On solutions of equations with measurable coefficients driven by $\alpha$- stable processes

Abstract

We prove the existence of solutions for the stochastic differential equation $dX_t=b(t,X_{t-})dZ_t+a(t,X_t)dt, X_0\in\R, t\ge 0,$ with only measurable coefficients $a$ and $b$ satisfying the condition $0<\mu\le |b(t,x)|\le \nu$ and $|a(t,x)|\le K$ for all $t\ge 0, x\in\R$ where $\mu, \nu, $ and $K$ are some constants. The driving process $Z$ is a symmetric stable process of index $1<\alpha<2$. This generalizes the result of N. V. Krylov \cite{Krylov} for the case of $\alpha=2$, that is when $Z$ is a Brownian motion. The proof is based on integral estimates of Krylov type for the given equation which are also derived in the note and are of independent interest. Moreover, unlike in \cite{Krylov}, we use a different approach to derive the corresponding integral estimates.