On semimartingale local time inequalities and applications in SDE's

TL;DR

This paper extends local time inequalities for semimartingales to discontinuous cases using balayage formula, and explores implications for pathwise uniqueness in certain stochastic differential equations.

Contribution

It generalizes the comparison theorem of local times to include discontinuous semimartingales and investigates pathwise uniqueness in related SDEs.

Findings

Extended local time inequality to discontinuous semimartingales

Established conditions for pathwise uniqueness in SDEs involving local time

Utilized balayage formula to derive new inequalities

Abstract

Using the balayage formula, we prove an inequality between the measures associated to local times of semimartingales. Our result extends the "comparison theorem of local times" of Ouknine $(1988)$, which is useful in the study of stochastic differential equations. The inequality presented in this paper covers the discontinuous case. Moreover, we study the pathwise uniqueness of some stochastic differential equations involving local time of unknown process.