Date of Award
Doctor of Philosophy
Krishna B. Athreya
We study the discrete time stochastic system on IR[subscript]sp0 d, x[subscript] n + 1 = A ([xi][subscript] n) x[subscript] n, where (with further specifications given in the text) [xi][subscript] n is a Feller stationary ergodic Markov chain taking values on a connected C[superscript][infinity] Riemannian manifold W and A is a mapping from W into Gl (d, IR).;This investigation is performed via the tools of geometric control theory and Lyapunov exponents. The two main results are: (1) Under some assumptions to be specified, the Markov chain (x[subscript] n, [xi][subscript] n) possesses a unique invariant (probability) measure. (2) Using the existence of this unique invariant probability measure, the Lyapunov exponent associated with this stochastic system is almost surely unique and independent of the intial value.;This last result is used to initiate a study of the asymptotic stability behavior of the sytem, using both Lyapunov and moment Lyapunov exponents.;To illustrate the above, a computer simulation is performed on a discretized version of the linear oscillator with damping and restoring force.;As it is the case for similar studies on continuous time systems, one of the assumptions required to obtain the above results amounts to a Lie algebra condition on some collection of vector fields arising from the dynamics of the system.
Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/
Patrick René Homblé
Homblé, Patrick René, "On the stability of linear stochastic difference equations " (1990). Retrospective Theses and Dissertations. 9442.