Asymptotics of survival probabilities and lower tail probability problem

Svetlana Boyarchenko; Sergei Levendorskiĭ

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arXiv · 2501.04218

Asymptotics of survival probabilities and lower tail probability problem

Svetlana Boyarchenko·Sergei Levendorskiĭ

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Abstract

The present paper is an addendum to the paper ``Lévy models amenable to efficient calculations", where we introduced a general class of Stieltjes-Lévy processes (SL-processes) and signed SL processes defined in terms of certain Stieltjes-Lévy 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 Lévy process and its extremum processes, and proved that essentially all popular classes of Lévy 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 $ϕ^\pm_q$ 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.

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Svetlana Boyarchenko, Sergei Levendorskiĭ. 2025-01-08. Asymptotics of survival probabilities and lower tail probability problem. https://arxiv.org/abs/2501.04218

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