Strong solution of stochastic differential equations with discontinuous and unbounded coefficients

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In this paper we study the existence and uniqueness of strong solution of the following d d -dimensional stochastic differential equation (SDE) driven by Brownian motion: d X t = b ( t , X t ) d t + σ ( t , X t ) d B t , X 0 = x , \begin{equation*} dX_t=b(t,X_t)dt+\sigma (t,X_t)dB_t, X_0=x, \end{equation*} where B B is a d d -dimensional standard Brownian motion; the diffusion coefficient σ \sigma is a Hölder continuous and uniformly nondegenerate d × d d\times d matrix-valued function and the drift coefficient b b may be discontinuous and unbounded, not necessarily in L p q \mathbb {L}_p^q , extending the previous works to discontinuous and unbounded drift coefficient situation. The idea is to combine the Zvonkin’s transformation with the Lyapunov function approach. Zvonkin’s transformation is a one-to-one (and quasi-isometric) transformation of a phase space that allows us to pass from a diffusion process with nonzero drift coefficient to a process without drift. 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 D and the associated partial differential equation on this domain. As an interesting by-product, we establish a localized version of the Krylov estimates and a localized version of the stability result of the SDEs of discontinuous coefficients.