Absolute Continuity under Time Shift for Ornstein-Uhlenbeck type Processes with Delay or Anticipation

TL;DR

This paper investigates the absolute continuity of Ornstein-Uhlenbeck type processes with delay or anticipation under time shifts, establishing existence, uniqueness, and Radon-Nikodym derivatives for these processes.

Contribution

It provides new results on existence, uniqueness, and explicit Radon-Nikodym densities for delayed or anticipative Ornstein-Uhlenbeck processes under time shifts.

Findings

Proved existence and uniqueness with boundedness conditions.

Calculated Radon-Nikodym density under time shifts.

Established absolute continuity properties of the processes.

Abstract

The paper is concerned with one-dimensional two-sided Ornstein-Uhlenbeck type processes with delay or anticipation. We prove existence and uniqueness requiring almost sure boundedness on the left half-axis in case of delay and almost sure boundedness on the right half-axis in case of anticipation. For those stochastic processes $(X,P_{\mu})$ we calculate the Radon-Nikodym density under time shift of trajectories, $P_{\mu}(dX_{\cdot -t})/P_{\mu}(dX)$, $t\in {\Bbb R}$.