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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.15662 |
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| _version_ | 1866910379501682688 |
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| author | Almuzaini, Atiqah Ma, Jin |
| author_facet | Almuzaini, Atiqah Ma, Jin |
| contents | In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hausdorff distance of all elements to the "generating" cone is finite. Such space should in particular include the so-called "upper sets", and has many useful cases in finance, such as the well-known set-valued risk measures, as well as the solvency cone in some super-hedging problems. We shall argue that, for such a special class of unbounded sets, under some conditions, the cancellation law is still valid, eliminating a major obstacle for extending the set-valued analysis to non-compact sets. We shall establish some basic algebraic and topological properties of such spaces, and show that some standard techniques will again be valid in studying the set-valued SDEs with unbounded (drift) coefficients which, to the best of our knowledge, is new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set-Valued Stochastic Differential Equations with Unbounded Coefficients Almuzaini, Atiqah Ma, Jin Probability In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hausdorff distance of all elements to the "generating" cone is finite. Such space should in particular include the so-called "upper sets", and has many useful cases in finance, such as the well-known set-valued risk measures, as well as the solvency cone in some super-hedging problems. We shall argue that, for such a special class of unbounded sets, under some conditions, the cancellation law is still valid, eliminating a major obstacle for extending the set-valued analysis to non-compact sets. We shall establish some basic algebraic and topological properties of such spaces, and show that some standard techniques will again be valid in studying the set-valued SDEs with unbounded (drift) coefficients which, to the best of our knowledge, is new. |
| title | Set-Valued Stochastic Differential Equations with Unbounded Coefficients |
| topic | Probability |
| url | https://arxiv.org/abs/2403.15662 |