Markov chain models and stochastic differential equations.

Abstract

The trajectories of motion of dynamic systems subject to Gaussian White Noise inputs have in the past been studied by application of the Fokker-Planck-Kolmogorov partial differential equation. This thesis sets forth an alternate approach where step functions are used to simulate Gaussian White Noise inputs over intervals of time. Systems considered are of the class dy&d1; =f&d1;1 y&d1;, u&d1;,t dt+f&d1;2 y&d1; ,u&d1;,t n&d1; tdt where n¯(t) is a Gaussian White Noise vector, y¯(t) is a state vector, u(t) is a piecewise continuous vector function and f¯ 1, f¯2 are continuous vector functions which may be linear or nonlinear. A mathematical procedure is formulated to obtain a Markov Chain model for {y¯(t)} or any of the individual components {y1(t)}, {y 2(t)}, ... , {yr(t)} as a separate and distinct stochastic processes. Finally, a stability test is developed to predict whether the range of values of {y¯(t)} is finite in the Euclidean space Rr for all t > 0.