesday 7 March 11:00-12:30 + 14:00-15:30
8. Tuesday 14 March 11:00-12:30 + 14:00-15:30
9. Tuesday 21 March 11:00-12:30 + 14:00-15:30
10. Tuesday 28 March 11:00-12:30 + 14:00-15:30
    Tuesday 11 April 11:30-12:30 (seminar by Simone Dovetta) + 14:00-16:00

What else?

For more information, do not hesitate to contact us via email.