On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider a one-dimensional stochastic equation =(,−)+(,), ≥0, with respect to a symmetric stable process of index 0<≤2. It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation =(−) with respect to the semimartingale =(,) and corresponding matrix . In the case of 1≤<2 we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. Theexistence proofs are established using the method of Krylov's estimatesfor processes satisfying the 2-dimensional equation. On another hand,the Krylov's estimates are based on some analytical facts of independentinterest that are also proved in the paper