On stochastic equations with measurable coefficients driven by symmetric stable processes
Summary: We consider a one-dimensional stochastic equation \[ dX_t = b(t, X_{t-})dZ_t + a(t, X_t)dt,\;t \geq 0, \] with respect to a symmetric stable process \(Z\) of index \(0 < \alpha \leq 2\). It is shown that solving this equation is equivalent to solving of a two-dimensional stochastic equation \[ dL_t = B(L_{t-})dWt \] with respect to the semimartingale \(W = (Z, t)\) and the corresponding matrix \(B\). In the case of \(1 \leq \alpha < 2\), we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. Existence proofs are established using the method of Krylov's estimates for processes satisfying the two-dimensional equation. Also, the Krylov estimates are based on some analytical facts of independent interest that are also proved.
scientific article; zbMATH DE number 6031633
scientific article