Strong Uniqueness of Singular Stochastic Delay Equations

TL;DR

This paper introduces a novel method using Malliavin calculus and local time variational calculus to establish the strong uniqueness of solutions for a broad class of stochastic delay equations with irregular drift, extending previous finite-dimensional results.

Contribution

The paper presents a new approach for proving strong uniqueness of solutions to stochastic delay equations with discontinuous drift, generalizing prior finite-dimensional results to infinite-dimensional settings.

Findings

Established strong uniqueness for stochastic delay equations with irregular drift.

Extended finite-dimensional results to infinite-dimensional stochastic delay equations.

Demonstrated applicability of Malliavin calculus and local time variational calculus to these equations.

Abstract

In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper can be regarded as an infinite-dimensional generalization of those of A. Y. Veretennikov [42] in the case of certain stochastic delay equations with irregular drift coefficients. The approach proposed in this work rests on Malliavin calculus and arguments of a "local time variational calculus", which may also be used to study other types of stochastic equations as e.g. functional It\^{o}-stochastic differential equations in connection with path-dependent Kolmogorov equations [15].