Differential equations driven by rough paths

By Terry Lyons, M. Caruana, Thierry Lévy, École d'été de probabilités de Saint-Flour. 2007

639 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/04ef954a073f026c78c79d426301f01d4a781a0a

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In Computational Theory and Mathematics

In Condensed Matter Physics

In Mathematical Physics

Citations in the corpus

Uniqueness for the signature of a path of bounded variation and the reduced path group (2005, background): “By the extension theorem of [7] and the continuity of the iterated integrals in the path we see that the signatures of a convergent sequence of paths will converge in p-variation for all p > 1.”
The Signature of a Rough Path: Uniqueness (2014, background): “37 [23]) and G(n) = π(n) ( G(∗) ) denote the free nilpotent Lie group of step n (p.”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “From the point of view of existence and uniqueness results, the appropriate way to measure the regularity of the Vi’s turns out to be the notion of Lipschitz-γ (short: Lip-γ) in the sense of Stein (see [12] and [26]).”
Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity (2018, neutral): “3 in [22], see also [19]) once we show that for 0 ≤ s ≤ u ≤ t ≤ T ⌊p⌋ ∑”
Lectures on singular stochastic PDEs (2015, background): “Nowadays there are many expositions of this theory [27, 9, 26, 8] and we refer the reader to the literature for more details.”
Controlled differential equations as Young integrals: A simple approach (2010, neutral): “In this subject, see [11, 16, 17, 21–23].”
Partial differential equations driven by rough paths (2008, background): “For a more complete exposition of this theory, we refer the reader to [21], [20], and [6].”
Graph Neural Controlled Differential Equations for Traffic Forecasting (2021, background): “The well-posedness 1 of NCDEs was already proved in (Lyons, Caruana, and L ´ evy 2007, Theorem 1.3) under the mild condition of the Lipschitz continuity.”
Neural Controlled Differential Equations for Irregular Time Series (2020, background): “The behaviour of controlled differential equations are typically described through the signature transform (also known as path signature or simply signature ) [20] of the driving process X .”
Joint calibration to SPX and VIX options with signature‐based models (2023, background): “Observe that multiindices can be identified with words, as it is done for instance in Lyons et al. (2020). Next, for each |I| ≥ 1 we set eI := ei1 ⊗ · · · ⊗ ein .”
Discretely sampled signals and the rough Hoff process (2013, background): “See [11, 20] for a discussion on the contrast with the classical notion of Lipschitzness.”
Integrability Estimates for Gaussian Rough Differential Equations (2011, result): “See [12] and [26], and note the contrast with classical Lipschitzness.”
The infinitesimal generator of the stochastic Burgers equation (2018, background): “16 in [45] together with the Cauchy–Schwarz inequality yields”