On stochastic equations with measurable coefficients driven by symmetric stable processes

We consider a one-dimensional stochastic equation dX _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 dL _t= B (L _t-) dW _t with respect to the semimartingale W = (Z, t) and corresponding matrix B. In the case of 1 < 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.