Stochastic differential equations with generalized drift vector

It is proved that the paths of the continuous Markov process constructed in [3] are the solutions of the stochastic differential equation
$$ dx(t)=a(x(t))dt+b^{1/2}(x(t))\,dw(t), $$
where $b(x)$, $x\in R^m$, is uniformly nonsingular bounded and Hölder continuous diffusion matrix and $a(x)$, $x\in R^m$, is the drift vector which may be represented in the form $a(x)=q(x)N(x)\delta_S(x)$. Here $S$ is the $(m-1)$-dimensional surface in $R^m$, $q(x)$, $|q(x)|\le 1$ is real valued continuous function, $N(x)$ is the conormal vector to $S$ at the point $x$ and $\delta_S(x)$ is the generalized function on $R^m$ action of which on the basic function is reduced to the integration over the surface $S$.