Infinite Dimensional Stochastic Realizations of Continuous-Time Stationary Vector Processes

TL;DRAbstract

In this paper we consider the problem of representing a given stationary Gaussian process with nonrational spectral density and continuous time as the output of a stochastic dynamical system. Since the spectral density is not rational, the dynamical system must be infinite-dimensional, and therefore the continuous-time assumption leads to certain mathematical difficulties which require the use of Hilbert spaces of distributions. (This is not the case in discrete time.) We show that, under certain conditions, there correspond to each proper Markovian splitting sub-space, two standard realizations, one evolving forward and one evolving backward in time.

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