×

Differential equations driven by rough paths. Ecole d’Eté de Probabilités de Saint-Flour XXXIV – 2004. Lectures given at the 34th probability summer school, July 6–24, 2004. (English) Zbl 1176.60002

Lecture Notes in Mathematics 1908. Berlin: Springer (ISBN 978-3-540-71284-8/pbk; 978-3-540-71285-5/ebook). xviii, 109 p. (2007).
The book provides an introduction to and a survey of the key concepts and results of the theory of rough paths. It grew out of the lecture notes for the corresponding lectures given at the Ecole d’Eté de Probabilités de Saint-Flour XXXIV - 2004.
The lecture notes start with a summary of the book’s contents providing a concise description of the underlying ideas and their historical development. The book then consists of five chapters: 1. Differential Equations Driven by Moderately Irregular Signals; 2. The Signature of a Path; 3. Rough Paths; 4. Integration Along Rough Paths; 5. Differential Equations Driven by Rough Paths.
The first two chapters provide the building blocks of the theory of rough paths. Chapter 1 introduces the analytical component, that is the concept of \(p\)-variation of a Banach space-valued continuous function and the Young integral. In particular, it provides the main result that the integral \(\int_0^t Y_u dX_u\) is a well-defined object, if \(X\) and \(Y\) are of finite \(p\) and \(q\) variation, respectively, with \(p^{-1}+q^{-1} >1\). Chapter 2 is devoted to the algebraic component of the theory, the signature of a path. For a Banach space \(V\) and a path \(X:[0,T]\longrightarrow V\) with bounded variation, the \(n\)-th iterated integral of \(X\) over \([s,t]\) (\(n\geq1 \) being an integer and \((s,t)\) being such that \(0\geq s\geq t\geq T\)) is the tensor of \(V^{\otimes n}\) defined by \(X_{s,t}^n=\int_{s<u_1<\ldots<u_n<t}dX_{u_1} \otimes \ldots dX_{u_n}\). Then the signature of \(X\) over some interval \([s,t]\) is an infinite sequence of the form \(S(X)_{s,t}=(1,X_{s,t}^1,X_{s,t}^2,\ldots)\) and the infinite sequence space \(\bigoplus_{n\geq 0}V^{\otimes n}\) is called the extended tensor algebra of \(V\). The fundamental property of a signature is a certain multiplicativity property. Further topics treated are, for example, functions on the range of the signature of the paths, Lie elements and logarithm and exponential of these.
Chapter 3 connects the two components and provides a definition of a \(p\)-rough path, roughly speaking, as a multiplicative functional of degree \(\lfloor p\rfloor\) with finite \(p\)-variation. Brownian rough paths are then studied in more detail. Chapters 4 and 5 then present the theory of differential equations driven by rough paths, starting with a definition of integration of rough paths and ending with a result on the existence and uniqueness of solutions of the differential equations.
The theory and the proofs have appeared in the literature before, however, their presentation has been unified and refined for these lecture notes. The book is very well readable and will be useful for everyone interested in learning about the theory of rough paths.

MSC:

60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
34A99 General theory for ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
37H99 Random dynamical systems
PDFBibTeX XMLCite
Full Text: DOI