On the area between a Lévy process with secondary jump inputs and its reflected version
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913463575511040 |
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| author | Kella, Offer Mandjes, Michel |
| author_facet | Kella, Offer Mandjes, Michel |
| contents | We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08753 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the area between a Lévy process with secondary jump inputs and its reflected version Kella, Offer Mandjes, Michel Probability 60G51 (Primary) 60K25(Secondary) We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful. |
| title | On the area between a Lévy process with secondary jump inputs and its reflected version |
| topic | Probability 60G51 (Primary) 60K25(Secondary) |
| url | https://arxiv.org/abs/2311.08753 |