  In this paper we study the existence and uniqueness of the strong solution of
following d dimensional stochastic differential equation (SDE) driven by
Brownian motion: dX(t)=b(t,X(t))dt+a(t,X(t))dB(t), X(0)= x, where B is a
d-dimensional standard Brownian motion; the diffusion coefficient a is a Holder
continuous and uniformly non-degenerate matrix-valued function and the drift
coefficient b may be discontinuous and unbounded, not necessarily in Sobolev
space, extending the previous works to discontinuous and unbounded drift
coefficient situation. The idea is to combine the Zvonkin transformation with
the Lyapunov function approach. To this end, we need to establish a local
version of the connection between the solutions of the SDE up to the exit time
of a bounded connected open set D and the associated partial differential
equation on this domain. As an interesting byproduct, we establish a localized
version of the Krylov estimates (Theorem 4.1) and a localized version of the
stability result of the stochastic differential equations of discontinuous
coefficients (Theorem 4.5).
