  We develop a variant of rough path theory tailor-made for analyzing a class
of financial asset price models known as rough volatility models. As an
application, we prove a pathwise large deviation principle (LDP) for a certain
class of rough volatility models, which in turn describes the limiting behavior
of implied volatility for short maturity under those models. First, we
introduce a partial rough path space and an integration map on it and then
investigate several fundamental properties including local Lipschitz continuity
of the integration map from the partial rough path space to a rough path space.
Second, we construct a rough path lift of a rough volatility model. Finally, we
prove an LDP on the partial rough path space, and the LDP for rough volatility
then follows by the continuity of the solution map of rough differential
equations.
