s=:M^{r,\varphi}_t\end{aligned}$$ is an $\sF_t$-martingale with respect to $\mP$ after time $r$. We denote by $\sP^{\sigma,b}_{r,z}$ the set of all martingale solutions associated with $(\sigma,b)$ and starting from $(r,z)$.

(ii) (Well-posedness) One says that the martingale problem for $\sL^{a,b}_t$ is well-posed if for each $(r,z)\in\mR_+\times\mR^{2d}$, there is exactly one solution $\mP_{r,z}$ to the martingale problem for $\sL^{a,b}_t$ starting from $(r,z)$.

(iii) (Weak solution) A triple $((\tilde Z,\tilde W); (\tilde \Omega,\tilde \sF,\tilde \mP);(\tilde \sF_t)_{t\geq 0})$ is called a weak solution of SDE with starting point $(r,z)\in\mR_+\times\mR^{2d}$ if $(\tilde \sF_t)_{t\geq 0}$ satisfies the usual conditions, $\tilde W$ is a $d$-dimensional $\tilde \sF_t$-Brownian motion, and $\tilde Z=(\tilde X,\dot{\tilde  X})$ is an $\tilde \sF_t$-adapted $\mR^{2d}$-valued process satisfying that $$\int^t_r(\|a_s(\tilde Z_s)\|+|b_s(\tilde Z_s)|)\dif s<\infty, \ t\geq r,\ \  \tilde\mP-a.s.,$$ and $$\tilde Z_t=z+\int^t_r(\dot{\tilde  X}_s,b_s(\tilde Z_s))\dif s+\int^t_r(0,\sigma_s(\tilde Z_s)\dif\tilde  W_s).$$

It is well known that the martingale solutions and weak solutions are equivalent, for example, see [@Ka-Sh p.318, Proposition 4.11].

Krylov’s type estimate
----------------------

In this subsection we first show the following important estimate of Krylov’s type for weak solutions. \[Th41\] Suppose that $\sigma$ satisfies [**(UE)**]{} and $$\begin{aligned}
\label{EJ1}
\lim_{|z-z'|\to 0}\sup_{t\in[0,T]}\|\sigma_t(z)-\sigma_t(z')\|=0,\  T>0,\end{aligned}$$ and $b\in \cap_T\mL^q(T)$ for some $q>(2(2d+1),\infty]$. Then for any $p>2d+1$ and $T>0$, there is a constant $C>0$ depending 