Summary: Existence of solution theorems are obtained for stochastic differential inclusions given in terms of the so-called current velocities (symmetric mean derivatives, a direct analogs of ordinary velocity of deterministic systems) and quadratic mean derivatives (giving information on the diffusion coefficient) on the flat \(n\)-dimensional torus. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy some natural conditions, under which they have \(\varepsilon\)-approximations that point-wise converge to Borel measurable selections of the corresponding set-valued mappings. For Part I, see [the authors, ibid. 2, No. 3, 25--33 (2015; Zbl 1359.60084)].