Rough path analysis via fractional calculus

By Yaozhong Hu, David Nualart. 2006

89 citations recorded by the source corpus.

Using fractional calculus we define integrals of the form b a f(x t )dy t , where x and y are vector-valued Holder continuous functions of order p ∈ (1/3, 1/2) and f is a continuously differentiable function such that f' is A-Holder continuous for some A > 1/β - 2. Under some further smooth conditions on f the integral is a continuous functional of x, y, and the tensor product x ⊗ y with respect to the Holder norms. We derive some estimates for these integrals and we solve differential equations driven by the function y. We discuss some applications to stochastic integrals and stochastic differential equations.

Subfields: Computer Vision and Pattern Recognition, Control and Systems Engineering, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/25a3719fd87eaf0c203a40161f86f1e427dbb243

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Citations in the corpus

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, background): “6, and extends the result given in [24] to 1/3 <H < 1.”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “Lipschitz estimates for rough integration or differential equations, at least for  ∈ [2 3), appear in [113] and [82].”
Differential Equations Driven by Hölder Continuous Functions of Order Greater than 1/2 (2006, result): “Similar results are obtained for the case 1/3 < β < 1/2 in a forthcoming paper [1].”
Convergence rates for the full Gaussian rough paths (2011, background): “• Wong-Zakai rates for the Brownian rough path (level 2) were first discussed in [14].”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “• Rough formulations of evolution heat equation with multiplicative noise (with varying degree of success) have been considered by Gubinelli and Tindel [33], Deya, Gubinelli and Tindel [16], Teichmann [61], Hu and Nualart [46] and Garrido-Atienza, Lu and Schmalfuss [27].”
Averaging principle for fast-slow system driven by mixed fractional Brownian rough path (2020, methodology): “2), it is not difficult to show that there exists a triplet (u, Z, ṽ) ∈ M m+n,d+d (This section follows the notations proposed by Hu and Nualart [15]).”