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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.09836 |
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| _version_ | 1866916218973192192 |
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| author | Choulli, Tahir Alsheyab, Safa' |
| author_facet | Choulli, Tahir Alsheyab, Safa' |
| contents | This paper considers a pair $(\mathbb{F},τ)$, where $\mathbb{F}$ is a filtration representing the "public" flow of information which is available to all agents overtime, and $τ$ is a random time which might not be an $\mathbb{F}$-stopping time. This setting covers the case of credit risk framework where $τ$ models the default time of a firm or client, and the setting of life insurance where $τ$ is the death time of an agent. It is clear that random times can not be observed before their occurrence. Thus the larger filtration $\mathbb{G}$, which incorporates $\mathbb{F}$ and makes $τ$ observable, results from the progressive enlargement of $\mathbb{F}$ with $τ$. For this informational setting, governed by $\mathbb{G}$, we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of $\mathbb{F}$-observable processes. The second direction lies in deriving the {\it mathematical structures} of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under $\mathbb{F}$. These three aspects allow us to quantify deeply how $τ$ impact the optimal stopping problem, while they are also vital for studying reflected backward stochastic differential equations which arise {\it naturally} from pricing and hedging of vulnerable claims. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09836 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal stopping problem under random horizon Choulli, Tahir Alsheyab, Safa' Probability This paper considers a pair $(\mathbb{F},τ)$, where $\mathbb{F}$ is a filtration representing the "public" flow of information which is available to all agents overtime, and $τ$ is a random time which might not be an $\mathbb{F}$-stopping time. This setting covers the case of credit risk framework where $τ$ models the default time of a firm or client, and the setting of life insurance where $τ$ is the death time of an agent. It is clear that random times can not be observed before their occurrence. Thus the larger filtration $\mathbb{G}$, which incorporates $\mathbb{F}$ and makes $τ$ observable, results from the progressive enlargement of $\mathbb{F}$ with $τ$. For this informational setting, governed by $\mathbb{G}$, we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of $\mathbb{F}$-observable processes. The second direction lies in deriving the {\it mathematical structures} of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under $\mathbb{F}$. These three aspects allow us to quantify deeply how $τ$ impact the optimal stopping problem, while they are also vital for studying reflected backward stochastic differential equations which arise {\it naturally} from pricing and hedging of vulnerable claims. |
| title | Optimal stopping problem under random horizon |
| topic | Probability |
| url | https://arxiv.org/abs/2301.09836 |