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 ψ(ξ), ξ ∈ Bbb R, d ≥ 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ⁰ = x ₀ ∈ ℝ ^d when (Re ψ(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.