Toward a Theory of Nonlinear Stochastic Realization
Abstract
The following is a central problem in stochastic systems theory: Given a stationary stochastic process {y(t);t is a member of the set of real numbers}, find a (possibly infinite-dimensional) vector Markov process {x(t);t is a member of the set of real numbers, called the state process, and a function f so that y(t) f(x(t)) for all t is a real number. Moreover, find a stochastic differential equation driven by a Wiener process and having the state process x as its unique solution. The problem of characterizing the family of all such representations is known as the stochastic realization problem. There is by now a rather comprehensive theory of stochastic realization for the case that {y(t);t is a real number} is Gaussian, in which case the representations can be taken to be linear, i.e. both f and the stochastic differential equation are linear. This linear theory can be applied to non-Gaussian processes also, but then we need to give up the requirement that x is Markov and that it is generated by a Wiener process, replacing these concepts by "wide sense Markov" and "orthogonal increment process" respectively. If we are not willing to do so, a nonlinear stochastic realization theory is needed. That is the topic of this paper. In this paper we shall apply Wiener's theory of homogeneous chaos to the nonlinear stochastic realization problem. For simplicity and ease of notation we shall assume that the process y is scalar, although the machinery which we develop is sufficient to accommodate also the vector case. Other assumptions, such as y admitting an innovation representation, are however crucial to our approach. (In this respect, it might be more appropriate to consider a process y with stationary increments, and indeed with minor modifications we could have done so.) In the extension of this work we see the possibility of making contact with nonlinear filtering and that is partially a motivation for this work.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1981
- Accession Number
- ADA459661
Entities
People
- Anders Lindquist
- Giorgio Picci
- Sanjoy Mittert
Organizations
- Massachusetts Institute of Technology