Global regularity of the value function in a stopper vs. singular-controller game
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918068988411904 |
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| author | Bovo, Andrea Milazzo, Alessandro |
| author_facet | Bovo, Andrea Milazzo, Alessandro |
| contents | We study a class of zero-sum stochastic games between a stopper and a singular-controller, previously considered in [Bovo and De Angelis (2025)]. The underlying singularly-controlled dynamics takes values in $\mathcal{O}\subseteq\mathbb{R}$. The problem is set on a finite time-horizon and is connected to a parabolic variational inequality of min-max type with spatial-derivative and obstacle constraints.
We show that the value function of the problem is of class $C^1$ in the whole domain $[0,T)\times\mathcal{O}$ and that the second-order spatial derivative and the second-order mixed derivative are continuous everywhere except for a (potential) jump across a non-decreasing curve (the stopping boundary of the game). The latter discontinuity is a natural consequence of the partial differential equation associated to the problem. Beyond its intrinsic analytical value, such a regularity for the value function is a stepping stone for further exploring the structure and properties of the free-boundaries of the stochastic game, which in turn determine the optimal strategies of the players. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global regularity of the value function in a stopper vs. singular-controller game Bovo, Andrea Milazzo, Alessandro Optimization and Control Analysis of PDEs Probability 35R35, 49N60, 60G40, 91A05, 91A15, 93E20 We study a class of zero-sum stochastic games between a stopper and a singular-controller, previously considered in [Bovo and De Angelis (2025)]. The underlying singularly-controlled dynamics takes values in $\mathcal{O}\subseteq\mathbb{R}$. The problem is set on a finite time-horizon and is connected to a parabolic variational inequality of min-max type with spatial-derivative and obstacle constraints. We show that the value function of the problem is of class $C^1$ in the whole domain $[0,T)\times\mathcal{O}$ and that the second-order spatial derivative and the second-order mixed derivative are continuous everywhere except for a (potential) jump across a non-decreasing curve (the stopping boundary of the game). The latter discontinuity is a natural consequence of the partial differential equation associated to the problem. Beyond its intrinsic analytical value, such a regularity for the value function is a stepping stone for further exploring the structure and properties of the free-boundaries of the stochastic game, which in turn determine the optimal strategies of the players. |
| title | Global regularity of the value function in a stopper vs. singular-controller game |
| topic | Optimization and Control Analysis of PDEs Probability 35R35, 49N60, 60G40, 91A05, 91A15, 93E20 |
| url | https://arxiv.org/abs/2506.19129 |