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Lithuanian Mathematical Journal

, Volume 47, Issue 4, pp 423–435 | Cite as

On driftless one-dimensional SDE’s with respect to stable Levy processes

  • V. P. Kurenok
Article

Abstract

The time-dependent SDE dX t = b(t, X t)dZ t with X 0 = x 0 ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation \(dA_t = \left| b \right|^\alpha (t,\bar Z \circ A_t )dt\) in class of time change processes, where \(\bar Z\) is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2.

Keywords

one-dimensional stochastic equations measurable coefficients symmetric stable processes time change equation monotone convergence 

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

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

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