On stochastic It\^o processes with drift in $L_d$

For It^o stochastic processes in ${\mathbb{R}}^d$ with drift in $L_d$ Aleksandrov’s type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding elliptic and parabolic operators in $L_p$ and $L_{p+1}$, respectively, where $p\ge d$.