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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 ∈ IR, d ≥ 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0 = x 0 ∈ IR d 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.doi:10.1515/156939706779801705 fatcat:36egleynbrcmhl4lyghuunajdm