Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators

TL;DR

This paper investigates time-inhomogeneous one-dimensional SDEs involving local time on curves, establishing existence, uniqueness, and their connection to parabolic PDEs with transmission conditions, along with solution regularity and Markov properties.

Contribution

It introduces new existence and uniqueness results for such SDEs and links them to PDEs with transmission conditions, expanding understanding of their regularity and Markovian nature.

Findings

Proved existence and uniqueness of solutions under mild assumptions

Established a Feynman-Kac representation for these SDEs

Characterized solutions as time-inhomogeneous Markov Feller processes

Abstract

In this paper we study time-inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDE under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDE and ensure the validity of a Feynman-Kac representation formula. These results are then used to characterize the solutions of these SDE as time-inhomogeneous Markov Feller processes.