Stochastic equations with multidimensional drift driven by Levy processes

The stochastic equation dXt = dLt + a(t,Xt)dt, t ≥ 0, is considered where L is a d-dimensional Levy process with the characteristic exponent ψ(ξ), ξ ∈ IR, d ≥ 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X0 = x0 ∈ IRd when (Reψ(ξ))−1 = o(|ξ|−1) as |ξ | → ∞. 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