On the optimal stopping of Gauss-Markov bridges with random pinning points
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913854393417728 |
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| author | Azze, Abel D'Auria, Bernardo |
| author_facet | Azze, Abel D'Auria, Bernardo |
| contents | We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_03636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the optimal stopping of Gauss-Markov bridges with random pinning points Azze, Abel D'Auria, Bernardo Probability Applications 60G40, 60J60 We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes. |
| title | On the optimal stopping of Gauss-Markov bridges with random pinning points |
| topic | Probability Applications 60G40, 60J60 |
| url | https://arxiv.org/abs/2505.03636 |