On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

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.