Multivariate Tempered Space-Fractional Negative Binomial Process and Risk Models with Shocks
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909209134628864 |
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| author | Pathak, Ashok Kumar Soni, Ritik |
| author_facet | Pathak, Ashok Kumar Soni, Ritik |
| contents | In this paper, we first define the multivariate tempered space-fractional Poisson process (MTSFPP) by time-changing the multivariate Poisson process with an independent tempered α-stable subordinator. Its distributional properties, the mixture tempered time and space variants and their PDEs connections are studied. Then we define the multivariate tempered space-fractional negative binomial process (MTSFNBP) and explore its key features. The Lévy measure density for the MTSFNBP is also derived. We present a bivariate risk model with a common shock driven by the tempered space-fractional negative binomial process. We demonstrate that the total claim amount process is stochastically equivalent to a univariate generalized Cramer-Lundberg risk model. In addition, some important ruin measures such as ruin probability, joint distribution of time to ruin and deficit at ruin along with governing integro-differential equations are obtained. Finally we show that the underlying risk process exhibits the long range dependence property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariate Tempered Space-Fractional Negative Binomial Process and Risk Models with Shocks Pathak, Ashok Kumar Soni, Ritik Probability 60G22, 60G51, 60E05, 60G55, 91B05 In this paper, we first define the multivariate tempered space-fractional Poisson process (MTSFPP) by time-changing the multivariate Poisson process with an independent tempered α-stable subordinator. Its distributional properties, the mixture tempered time and space variants and their PDEs connections are studied. Then we define the multivariate tempered space-fractional negative binomial process (MTSFNBP) and explore its key features. The Lévy measure density for the MTSFNBP is also derived. We present a bivariate risk model with a common shock driven by the tempered space-fractional negative binomial process. We demonstrate that the total claim amount process is stochastically equivalent to a univariate generalized Cramer-Lundberg risk model. In addition, some important ruin measures such as ruin probability, joint distribution of time to ruin and deficit at ruin along with governing integro-differential equations are obtained. Finally we show that the underlying risk process exhibits the long range dependence property. |
| title | Multivariate Tempered Space-Fractional Negative Binomial Process and Risk Models with Shocks |
| topic | Probability 60G22, 60G51, 60E05, 60G55, 91B05 |
| url | https://arxiv.org/abs/2405.13813 |