Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910091926568960 |
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| author | Bogso, Antoine Marie Pellat, Rhoss Likibi Kamdem, Wilfried Kuissi Pamen, Olivier Menoukeu |
| author_facet | Bogso, Antoine Marie Pellat, Rhoss Likibi Kamdem, Wilfried Kuissi Pamen, Olivier Menoukeu |
| contents | We study a stochastic control problem for nonlinear systems governed by
stochastic differential equations with irregular drift. The drift
coefficient is assumed to decompose as $b(t,x,a)=b_1(t,x)+b_2(x)b_3(t,a)$, where $b_1$ is bounded and Borel
measurable, $b_2$ has bounded variation, and $b_3$ is bounded and
smooth.
Under these minimal regularity assumptions, we establish a
Pontryagin-type stochastic maximum principle. The analysis relies on
new results for SDEs with random drift of bounded variation, including
existence, uniqueness, and Malliavin-Sobolev differentiability of the
state process. A key ingredient is an explicit representation of the
first variation process obtained via integration with respect to the
space-time local time of bounded variation processes. By combining a
suitable approximation scheme with Ekeland's variational principle, and
using a Garcia-Rodemich-Rumsey inequality to obtain a uniform control
of the first variation, we derive the maximum principle. As an application, we derive an optimal corridor-type capital adjustment policy for an insurance surplus model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach Bogso, Antoine Marie Pellat, Rhoss Likibi Kamdem, Wilfried Kuissi Pamen, Olivier Menoukeu Optimization and Control Probability 60H10, 60H07, 49J55 We study a stochastic control problem for nonlinear systems governed by stochastic differential equations with irregular drift. The drift coefficient is assumed to decompose as $b(t,x,a)=b_1(t,x)+b_2(x)b_3(t,a)$, where $b_1$ is bounded and Borel measurable, $b_2$ has bounded variation, and $b_3$ is bounded and smooth. Under these minimal regularity assumptions, we establish a Pontryagin-type stochastic maximum principle. The analysis relies on new results for SDEs with random drift of bounded variation, including existence, uniqueness, and Malliavin-Sobolev differentiability of the state process. A key ingredient is an explicit representation of the first variation process obtained via integration with respect to the space-time local time of bounded variation processes. By combining a suitable approximation scheme with Ekeland's variational principle, and using a Garcia-Rodemich-Rumsey inequality to obtain a uniform control of the first variation, we derive the maximum principle. As an application, we derive an optimal corridor-type capital adjustment policy for an insurance surplus model. |
| title | Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach |
| topic | Optimization and Control Probability 60H10, 60H07, 49J55 |
| url | https://arxiv.org/abs/2505.09309 |