Risk sensitive identification of linear stochastic systems

Abstract

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set E_K, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°_K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes.