Geodesic
The analysis of stability of a~linear oscillator with multiplicative noise
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 127-145

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In this paper, we investigate a linear SDE of second order in the Ito sense with a multiplicative noise with real parameters. This equation was reduced to a two-dimensional linear SDEs system of first order with the help of replacement of variables. This linear SDEs system is linearization of an arbitrary two-dimensional nonlinear system. We investigate the stability of a trivial solution to a linear system SDE. We obtain conditions for parameters of the system for various modes of stability. We compare the known numerical methods on the solution of an oscillatory system. 
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     author = {T. A. Averina and A. A. Alifirenko},
     title = {The analysis of stability of a~linear oscillator with multiplicative noise},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {127--145},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2007_10_2_a1/}
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T. A. Averina; A. A. Alifirenko. The analysis of stability of a~linear oscillator with multiplicative noise. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 127-145. http://geodesic.mathdoc.fr/item/SJVM_2007_10_2_a1/