The stochastic harmonic oscillator with finite correlation times

The moments of an undamped stochastic harmonic oscillator are examined using iteration procedures described in earlier papers. The noise is similar to white noise, but has finite correlation time which is taken to be less than the period of the oscillator. When the time of observation is finite, the first moment is explicitly given, attempting first-order approximation. For infinite time, we establish certain conditions which assure boundedness of the first and second moments. All the results obtained are applied to the white noise when its correlation time is considered finite.     

On stochastic differential equations with non-white noise having small correlation times

This is an extension of an earlier paper by the second author. In the previous work, an algorithm was derived for the moments of an homogeneous equation with noise similar to the white noise, but has small, finite correlation time. In the present work this algorithm is extended for an inhomogeneous equation with the same characteristics. Previous notation has been kept throughout the paper.     

The stochastic harmonic oscillator with linear damping

The moments of the stochastic harmonic oscillator are examined, in the presence of linear damping. The procedures we follow are those described in earlier papers for stochastic linear systems and for the undamping case. The noise is approximated from the white noise process and has small but finite correlation time. The first moment is shown to be bounded and tending to zero, as time approaches infinity. The second moment is also shown to be bounded, but only if the damping factor satisfies a proper condition. Results are compared with those found in the case where there is no damping and with those obtained in other works, using different methods.