Sample Stability of Second Order Linear Differential Equations with Wide Band Noise Coefficients

Abstract

The sample stability boundaries for second order linear Ito stochastic differential equations are determined. Through the use of correction terms, the results are applied to ordinary stochastic differential equations with wide band random coefficients. The theoretical conclusions are compared with digital computer simulation. A number of interesting examples are presented which illustrate the extent of the analytical results obtained. In particular, examples are presented for which, with probability one; a deterministically stable system is destabilized; a deterministically unstable system is stabilized ; the stability of the system is unaffected by the introduction of wide band random coefficients. Finally, an example is presented for which an increase in damping, above critical, causes a decrease in the region of stability, and a system is presented whose sample solutions are almost all deterministic.