ERCIM News No.40 - January 2000

## Stochastic Realization and Identification

by Andrea Gombani

One of the research themes of the Systems Theory Group at LADSEB is Stochastic Realization and its main object is to construct finite dimensional representations of an infinite (or, in practice, very long) sequence of random variables, called stochastic processes. Although the subject definitely belongs to the realm of fundamental research, laying on the border between mathematics and theoretical engineering, in recent years it has lead to a surprising variety of applications in identification, signal processing, physics and finance.The research in this field was triggered, like most of systems theory, by Kalman in the early sixties; at LADSEB the group was started in the early seventies by Giorgio Picci. The main aim of the research has been to condense the information contained in very long random signals (for example a time series of economic data, or the disturbance affecting data over a telephone line) into a small number of variables (called state). This representation is quite important since it enables the use of powerful existing techniques (like filtering and control) to analyse and handle the signals.

More precisely, if y(t) is a linear gaussian stochastic process, any Markov process x(t) which makes the past and future of y conditionally independent is called a state of the process. If the state has dimension as small as possible, it is called minimal realization. The problem then is to characterize all minimal realizations. There exist several ways to do this: by the set of solutions of a Riccati equation, using splitting subspaces in a Hilbert space of random variables or coinvariant subspaces in a Hardy space of analytic functions on the disk or the left half-plane.

Therefore the tools are essentially mathematical and they encompass theory of stochastic processes, linear algebra and, above all, functional analysis. In fact, it is through the use of Hardy spaces and functional models in this setup, that the major steps forward have been made. Most of the research work has been carried out jointly with the Electrical Engineering Department of the University of Padua (with which LADSEB has a longstanding cooperation) and the Division of Optimization and Systems Theory of the Royal Institute of Technology (KTH), Stockholm (also affiliated to ERCIM through SICS) and, more recently, with the Mathematics Department of the Ben Gurion University (BGU), Beer-Sheva, Israel.

A main motivation for research in this field has been the need to understand some fundamental issues in stochastic system theory. Some early applications have been in time series analysis and signal processing. However, in recent years there has been a growing number of applications of this theory in a number of different areas. These include the use in subspace methods for systems identification (again with KTH), leading to the development of new algorithms for identification of linear systems; geometric control theory (with BGU), also leading to new algorithms, this time regarding the disturbance decoupling problem; the Modelling of surface acoustic wave filters for channel selection in the GSM mobile phones (under development with the MIAOU project at INRIA); factor analysis and compartmental systems (partly in cooperation with CWI); the volatility in the term structure of interest rates, like options on bonds (joint work with the Finance Department of the Stockholm Business School). In this last application, an algorithm for pricing illiquid options (eg those which have a thin market or no market at all, eg new options) in terms of the prices of liquid ones (i.e. those treated on the market in large volumes) is currently being tested.

This research is done within the Systems Theory Group at LADSEB (see also Complexity of Hidden Markov Models, by Lorenzo Finesso, in this issue); this Group participates in the ERCIM Working Group on Control and Systems and in two EU networks (ERNSI and DYNSTOCH).

Links:

http://www.ladseb.pd.cnr.it/control/control.html

Please contact:

Andrea Gombani - LADSEB-CNR

Tel: +39 049 829 5756

E-mail: gombani@ladseb.pd.cnr.it

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