On External Behavior Equivalence of Continuous-Time Stochastic Linear Control Systems

In this paper we consider continuous time stochastic linear control systems and study the notion of stochastic external behaviour equivalence. Starting from our earlier work on equivalences given for discrete time stochastic control systems, we first extend the notion of stochastic external behaviour equivalence and the corresponding geometric conditions to the continuous-time framework. We then address model reduction and we show that the system with minimal dimension in the state space which has equivalent stochastic external behavior of a system, can be easily obtained by factoring out the unobservable dynamics from the given system. Connections with equivalence notions given for nondeterministic linear control systems and with stochastic linear realization theory are also formally discussed.