Asymptotics of survival probabilities and lower tail probability problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917887077253120 |
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| author | Boyarchenko, Svetlana Levendorskiĭ, Sergei |
| author_facet | Boyarchenko, Svetlana Levendorskiĭ, Sergei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of survival probabilities and lower tail probability problem Boyarchenko, Svetlana Levendorskiĭ, Sergei Probability 60G51, 60G52, 60-08, 65C05, 91G05, 91G20, 97M30 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. |
| title | Asymptotics of survival probabilities and lower tail probability problem |
| topic | Probability 60G51, 60G52, 60-08, 65C05, 91G05, 91G20, 97M30 |
| url | https://arxiv.org/abs/2501.04218 |