Well-Posedness, Stability, and Sensitivities for Stochastic Delay Equations: A Generalized Coupling Approach

TL;DR

This paper introduces a generalized coupling method to analyze stochastic delay equations with non-smooth coefficients, proving existence, uniqueness, ergodic behavior, and sensitivity stabilization rates.

Contribution

It presents a novel coupling approach for stochastic delay equations with H"older continuous coefficients, addressing cases where PDE methods are inapplicable.

Findings

Existence and uniqueness of weak solutions established.

Weak ergodic rates for segment processes proven.

Sensitivity stabilization rates derived under smoothness assumptions.

Abstract

We develop a new generalized coupling approach to the study of stochastic delay equations with H\"older continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solutions, and establish weak ergodic rates for the corresponding segment processes. We also prove, under additional smoothness assumptions on the coefficients, stabilization rates for the sensitivities in the initial value of the corresponding semigroups