International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 258415, 17 pages
http://dx.doi.org/10.1155/2012/258415
Research Article

On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

V. P. Kurenok

Department of Electrical and Systems Engineering, Washington University in St. Louis, One Brookings Drive 1, St. Louis, MO 63130-4899, USA

Received 16 August 2011; Accepted 4 January 2012

Academic Editor: Henri Schurz

Copyright © 2012 V. P. Kurenok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a one-dimensional stochastic equation 𝑑 𝑋 𝑡 = 𝑏 ( 𝑡 , 𝑋 𝑡 − ) 𝑑 𝑍 𝑡 + 𝑎 ( 𝑡 , 𝑋 𝑡 ) 𝑑 𝑡 , 𝑡 ≥ 0 , with respect to a symmetric stable process 𝑍 of index 0 < 𝛼 ≤ 2 . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation 𝑑 𝐿 𝑡 = 𝐵 ( 𝐿 𝑡 − ) 𝑑 𝑊 𝑡 with respect to the semimartingale 𝑊 = ( 𝑍 , 𝑡 ) and corresponding matrix 𝐵 . In the case of 1 ≤ 𝛼 < 2 we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper.