Exponential moments of first passage times and related quantities for   L\'evy processes
  For a L\'evy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$. 
Frank Aurzada, Alexander Iksanov and Matthias Meiners
math.PR