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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.

##### MSC:
 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
##### Keywords:
rough paths; differential equations; $$p$$-variation; continuity
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