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2020 | Buch

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A Course on Rough Paths

With an Introduction to Regularity Structures

verfasst von: Prof. Peter K. Friz, Prof. Martin Hairer

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

With many updates and additional exercises, the second edition of this book continues to provide readers with a gentle introduction to rough path analysis and regularity structures, theories that have yielded many new insights into the analysis of stochastic differential equations, and, most recently, stochastic partial differential equations.

Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property.

Essentially self-contained, this textbook puts the emphasis on ideas and short arguments, rather than aiming for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis and probability courses, with little more than Itô-integration against Brownian motion required for most of the text.

From the reviews of the first edition:

"Can easily be used as a support for a graduate course ... Presents in an accessible way the unique point of view of two experts who themselves have largely contributed to the theory" - Fabrice Baudouin in the Mathematical Reviews

"It is easy to base a graduate course on rough paths on this … A researcher who carefully works her way through all of the exercises will have a very good impression of the current state of the art" - Nicolas Perkowski in Zentralblatt MATH

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

We give a short overview of the scopes of both the theory of rough paths and the theory of regularity structures. The main ideas are introduced and we point out some analogies with other branches of mathematics.

Peter K. Friz, Martin Hairer

Chapter 2. The space of rough paths

Abstract

We define the space of (H¨older continuous) rough paths, as well as the subspace of “geometric” rough paths which preserve the usual rules of calculus. The latter can be interpreted in a natural way as paths with values in a certain nilpotent Lie group. At the end of the chapter, we give a short discussion showing how these definitions should be generalised to treat paths of arbitrarily low regularity.

Peter K. Friz, Martin Hairer

Chapter 3. Brownian motion as a rough path

Abstract

In this chapter, we consider the most important example of a rough path, which is the one associated to Brownian motion. We discuss the difference, at the level of rough paths, between Ito and Stratonovich Brownian motion. We also provide a natural example of approximation to Brownian motion which converges to neither of them.

Peter K. Friz, Martin Hairer

Chapter 4. Integration against rough paths

Abstract

The aim of this chapter is to give a meaning to the expression R Yt dXt for a suitable class of integrands Y , integrated against a rough path X. We first discuss the case originally studied by Lyons where Y = F(X).

Peter K. Friz, Martin Hairer

Chapter 5. Stochastic integration and Itô’s formula

Abstract

In this chapter, we compare the integration theory developed in the previous chapter to the usual theories of stochastic integration, be it in the Itˆo or the Stratonovich sense.

Peter K. Friz, Martin Hairer

Chapter 6. Doob–Meyer type decomposition for rough paths

Abstract

A deterministic Doob–Meyer type decomposition is established. It is closely related to the question to what extent Y' is determined by Y , given that (Y, Y') ∈ D2α X . The crucial property is true roughness of X, a deterministic property that guarantees that X varies in all directions, all the time.

Peter K. Friz, Martin Hairer

Chapter 7. Operations on controlled rough paths

Abstract

At first sight, the notation R Y dX introduced in Chapter 4 is ambiguous since the resulting controlled rough path depends in general on the choices of both the secondorder process X and the derivative process Y 0 . Fortunately, this “lack of completeness” in our notations is mitigated by the fact that in virtually all situations of interest, Y is constructed by using a small number of elementary operations described in this chapter. For all of these operations, it turns out to be intuitively rather clear how the corresponding derivative process is constructed.

Peter K. Friz, Martin Hairer

Chapter 8. Solutions to rough differential equations

Abstract

We show how to solve differential equations driven by rough paths by a simple Picard iteration argument. This yields a pathwise solution theory mimicking the standard solution theory for ordinary differential equations. We start with the simple case of differential equations driven by a signal that is sufficiently regular for Young’s theory of integration to apply and then proceed to the case of more general rough signals.

Peter K. Friz, Martin Hairer

Chapter 9. Stochastic differential equations

Abstract

We identify the solution to a rough differential equation driven by the Itˆo or Stratonovich lift of Brownian motion with the solution to the corresponding stochastic differential equation. In combination with continuity of the Itˆo–Lyons maps, a quick proof of the Wong–Zakai theorem is given. Applications to Stroock–Varadhan support theory and Freidlin–Wentzell large deviations are briefly discussed.

Peter K. Friz, Martin Hairer

Chapter 10. Gaussian rough paths

Abstract

We investigate when multidimensional stochastic processes can be viewed – in a “canonical” fashion – as random rough paths. Gaussianity only enters through equivalence of moments.

Peter K. Friz, Martin Hairer

Chapter 11. Cameron–Martin regularity and applications

Abstract

A continuous Gaussian process gives rise to a Gaussian measure on path-space. Thanks to variation regularity properties of Cameron–Martin paths, powerful tools from the analysis on Gaussian spaces become available. A general Fernique type theorem leads us to integrability properties of rough integrals with Gaussian integrator akin to those of classical stochastic integrals. We then discuss Malliavin calculus for differential equations driven by Gaussian rough paths. As application a version of H¨ormander’s theorem in this non-Markovian setting is established.

Peter K. Friz, Martin Hairer

Chapter 12. Stochastic partial differential equations

Abstract

Second order stochastic partial differential equations are discussed from a rough path point of view. In the linear and finite-dimensional noise case we follow a Feynman– Kac approach which makes good use of concentration of measure results, as those obtained in Section 11.2. Alternatively, one can proceed by flow decomposition and this approach also works in a number of nonlinear situations.

Peter K. Friz, Martin Hairer

Chapter 13. Introduction to regularity structures

Abstract

We give a short introduction to the main concepts of the general theory of regularity structures. This theory unifies the theory of (controlled) rough paths with the usual theory of Taylor expansions and allows to treat situations where the underlying space is multidimensional.

Peter K. Friz, Martin Hairer

Chapter 14. Operations on modelled distributions

Abstract

The original motivation for the development of the theory of regularity structures was to provide robust solution theories for singular stochastic PDEs like the KPZ equation or the dynamical Φ 4 3 model.

Peter K. Friz, Martin Hairer

Chapter 15. Application to the KPZ equation

Abstract

We show how the theory of regularity structures can be used to build a robust solution theory for the KPZ equation. We also give a very short survey of the original approach to the same problem using controlled rough paths and we discuss how the two approaches are linked.

Peter K. Friz, Martin Hairer

Backmatter

Titel: A Course on Rough Paths
verfasst von: Prof. Peter K. Friz
Prof. Martin Hairer
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-41556-3
Print ISBN: 978-3-030-41555-6
DOI: https://doi.org/10.1007/978-3-030-41556-3