Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

V. P. Kurenok ()
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V. P. Kurenok: University of Wisconsin-Green Bay

Journal of Theoretical Probability, 2007, vol. 20, issue 4, 859-869

Abstract: 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.

Keywords: One-dimensional Levy processes; Time-dependent drift; Krylov’s estimates; Weak convergence (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0086-x

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