Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912040653684736 |
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| author | Shashkov, Timofei |
| author_facet | Shashkov, Timofei |
| contents | Let $\mathbf{B}(t)=(B_1(t), B_2(t))$, $t\geq 0$ be a two-dimensional Brownian motion with independent components and define the $\mathbfγ$-reflected process $$\mathbf{X}(t)=(X_1(t),X_2(t))=\left(B_1(t)-c_1t-γ_1\inf_{s_1\in[0,t]}(B_1(s_1)-c_1s_1),B_2(t)-c_2t-γ_2\inf_{s_2\in[0,t]}(B_2(s_2)-c_2s_2)\right),$$ with given finite constants $c_1,c_2$ and $γ_1,γ_2\in[0,2)$. The goal of this paper is to derive the asymptotics of the ruin probability $$\mathbb{P}\{\exists_{t\in[0,T]}: X_1(t)>u,X_2(t)>au\}$$ as $u\to\infty$ and $T>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax Shashkov, Timofei Probability Primary 60G15, secondary 60G70 Let $\mathbf{B}(t)=(B_1(t), B_2(t))$, $t\geq 0$ be a two-dimensional Brownian motion with independent components and define the $\mathbfγ$-reflected process $$\mathbf{X}(t)=(X_1(t),X_2(t))=\left(B_1(t)-c_1t-γ_1\inf_{s_1\in[0,t]}(B_1(s_1)-c_1s_1),B_2(t)-c_2t-γ_2\inf_{s_2\in[0,t]}(B_2(s_2)-c_2s_2)\right),$$ with given finite constants $c_1,c_2$ and $γ_1,γ_2\in[0,2)$. The goal of this paper is to derive the asymptotics of the ruin probability $$\mathbb{P}\{\exists_{t\in[0,T]}: X_1(t)>u,X_2(t)>au\}$$ as $u\to\infty$ and $T>0$. |
| title | Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax |
| topic | Probability Primary 60G15, secondary 60G70 |
| url | https://arxiv.org/abs/2403.02941 |