A dynamic programming principle for multiperiod control problems with bicausal constraints
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929570207236096 |
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| author | Mirmominov, Ruslan Wiesel, Johannes |
| author_facet | Mirmominov, Ruslan Wiesel, Johannes |
| contents | We consider multiperiod stochastic control problems with non-parametric uncertainty on the underlying probabilistic model. We derive a new metric on the space of probability measures, called the adapted $(p, \infty)$--Wasserstein distance $\mathcal{AW}_p^\infty$ with the following properties: (1) the adapted $(p, \infty)$--Wasserstein distance generates a topology that guarantees continuity of stochastic control problems and (2) the corresponding $\mathcal{AW}_p^\infty$-distributionally robust optimization (DRO) problem can be computed via a dynamic programming principle involving one-step Wasserstein-DRO problems. If the cost function is semi-separable, then we further show that a minimax theorem holds, even though balls with respect to $\mathcal{AW}_p^\infty$ are neither convex nor compact in general. We also derive first-order sensitivity results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A dynamic programming principle for multiperiod control problems with bicausal constraints Mirmominov, Ruslan Wiesel, Johannes Optimization and Control Mathematical Finance We consider multiperiod stochastic control problems with non-parametric uncertainty on the underlying probabilistic model. We derive a new metric on the space of probability measures, called the adapted $(p, \infty)$--Wasserstein distance $\mathcal{AW}_p^\infty$ with the following properties: (1) the adapted $(p, \infty)$--Wasserstein distance generates a topology that guarantees continuity of stochastic control problems and (2) the corresponding $\mathcal{AW}_p^\infty$-distributionally robust optimization (DRO) problem can be computed via a dynamic programming principle involving one-step Wasserstein-DRO problems. If the cost function is semi-separable, then we further show that a minimax theorem holds, even though balls with respect to $\mathcal{AW}_p^\infty$ are neither convex nor compact in general. We also derive first-order sensitivity results. |
| title | A dynamic programming principle for multiperiod control problems with bicausal constraints |
| topic | Optimization and Control Mathematical Finance |
| url | https://arxiv.org/abs/2410.23927 |