  The theory of one-dimensional stochastic differential equations driven by
Brownian motion is classical and has been largely understood for several
decades. For stochastic differential equations with jumps the picture is still
incomplete, and even some of the most basic questions are only partially
understood. In the present article we study existence and uniqueness of weak
solutions to \[
  {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t
  \]driven by a (symmetric) $\alpha$-stable L\'evy process, in the spirit of
the classical Engelbert-Schmidt time-change approach. Extending and completing
results of Zanzotto we derive a complete characterisation for existence und
uniqueness of weak solutions for $\alpha\in(0,1)$. Our approach is not based on
classical stochastic calculus arguments but on the general theory of Markov
processes. We proof integral tests for finiteness of path integrals under
minimal assumptions.
