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arXiv · 1811.05173

Optimal extension to Sobolev rough paths

Chong Liu·David J. Prömel·Josef Teichmann
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Abstract

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $\alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.

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Chong Liu, David J. Prömel, Josef Teichmann. 2018-11-13. Optimal extension to Sobolev rough paths. https://arxiv.org/abs/1811.05173

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