Francesco Caravenna

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

PhD Course, Fall 2024

Stochastic Analysis and Applications

Francesco Caravenna

Links

• Description of the course
• E-learning webpage with videos [Guest access with password SAA-2425]

Overview

This course presents modern topics in differential equations driven by irregular functions (continuous yet non-differentiable) that lead to the foundational concept of rough paths. This generalization offers new insights into the classical theory of stochastic integration with respect to Brownian motion. We may also explore applications to singular (stochastic) partial differential equations.

The course’s analytic core requires few prerequisites. Applications in stochastic integration require measure-theoretic probability, and familiarity with Brownian motion is helpful (interested students lacking some background are encouraged to contact me).

Program

• Introduction to singular differential equations, the sewing bound
• Difference equations: the Young case
• Difference equations: the rough case
• Stochastic differential equations
• The sewing lemma and the Young integral
• Rough paths and rough integration
• Examples and applications

References

• Peter Friz and Martin Hairer. A Course on Rough Paths. Second Edition, Springer (2020)
• Francesco Caravenna, Massimiliano Gubinelli, Lorenzo Zambotti. Ten Lectures on Rough Paths. Lecture Notes.

Lecture Notes

• Introduction
• Chapter 1. The Sewing Bound
• Chapter 2. Difference equations: Young case
• Chapter 3. Difference equations: rough case
• Chapter 4. Stochastic Differential Equations
• Chapter 5. Wong-Zakai
• Chapter 6. The Sewing Lemma
• Chapter 7. The Young integral
• Chapter 8. Rough paths
• Chapter 9. Geometric rough paths
• Chapter 10. Rough integration
• Chapter 11. Rough integral equations

Lectures

• Lecture 1 (6 Nov 2024). The sewing bound.
  Introduction and plan of the course. Reformulation of controlled ODEs as finite difference equations. Function spaces of Hölder type. Local uniqueness in the linear case. The sewing bound (statement). [Notes 1]
• Lecture 2 (13 Nov 2024). Young difference equations: a priori estimates and uniqueness.
  Reminders from the previous lecture. Proof of the sewing bound. Discrete sewing bound. Well-posedness (statement) for difference equations driven by X of class Cα with α>1/2 (Young case). A priori estimates. Uniqueness (sketch of the proof). [Notes 2]
• Lecture 3 (20 Nov 2024). Young difference equations: existence and continuity.
  Reminders from the previous lecture. Young difference equations (driven by X of class Cα with α>1/2): proof of uniqueness and existence. Discussion of continuity of solution map. Introduction to rough difference equations and rough paths. [Notes 3]
• Lecture 4 (27 Nov 2024). Rough paths and rough difference equations.
  Rough paths of regularity 1/3 < α ≤ 1/2: definition and discussion. Rough difference equations: a priori estimates, proof of uniqueness (sketch). [Notes 4]
• Lecture 5 (4 Dec 2024). Rough difference equations: existence and continuity.
  Rough difference equations: existence of solutions (with proof) and continuity of the solution map (statement and discussions). Introduction to stochastic difference equations and their relations with rough difference equations. The Ito rough path. [Notes 5]
• Lecture 6 (11 Dec 2024). Stochastic Differential Equations.
  The "Ito rough path" is a.s. a rough path (Theorem 1). Solutions of stochastic differential equations are a.s. solutions of the corresponding rough difference equations (Theorem 2). Refined Kolmogorov criterion: deterministic and random part. Local expansions of stochastic integrals (Theorem 3). [Notes 6]
• Lecture 7 (18 Dec 2024). Ito, Stratonovich and Wong-Zakai.
  Proof of local expansions of stochastic integrals (Theorem 3, part b). SDEs and RDEs with a drift. Stratonovich SDEs and RDEs. The Theorem of Wong-Zakai. [Notes 7]
• Lecture 8 (8 Jan 2025). The sewing lemma and the Young integral.
  Reformulation of controlled ODEs as integral equations. Abstract setting: germ, integral and remanider. The sewing lemma. The Young integral. [Notes 8]
• Lecture 9 (15 Jan 2025). Generalized integral, rough paths and rough integral.
  Reminders on the Sewing Lemma and the Young integral. Generalized notion of integral beyond Young. Rough paths revisited. Controlled paths. Rough integral. Continuity properties of the rough integral (hints). [Notes 9]
• Lecture 10 (22 Jan 2025). Seminar "Invariance principles for lifted random processes" by Tal Orenshtein. [Notes 10]

When

A dozen lectures from March to May 2020.

1. Thu 19 March 14:30
2. Wed 25 March 14:30
3. Thu 2 April 14:30
4. Wed 8 April 14:30
5. Thu 16 April 14:30
6. Thu 23 April 14:30
7. Wed 29 Apr 14:30
8. Wed 6 May 14:30
9. Wed 13 May 14:30
10. Wed 20 May 14:30
11. Wed 27 May 14:30