  The classical result by It\^o on the existence of strong solutions of
stochastic differential equations (SDEs) with Lipschitz coefficients can be
extended to the case where the drift is only measurable and bounded. These
generalizations are based on techniques presented by Zvonkin and Veretennikov,
which rely on the uniform ellipticity of the diffusion coefficient. In this
paper we study the case of degenerate ellipticity and give sufficient
conditions for the existence of a solution. The conditions on the diffusion
coefficient are more general than previous results and we gain fundamental
insight into the geometric properties of the discontinuity of the drift on the
one hand and the diffusion vector field on the other hand. Besides presenting
existence results for the degenerate elliptic situation, we give an example
illustrating the difficulties in obtaining more general results than those
given. The particular types of SDEs considered arise naturally in the framework
of combined optimal control and filtering problems.
