On time-reversibility of linear stochastic models

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.

The Role of the Time-Arrow in Mean-Square Estimation of Stochastic Processes

“…Thus, in the usual slight abuse of notation, z : x k → x k−1 is the "delay operator" which is opposite to the way this is most often used in signal processing literature. 4…”

“…The forward and reversed processes have similar realizations (cf. [4]). Indeed, we can easily see that…”

“…The paradox of the apparent directionality of physics originating in physical laws that are time-symmetric is a key conundrum; Feynman states that there is a fundamental law which says, that "uxels only make wuxels and not vice versa," but we have not found this yet. Thus, the time reversibility of physical models, as well as the lack, are of great interest, see, e.g., [4], [5], [6]. In a similar vain, it is expected that the time-arrow will draw increasingly more attention in modeling of engineering systems as well.…”

Optimal Estimation With Missing Observations via Balanced Time-Symmetric Stochastic Models

“…In Section III we utilize this mechanism in the context of general output processes and, similarly, introduce a pair of time-opposite models. These two introductory sections, II and III, deal with stationary models for simplicity and are largely based on [20]. The corresponding generalizations to time-varying systems are given in Section IV and in the appendix, in continuous and discrete-time, respectively.…”