Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.16847 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913239097409536 |
|---|---|
| author | Arharas, Ihsan Bouhadou, Siham Hilbert, Astrid Ouknine, Youssef |
| author_facet | Arharas, Ihsan Bouhadou, Siham Hilbert, Astrid Ouknine, Youssef |
| contents | We consider the optimal stopping time problem under model uncertainty $R(v)= {\text{ess}\sup\limits}_{ \mathbb{P} \in \mathcal{P}} {\text{ess}\sup\limits}_{τ\in \mathcal{S}_v} E^\mathbb{P}[Y(τ) \vert \mathcal{F}_v]$, for every stopping time $v$, set in the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, and particularly allows for general payoff processes that are not necessarily right-continuous. Under weaker integrability, and regularity assumptions on the reward family $Y=(Y(v), v\in \mathcal{S})$, we show the existence of an optimal stopping time. We then proceed to find sufficient conditions for the existence of an optimal model. For this purpose, we present a universal Doob-Meyer-Mertens's decomposition for the Snell envelope family associated with $Y$ in the sense that it holds simultaneously for all $\mathbb{P} \in \mathcal{P}$. This decomposition is then employed to prove the existence of an optimal probability model and study its properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16847 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Stopping Under Model Uncertainty in a General Setting Arharas, Ihsan Bouhadou, Siham Hilbert, Astrid Ouknine, Youssef Probability 60G40, 60H30, 60G07 We consider the optimal stopping time problem under model uncertainty $R(v)= {\text{ess}\sup\limits}_{ \mathbb{P} \in \mathcal{P}} {\text{ess}\sup\limits}_{τ\in \mathcal{S}_v} E^\mathbb{P}[Y(τ) \vert \mathcal{F}_v]$, for every stopping time $v$, set in the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, and particularly allows for general payoff processes that are not necessarily right-continuous. Under weaker integrability, and regularity assumptions on the reward family $Y=(Y(v), v\in \mathcal{S})$, we show the existence of an optimal stopping time. We then proceed to find sufficient conditions for the existence of an optimal model. For this purpose, we present a universal Doob-Meyer-Mertens's decomposition for the Snell envelope family associated with $Y$ in the sense that it holds simultaneously for all $\mathbb{P} \in \mathcal{P}$. This decomposition is then employed to prove the existence of an optimal probability model and study its properties. |
| title | Optimal Stopping Under Model Uncertainty in a General Setting |
| topic | Probability 60G40, 60H30, 60G07 |
| url | https://arxiv.org/abs/2303.16847 |