Asymptotic expansions are obtained for finite-dimensional
symmetric stable distributions.
They are used to prove the existence of continuous
transition probability densities of stable and stable-like
jump-diffusions, and to construct local multiplicative asymptotics
and global two-sided estimates for these densities.
As a consequence, the distribution of the first passage times
for stable jump-diffusions is estimated and the
integral test for the limsup behaviour of their sample paths
as $t\to 0$ is provided. 1991 Mathematics Subject Classification: 60E07, 60G17, 60J35, 47D07.
