ABSTRACT
Abstract We consider a one-dimensional homogeneous stochastic differential equation of the form
dXt = b{Xt)dt + o{Xt)dBu X0 = x, where b and a are supposed to be measurable functions and a ^ 0. No assumptions of boundedness (or boundedness away from zero) are imposed. We introduce a class of points called isolated singular points and investigate the weak existence as well as the uniqueness in law of the solution in the neighbourhood of such a point. A complete qualitative classification of these points is presented. There are 63 different types. The constructed classification allows us to find out whether a solution can reach an isolated singular point, whether it can leave this point, and so on. It has been found that, for 59 types, there exists a unique solution in the neighbourhood of the corresponding isolated singular point. Moreover, the solution is a strong Markov process. The remaining 4 types of isolated singular points (we call them branch types) disturb the uniqueness. One can construct various "bad" solutions in the neighbourhood of a branch point. In particular, there exist non-Markov solutions. As an application of the obtained results, we consider equations of the form
dXt = ix\Xt\adt + v\Xt\*dBu X0 = x,
and present the classification for this case.
