Convergence of Eigenvalues in State-Discretization of Linear Stochastic Systems

The transition operator that describes the time evolution of the state probability distribution for continuous-state linear systems is given by an integral operator. A state-discretization approach is proposed, which consists of a finite rank approximation of this integral operator. As a result of the state-discretization procedure, a Markov chain is obtained, in which case the transition operator is represented by a transition matrix. Spectral properties of the integral operator for the continuous-state case are presented. The relationships between the integral operator and the finite rank approximation are explored. In particular, the limiting properties of the eigenvalues of the transition matrices of the resulting Markov chains are studied in connection to the eigenvalues of the original continuous-state integral operator.