Key words stochastic differential inclusion, locally Lipschitzean condition, semicontinuity of multifunctions, existence of solutions, stochastic fluid oscillations MSC (2000) 60H10, 34A60In this paper on stochastic differential inclusions with set-valued drift, we will consider the possible existence of strong solutions in the case of a kind of lower semicontinuous multifunctions as well as of upper semicontinuous ones. The dispersion matrix conditions are weakened in the classical way. To show existence results for such inclusions, we will use a locally Lipschitzean approximation of lower semicontinuous multifunctions. A set-valued stochastic system with an upper semicontinuous drift will be used to describe the motion of a fluid oscillator with small random perturbations.

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