Stochastic equations with multidimensional drift driven by Levy processes

The stochastic equation dX(t) = dL(t) + a(t,X-t)dt, t >= 0, is considered where L is a d-dimensional Levy process with the characteristic exponent psi(xi), xi is an element of Bbb R, d >= 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X-o = x(o) is an element of R-d when (Re psi(xi)(1) = 0(vertical bar xi vertical bar) as vertical bar xi vertical bar -> infinity. The proof idea is based on Krylov's estimates for Levy processes with time -dependent drift and some variants of those estimates are derived in this note.