On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

Abstract EN

We consider a one-dimensional stochastic equation dXt=b(t,Xt−)dZt+a(t,Xt)dt, 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 dLt=B(Lt−)dWt 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.