On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider a one-dimensional stochastic equation ð ‘‘ ð ‘‹ ð ‘¡ = ð ‘ ( ð ‘¡ , ð ‘‹ ð ‘¡ − ) ð ‘‘ ð ‘ ð ‘¡ + ð ‘Ž ( ð ‘¡ , ð ‘‹ ð ‘¡ ) ð ‘‘ ð ‘¡ , ð ‘¡ ≥ 0 , with respect to a symmetric stable process ð ‘ of index 0 < ð ›¼ ≤ 2 . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation ð ‘‘ ð ¿ ð ‘¡ = ð µ ( ð ¿ ð ‘¡ − ) ð ‘‘ ð ‘Š ð ‘¡ with respect to the semimartingale ð ‘Š = ( ð ‘ , ð ‘¡ ) and corresponding matrix ð µ . In the case of 1 ≤ ð ›¼ < 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.