On stochastic differential equations driven by a Cauchy process and other stable Lévy motions

Abstract

We consider the class of one-dimensional stochastic differential equations

$$dX_t = b(X_{t-})dZ_t, \quad t \geq 0,$$

where $b$ is a Borel measurable real function and $Z$ is a strictly $\alpha$-stable Lévy process $(0 < \alpha \leq 2)$. Weak solutions are investigated improving previous results of the author in various ways.

In particular, for the equation driven by a strictly 1-stable Lévy process, a sufficient existence condition is proven.

Also we extend the weak existence and uniqueness exact criteria due to Engelbert and Schmidt for the Brownian case (i.e., $\alpha = 2$) to the class of equations with $\alpha$ such that $1 < \alpha \leq 2$. The results employ some representation properties with respect to strictly stable Lévy processes.