Asymptotics of survival probabilities and lower tail probability problem

Abstract: The present paper is an addendum to the paper ``L\'evy models amenable to efficient calculations", where we introduced a general class of Stieltjes-L\'evy processes (SL-processes) and signed SL processes defined in terms of certain Stieltjes-L\'evy measures. We demonstrated that SL-processes enjoyed all properties that we used earlier to develop efficient methods for evaluation of expectations of functions of a L\'evy process and its extremum processes, and proved that essentially all popular classes of L\'evy processes are SL-processes; sSL-processes fail to possess one important property. In the present paper, we use the properties of (s)SL-processes to derive new formulas for the Wiener-Hopf factors $\phi^\pm_q$ for small $q$ in terms of the absolute continuous components of SL-measures and their densities, and calculate the leading terms of the survival probability also in terms of the absolute continuous components of SL-measures and their densities. The lower tail probability is calculated for more general classes of SINH-regular processes constructed earlier.