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On local time at time varying curve
Anass Ben Taleb · 2018-10-23 · via math.PR updates on arXiv.org

Let $ \left(X_{t} \right)_{t\geq 0} $ be a continuous semimartingale. Let $ L^{z}_{t}\left(X\right) $ its family of local times. In \cite{YOR} Yor showed that the family $ \left( L^{z}_{t}\left(X\right) \right)_{ z \in \mathbb{R}, t \geq 0} $ has a version that is continuous in $t$, càd-làg in $z$. In this paper we extend this result by showing that for a 'regular' family of curves parametrized by $z$, the corresponding family of local times has a 'regular' version. This result was used in \cite{BENT} to prove a change of variable formula for continuous semimartingales.

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