Continuity of Local Time: An applied perspective

TL;DR

This paper explores how the mathematical property of local time continuity in Brownian motion influences practical applications across physics, biology, and finance, especially in systems with discontinuous dispersion coefficients.

Contribution

It extends previous theoretical results to demonstrate the role of local time continuity in real-world problems involving discontinuities in stochastic models.

Findings

Established a link between macro-scale deterministic continuity and micro-scale local time continuity.

Provided explicit principles connecting physical phenomena to stochastic local time properties.

Extended theoretical results to applied contexts with discontinuous dispersion coefficients.

Abstract

Continuity of local time for Brownian motion ranks among the most notable mathematical results in the theory of stochastic processes. This article addresses its implications from the point of view of applications. In particular an extension of previous results on an explicit role of continuity of (natural) local time is obtained for applications to recent classes of problems in physics, biology and finance involving discontinuities in a dispersion coefficient. The main theorem and its corollary provide physical principles that relate macro scale continuity of deterministic quantities to micro scale continuity of the (stochastic) local time.