Journal of Theoretical Probability

, Volume 20, Issue 4, pp 859–869 | Cite as

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes



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∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0=x 0∈ℝ when (ℛeψ(ξ))−1=o(|ξ|−1) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 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.


One-dimensional Levy processes Time-dependent drift Krylov’s estimates Weak convergence 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Natural and Applied SciencesUniversity of Wisconsin-Green BayGreen BayUSA

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