Symmetric Stable Laws and Stable‐Like Jump‐Diffusions

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.