A Course on Rough Paths
With an Introduction to Regularity Structures
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Beschreibung
Lyons’ rough path analysis has provided new insights in the analysis of stochastic differential equations and stochastic partial differential equations, such as the KPZ equation. This textbook presents the first thorough and easily accessible introduction to rough path analysis.When applied to stochastic systems, rough path analysis provides a means to construct a pathwise solution theory which, in many respects, behaves much like the theory of deterministic differential equations and provides a clean break between analytical and probabilistic arguments. It provides a toolbox allowing to recover many classical results without using specific probabilistic properties such as predictability or the martingale property. The study of stochastic PDEs has recently led to a significant extension – the theory of regularity structures – and the last parts of this book are devoted to a gentle introduction.Most of this course is written as an essentially self-contained textbook, with an emphasis on ideas and short arguments, rather than pushing for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis courses and has some interest in stochastic analysis. For a large part of the text, little more than Itô integration against Brownian motion is required as background.
Lyons’ rough path analysis has provided new insights in the analysis of stochastic differential equations and stochastic partial differential equations, such as the KPZ equation. This textbook presents the first thorough and easily accessible introduction to rough path analysis.
When applied to stochastic systems, rough path analysis provides a means to construct a pathwise solution theory which, in many respects, behaves much like the theory of deterministic differential equations and provides a clean break between analytical and probabilistic arguments. It provides a toolbox allowing to recover many classical results without using specific probabilistic properties such as predictability or the martingale property. The study of stochastic PDEs has recently led to a significant extension – the theory of regularity structures – and the last parts of this book are devoted to a gentle introduction.
Most of this course is written as an essentially self-contained textbook, with an emphasis on ideas and short arguments, rather than pushing for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis courses and has some interest in stochastic analysis. For a large part of the text, little more than Itô integration against Brownian motion is required as background.
Provides a self-contained and easily accessible introduction to rough path analysis with many exercises Focuses on the simplest setting applicable to analysis of stochastic differential equations Includes recent applications to stochastic partial differential equations Includes supplementary material: sn.pub/extras
Autor*in
Peter K. Friz
Themen in »A Course on Rough Paths«
Gaussian Processes Regularity Structures Robust Stochastic Integration Rough Paths Stochastic Analysis Stochastic Differential Equations Stochastic Partial Differential Equations ordinary differential equations partial differential equations
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“The book under review is a concise introduction to the theory of rough paths, their applications in stochastic analysis, and to recent extensions allowing to handle stochastic partial differential equations. … It is easy to base a graduate course on rough paths on this monograph. … 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 in rough paths … .” (Nicolas Perkowski, zbMATH 1327.60013, 2016)
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