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 dXt=b(Xt-)dZt where b is a Borel measurable real function and Z is a strictly α-stable Lévy process (0 < α ≤ 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., α = 2) to the class of equations with α such that 1 < α ≤ 2. The results employ some representation properties with respect to strictly stable Lévy processes.