On time-reversibility of linear stochastic models

TL;DR

This paper generalizes the concept of time-reversibility in linear stochastic models, integrating ideas from stochastic realization theory and moment problems to handle systems driven by arbitrary second-order stationary processes.

Contribution

It introduces a unified framework for time-reversal in stochastic realization theory applicable to a broader class of processes, extending previous methods.

Findings

Unified approach to time-reversal in stochastic models

Applicable to systems driven by arbitrary second-order stationary processes

Enhances understanding of stochastic system dualities

Abstract

Reversal of the time direction in stochastic systems driven by white noise has been central throughout the development of stochastic realization theory, filtering and smoothing. Similar ideas were developed in connection with certain problems in the theory of moments, where a duality induced by time reversal was introduced to parametrize solutions. In this latter work it was shown that stochastic systems driven by arbitrary second-order stationary processes can be similarly time-reversed. By combining these two sets of ideas we present herein a generalization of time-reversal in stochastic realization theory.