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

We consider the stochastic differential equation dXt= b(Xt)dZt, t[greater-or-equal, slanted]o, where b is a Borel measurable real function and Z is a symmetric [alpha]-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 [alpha]-stable Lévy motion with 1 < [alpha] [less-than-or-equals, slant] 2. In Section 2 we study the existence of weak solutions for the above equation. When 0 < [alpha] < 1 or 1 < [alpha] [less-than-or-equals, slant] 2 we prove a sufficient existence condition. In the case 1 < [alpha] [less-than-or-equals, slant] 2, we extend Engelbert and Schmidt's necessary and sufficient existence condition (for the equation driven by a Wiener process) to the above equation: we prove that, for every [alpha] there exists a nontrivial solution starting from [alpha], if and only if b [alpha] is locally integrable. In Section 3 we study "local" solutions. We also prove a result relating "local" and "global" solutions.