Stochastic equations with time-dependent drift driven by levy processes

V. P. Kurenok

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8 Scopus citations

Abstract

The stochastic equation dX t =dS t +a(t,X t )dt, t0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ â. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0=x 0 â when (e∈ 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.

Original languageEnglish
Pages (from-to)859-869
Number of pages11
JournalJournal of Theoretical Probability
Volume20
Issue number4
DOIs
StatePublished - Dec 2007

Keywords

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

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