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An extension theorem to rough paths. (English) Zbl 1134.60047
Can every continuous path of finite \(p\)-variation in a Banach space \(V\) be lifted to a geometric \(q\)-rough path, where \(q>p\)? The authors give an affirmative answer through this paper. In fact, as the major result, they prove the following theorem.
Theorem. Fix \(p \in [1,+\infty)\). Let \(V\) be a Banach space and \(K\) a closed subgroup of \(G^{([p])}(V)\). If \(x\) is a \((G^{([p])}(V) / K, \| \cdot\| _{G}^{([p])(V) / K})\) continuous path of finite \(p\)-variation, with \( p \notin N \setminus \{0,1\}\), then one can lift \(x\) to a weak geometric \(p\)-rough path.

60H99 Stochastic analysis
34F05 Ordinary differential equations and systems with randomness
34G99 Differential equations in abstract spaces
46N30 Applications of functional analysis in probability theory and statistics
60G17 Sample path properties
93C99 Model systems in control theory
Full Text: DOI Numdam EuDML
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