Abstract. Let X be of the form X t = t 0 b s dZ s + t 0 a s ds, t ≥ 0, where Z is a symmetric stable process of index α ∈ (1, 2) with Z 0 = 0. We obtain various L 2 -estimates for the process X. In particular, for m ∈ N, t ≥ 0, and any measurable, nonnegative function f we derive the inequalityAs an application of the obtained estimates, we prove the existence of solutions for the stochastic equation dX t = b(X t− )dZ t + a(X t )dt for any initial value x 0 ∈ R.

The stochastic equationWe prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial valueThe 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.

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The stochastic equation dX t = dS t + a(t, X t )dt, t ≥ 0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ), ξ ∈ IR. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0 = x 0 ∈ IR when (Reψ(ξ)) −1 = o(|ξ| −1 ) as |ξ| → ∞. These conditions coincide with those found by Tanaka-Tsuchiya-Watanabe (1974) in the case of a(t, x) = a(x). Our approach is based on Krylov's estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.

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