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

TL;DR

This paper establishes the existence of solutions for stochastic differential equations driven by symmetric stable processes with measurable coefficients, extending Krylov's results from Brownian motion to stable processes of index between 1 and 2.

Contribution

It generalizes Krylov's classical results for Brownian motion to the case of symmetric stable processes with measurable coefficients, introducing new integral estimates and methods.

Findings

Proves existence of solutions for SDEs driven by stable processes with measurable coefficients.

Develops integral estimates of Krylov type for these equations.

Introduces a novel approach for deriving integral estimates in this context.

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.