On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion

We consider the stochastic differential equation dXt= b(Xt)dZt, t⩾o, b is a Borel measurable real function and Z is a symmetric α-stable Lévy motion. In Section 1 we study the convergence of certain functionals of Z and in particular, we extend Engelbert and Schmidt 0–1 law (for functionals of the Wiener process) to functionals of a symmetric α-stable Lévy motion with 1 < α ⩽ 2. In Section 2 we study the existence of weak solutions for the above equation. When 0 < α < 1 or 1 < α ⩽ 2 we prove a sufficient existence condition. In the case 1 < α ⩽ 2, we extend Engelbert and Schmidtʹs necessary and sufficient existence condition (for the equation driven by a Wiener process) to the above equation: we prove that, for every α there exists a nontrivial solution starting from α, if and only if |b| α is locally integrable. In Section 3 we study “local” solutions. We also prove a result relating “local” and “global” solutions.