The current paper is devoted to the Cauchy problem for the white noise driven Ostrovsky equation with positive dispersion. We obtain the local well-posedness for the initial data u0(⋅,ω)∈Hs(R) (a.e. ω∈Ω) which is F0-measurable with s>−34 and Φ∈L20,s∩L20(L2(R),H˙s,−12+ϵ(R)), and the globally well-posedness for the initial data u0(⋅,ω)∈L2(R) (a.e. ω∈Ω) which is F0-measurable and Φ∈L20,0∩L20(L2(R),H˙0,−12+ϵ(R)). The key ingredients that we used in this paper are some bilinear estimates in Xs,b-type spaces given in subsection 2.1, Itô formula, the BDG inequality and stopping time techniques as well as frequency truncated technique. Our method can be applied to the case γ=0, which is just the KdV equation, thus, our result improves the local well-posedness result of A. de Bouard, A. Debussche and Y. Tsutsumi (1999) [3].