In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process.
[4][5][6][7] Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems.
Chen, the Lipschitz function theory of H. Whitney and core ideas of stochastic analysis.
The concepts and the uniform estimates have widespread application in pure and applied Mathematics and beyond.
It provides a toolbox to recover with relative ease many classical results in stochastic analysis (Wong-Zakai, Stroock-Varadhan support theorem, construction of stochastic flows, etc) without using specific probabilistic properties such as the martingale property or predictability.
At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and multidimensional path
The Signature is a homomorphism from the monoid of paths (under concatenation) into the grouplike elements of the free tensor algebra.
Martin Hairer used rough paths to construct a robust solution theory for the KPZ equation.
[8] He then proposed a generalization known as the theory of regularity structures[9] for which he was awarded a Fields medal in 2014.
taking values in a Banach space, need not be differentiable nor of bounded variation.
Rough paths give an almost sure pathwise definition of stochastic differential equations.
This continuity property and the deterministic nature of solutions makes it possible to simplify and strengthen many results in Stochastic Analysis, such as the Freidlin-Wentzell's Large Deviation theory[10] as well as results about stochastic flows.
In fact, rough path theory can go far beyond the scope of Itô and Stratonovich calculus and allows to make sense of differential equations driven by non-semimartingale paths, such as Gaussian processes and Markov processes.
(see Topological tensor product, note that rough path theory in fact works for a more general class of norms).
[12] A central result in rough path theory is Lyons' Universal Limit theorem.
is the solution to the differential equation driven by the geometric rough path
[13] This limiting geometric rough path can be used to make sense of differential equations driven by fractional Brownian motion with Hurst parameter
Once an enhancement has been chosen, the machinery of rough path theory will allow one to make sense of the controlled differential equation for sufficiently regular vector fields
Note that every stochastic process (even if it is a deterministic path) can have more than one (in fact, uncountably many) possible enhancements.
[14] Different enhancements will give rise to different solutions to the controlled differential equations.
[16] There are, in particular, many results on the solution to differential equation driven by fractional Brownian motion that have been proved using a combination of Malliavin calculus and rough path theory.
In fact, it has been proved recently that the solution to controlled differential equation driven by a class of Gaussian processes, which includes fractional Brownian motion with Hurst parameter
The Freidlin Wentzell's large deviation theory aims to study the asymptotic behavior, as
The Universal Limit Theorem guarantees that the Itô map sending the control path
[10] This strategy can be applied to not just differential equations driven by the Brownian motion but also to the differential equations driven any stochastic processes which can be enhanced as rough paths, such as fractional Brownian motion.
has sufficient regularity so that the stochastic differential equation has a unique solution in the sense of rough path.
The Universal Limit Theorem once again reduces this problem to whether the Brownian rough path
In fact, rough path theory gives the existence and uniqueness of
As in the case of Freidlin–Wentzell theory, this strategy holds not just for differential equations driven by the Brownian motion but to any stochastic processes that can be enhanced as rough paths.
[22][23] It is also possible to extend the core results in rough path theory to infinite dimensions, providing that the norm on the tensor algebra satisfies certain admissibility condition.