On the Strong Feller Property and Well-Posedness for SDEs with Functional, Locally Unbounded Drift

TL;DR

This paper investigates the well-posedness and strong Feller property of stochastic differential equations with functional, locally unbounded drift, using Zvonkin's transformation and probabilistic methods without requiring continuity assumptions.

Contribution

It extends existing methods to functional SDEs with unbounded drift, providing new results on well-posedness and regularity under minimal conditions.

Findings

Established well-posedness for a class of functional SDEs with unbounded drift.

Proved the strong Feller property for these equations.

Developed a framework avoiding continuity assumptions, relying on integrability conditions.

Abstract

We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability conditions and avoid continuity assumptions as far as possible. Our approach is mainly based on Zvonkin's transformation [18], an extended version of the probabilistic approach of Maslowski and Seidler [12] and the convergence concept for random variables in topological spaces in [2].