Controlling rough paths

By Massimiliano Gubinelli. 2003

697 citations recorded by the source corpus.

Abstract We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Holder 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 Holder topology.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b583332e186cb47d2c2b0513093ff707f67752a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

Citations in the corpus

On probability laws of solutions to differential systems driven by a fractional Brownian motion (2014, background): “, [20, 21]) that, whenever the vector fields V satisfy Hypothesis 1.”
Rough paths and 1d SDE with a time dependent distributional drift: application to polymers (2014, methodology): “We are then able to define the integral of a function v controlled by W (see [17, 18, 19]): Theorem 1.”
Young Integrals and SPDEs (2004, background): “The existence and uniqueness problem for ordinary dierential equations driven by an irregular path of H¨older continuity greater than 1/2 is now fairly well understood, either in the context of fractional integrals (Z¨ah98), or as a first step towards the definition of dierential equations driven by”
Averaging dynamics driven by fractional Brownian motion (2019, methodology): “ided that γα>κ. Furthermore, one has the bound |t→ f(t,xt)|−κ . |f|−κ,γ(1+|x|γα Tγα) , (3.13) on any interval of length T. Proof. This is an immediate consequence of the deterministic sewing lemma [Gub04]: Let Ξs,t be a deterministic two parameter process with |Ξst| ≤ Kˆ|t−s|η , |δΞsut| ≤ Kˆ Lip|”
Lectures on singular stochastic PDEs (2015, methodology): “The aim of these lectures is to explain how to apply controlled path ideas [12] to solve basic problems in singular stochastic parabolic equations.”
Solving the KPZ equation (2011, methodology): “Q02, LCL07, FV10b] or, for a simpliﬁed exposition covering most of the notions required for this work, see [Hai11]. We will mostly make use of the notations and terminology introduced by Gubinelli in [Gub04] since the estimates given in that work seem to be the ones that are most suitable for the pr”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, neutral): “There have been a number of (re)formulations of rough path theory by other authors including [33], [71], [46] and [82].”
Elements of Stochastic Calculus via Regularization (2006, methodology): “Lyons rough paths theory approach, see for instance [35, 34, 30, 27, 8] has considered in detail the problem of the existence of integrals of the type ∫ t 0 g(X)·dX .”
Differential equations driven by rough paths with jumps (2017, result): “[17, 11]) similar to what is now mainstream in rough integration.”
Yet another introduction to rough paths (2009, neutral): “Gubinelli, rather relies on algebraic considerations [Gub04].”
The Signature of a Rough Path: Uniqueness (2014, positive): “Gubinelli [30] for controlling rough paths, T.”
Multiplicative stochastic heat equations on the whole space (2015, background): “he following criterion for the convergence of the sequence fnis an adaptation of Theorem 3.23 in [Hai14b], which in turn can be viewed as a multidimensional generalisation of Gubinelli’s sewing lemma [Gub04]. Proposition 2.12 Let α<0. Assume that there exists a constant kAk such that sup n≥0 sup ”
Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity (2018, background): “Moreover, for g ∈ Cq+1(R,R) the function g′(X) is controlled by X with Gubinelli derivatives g′(X(t)) − g′(X(s)) = q−1 ∑ k=1 g(k+1)(X(s)) k!”