Differential equations driven by Gaussian signals

By Peter K. Friz, Nicolas Victoir. 2007

143 citations recorded by the source corpus.

We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful yet conceptually simple framework in which to analysize differential equations driven by Gaussian signals in the rough paths sense.

Subfields: Artificial Intelligence, Finance

View Paper: https://www.semanticscholar.org/paper/f2cb6894da2da23c58dbfe9e1d457a061388d92e

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Citations in the corpus

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, unknown): “Our exposition here follows in essence [57].”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “2, [15], Theorem 35, asserts that (Xt)t∈[0,T ] lifts to a geometric p-rough path provided p > 2ρ.”
Solving the KPZ equation (2011, result): “OL OF THE UNIVERSAL PROCESS If we could really decompose X as the sum of two processes with these properties, then we could make sense of the limit X" t !X t in almost exactly the same way as in [FV10a]. Unfortunately, while this argument is undoubtedly appealing, it doesn’t seem obvious how to”
Differential Equations Driven by Π-Rough Paths (2012, background): “The rough path representation of an even larger class of Gaussian processes has been explored by Friz and Victoir [3, 4] and others.”
Convergence rates for the full Gaussian rough paths (2011, background): “[8]), ∣∣∣∣∫ 1 0 Z(1) 0,u dZ 2 u ∣∣∣∣2 L2 = E [(∫ 1”
Geometric versus non-geometric rough paths (2012, unknown): “To put this another way , suppose X is a Brownian motion , or indeed any path for which there is a canonical geometric rough path lying above it [17,18]; in the case of Brownian motion , this corresponds to constructing Stratonovich integrals .”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “A theory of such Gaussian rough paths has been developed by a succession of authors [3, 5, 7, 10], and we will mostly work within their framework.”
A theory of regularity structures (2013, background): “In particular, due to the symmetry of finite-dimensional control systems under time reversal, this is often the case in the classical theory of rough paths, see [28,44,82].”
Integrability of (Non-)Linear Rough Differential Equations and Integrals (2011, background): “[11, 12]) of finite 1-variation of R, the covariance function of ψ (·, t).”
A (rough) pathwise approach to a class of non-linear stochastic partial differential equations (2009, background): “…noise For instance, one can take z as (the rough path lift of) fractional Brownian motion with Hurst parameter 1 / 4 < H < 1 / 2, cf. [11] or [18], a regime which is “rougher” than Brownian and notoriously difﬁcult to handle; or a diffusion with uniformly elliptic generator in divergence…”