Stochastic Delay Differential Equations with Jump Reflection: Invariant Measure

TL;DR

This paper studies multi-dimensional stochastic delay differential equations with jump reflection, proving existence of a unique invariant measure and exploring the relationship between the regulator and local time of solutions.

Contribution

It establishes the existence and uniqueness of an invariant measure for these complex stochastic equations under monotone conditions, advancing understanding of their long-term behavior.

Findings

Unique invariant measure exists for the segment process.

Relationship between regulator and local time is established.

Local time properties at large time are discussed.

Abstract

In this paper, we consider a class of multi-dimensional stochastic delay differential equations with jump reflection. Based on existence and uniqueness of the strong solution to the equation, we prove that the Markov semigroup generated by the segment process corresponding to the solution admits a unique invariant measure on the Skorohod space when the coefficients of equation satisfy a class of monotone conditions. Finally, we establish a relationship between the regulator and the local time of the solution and discuss a local time property at large time under the stationary setting.