On local time at time varying curve

TL;DR

This paper extends the continuity properties of local times of continuous semimartingales to families of curves, enabling more flexible analysis and applications such as change of variable formulas.

Contribution

It generalizes Yor's result by establishing regularity of local times along parametrized curves, broadening their applicability.

Findings

Family of local times is continuous in time and regular along curves.

Extended regularity results facilitate change of variable formulas.

Applicable to a broader class of stochastic processes and curves.

Abstract

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 $t$, c\`ad-l\`ag 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.