Modelling and Estimation for Finite State Reciprocal Processes

A reciprocal equation is a kind of descriptor linear discrete-index stochastic system which is well known be satisfied (pathwise) by all Gaussian reciprocal processes. From a system-theoretic point of view, it is a kind of `noncausal´ linear system, in the sense that the solution of it cannot be determined by only an `initial´ condition, indeed requiring the `terminal´ state as well, besides all the `input´ function between initial and terminal states. Also, nice properties are known of a reciprocal equation, such as the equivalence of it with a couple of ordinary (causal) dynamic systems running in opposite directions. For these reasons, here we assume a reciprocal equation as the target of stochastic realization for the class of finite state reciprocal processes, also named reciprocal chains. The central result of the present paper is showing that any canonical reciprocal chain, i.e. valued in the canonical base of REALRN , N being the cardinality of the set of chain´s states, satisfies (pathwise) a reciprocal equation in a N² dimensional canonical variable, or in other word a quadratic reciprocal equation, named `Augmented state reciprocal model´ (ASRM). Also, for a partially observed reciprocal chain, a linear-optimal smoother is derived. All the results here presented are based upon the idea that a reciprocal chain is a `combination´ of Markov bridges, to this purpose other forms, besides the ASRM, are presented in order to make clear the meaning of this `combination´, as well as to prove that the linear smoother can be actually implemented as N smoothers all operating independently on each Markov bridge component.