  We show that the SDE $dX_t = \sigma(X_{t-}) \, dL_t$, $X_0 \sim \mu$ driven
by a one-dimensional symnmetric $\alpha$-stable L\'evy process $(L_t)_{t \geq
0}$, $\alpha \in (0,2]$, has a unique weak solution for any continuous function
$\sigma: \mathbb{R} \to (0,\infty)$ which grows at most linearly. Our approach
relies on random time changes of Feller processes. We study under which
assumptions the random-time change of a Feller process is a conservative
$C_b$-Feller process and prove the existence of a class of Feller processes
with decomposable symbols. In particular, we establish new existence results
for Feller processes with unbounded coefficients. As a by-product, we obtain a
sufficient condition in terms of the symbol of a Feller process $(X_t)_{t \geq
0}$ for the perpetual integral $\int_{(0,\infty)} f(X_{s}) \, ds$ to be
infinite almost surely.
