Abstract
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 (ℛe ψ(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.
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Kurenok, V.P. Stochastic Equations with Time-Dependent Drift Driven by Levy Processes. J Theor Probab 20, 859–869 (2007). https://doi.org/10.1007/s10959-007-0086-x
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DOI: https://doi.org/10.1007/s10959-007-0086-x