The Local Time-Space Integral and Stochastic Differential Equations

TL;DR

This paper develops a two-parameter local time-space integral and a general time-dependent Itô-Tanaka formula, enabling the proof of existence and uniqueness of solutions for a broad class of time-inhomogeneous stochastic differential equations involving local time.

Contribution

It introduces a comprehensive local time-space integral and extends the Itô-Tanaka formula to the time-dependent case, addressing key barriers in the theory of SDEs with local time.

Findings

Established a general time-dependent Itô-Tanaka formula.

Proved existence and uniqueness of strong solutions for time-inhomogeneous SDEs with local time.

Connected and extended classical formulas in local time calculus.

Abstract

Processes which arise as solutions to stochastic differential equations involving the local time (SDELTs), such as skew Brownian motion, are frequent sources of inspiration in theory and applications. Existence and uniqueness results for such equations rely heavily on the It\^o-Tanaka formula. Recent interest in time-inhomogeneous SDELTs indicates the need for comprehensive existence and uniqueness results in the time-dependent case, however, the absence of a suitable time-dependent It\^o-Tanaka formula forms a major barrier. Rigorously developing a two-parameter integral with respect to local time, known as the local time-space integral, we connect together and extend many known formulae from the literature and establish a general time-dependent It\^o-Tanaka formula. Then, we prove the existence of a unique strong solution for a large class of time-inhomogeneous SDELTs.