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Controlling Rough Paths
TL;DR: In this article, the authors formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution, and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths.
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Abstract: We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons' theory of rough paths in Hoelder topology.
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Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
- 01 Dec 1992
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
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•Book
Multidimensional Stochastic Processes as Rough Paths
Peter K. Friz,Nicolas B. Victoir +1 more
- 01 Feb 2010
TL;DR: Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations as mentioned in this paper, and it has been used extensively in the analysis of partial differential equations.
A theory of regularity structures
TL;DR: In this article, a regularity structure is introduced to describe functions and distributions via a kind of "jet" or local Taylor expansion around each point, which allows to describe the local behaviour not only of functions but also of large classes of distributions.
786
Solving the KPZ equation
TL;DR: In this article, the authors introduce a new solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution, providing a pathwise notion of a solution, together with a very detailed approximation theory.
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Paracontrolled distributions and singular PDEs
TL;DR: In this article, the authors introduce an approach to study singular PDEs based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths, and illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise.
286
References
Differential equations driven by rough signals
TL;DR: In this paper, the authors provide a systematic approach to the treatment of differential equations of the type======dyt = Si fi(yt) dxti¯¯¯¯where the driving signal is a rough path.
1.3K
•Book
System Control and Rough Paths
Terry Lyons,Zhongmin Qian +1 more
- 06 Feb 2003
TL;DR: In this article, the authors introduce Lipschitz paths, Brownian rough paths, and Universal limit theoem for vector fields and flow equations, and integrate them along rough paths.
739
Approximations of the Brownian rough path with applications to stochastic analysis
Peter K. Friz,Nicolas Victoir +1 more
TL;DR: In this paper, a geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance.