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2019 | OriginalPaper | Buchkapitel

# Multiscale Systems, Homogenization, and Rough Paths

verfasst von: Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang

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## Abstract

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479–520], taking into account recent progress in p-variation and càdlàg rough path theory.
Fußnoten
1
Since our limit processes here—a Brownian motion—is continuous, there is no need to work with the Skorokhod topology on D.

2
In view of the genuine non-linearity of rough path spaces, we refrain from writing $$\Vert \mathbf {X}- \tilde{\mathbf {X}} \Vert _{p\text {-var},[0,1]}$$.

3
In coordinates, when $$\mathcal {B}= \mathbb {R}^m$$, we have $$DV (Y_s) V (Y_s) \mathbb {X}_{s,t} = \partial _\alpha V_\gamma (Y_s) V^\alpha _\beta (Y_s) \mathbb {X}_{s,t}^{\beta ,\gamma }$$ with summation over $$\alpha = 1, \ldots , d$$ and $$\beta , \gamma = 1, \ldots , m.$$.

4
Often $$B^n$$ has continuous BV sample paths. Every such process is (trivially) a semimartingale (under its own filtration); the Stratonovich SDE interpretation is the one consistent with the ODE interpretation, in the sense of a Riemann-Stieltjes integral equation.

5
Again it suffices to work with the uniform topology on both $${\pmb {\mathscr {C}}}$$ and $${\pmb {\mathscr {D}}}$$.

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Titel
Multiscale Systems, Homogenization, and Rough Paths
verfasst von
Ilya Chevyrev
Peter K. Friz
Alexey Korepanov
Ian Melbourne
Huilin Zhang