On control of diffusion processes on a surface in euclidean space

Controlled diffusion processes that begin on some hypersurface and stay on it always are considered in Euclidean space. Such a situation includes controlled diffusion processes in the whole space. The results thus make sense for this case, and it turns out that even then they are stronger in many respects than previously known results. Another new element here is differentiation in the space of strategies, implemented by introducing auxiliary parameters.