Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls
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| Format: | Preprint |
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2026
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| _version_ | 1866917490805702656 |
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| author | Ding, Kai Wen, Jiaqiang Xiong, Jie Zhang, Xin |
| author_facet | Ding, Kai Wen, Jiaqiang Xiong, Jie Zhang, Xin |
| contents | This paper studies finite-horizon stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps, where the weighting matrices may be random and indefinite. Under a uniform convexity condition on the cost functional, we prove that the associated stochastic Riccati equation (SRE) with jumps admits a unique strongly regular solution. As a consequence, the open-loop optimal control admits a closed-loop representation. The proof does not rely on a global representation of the form $P=\mathbf Y\mathbf X^{-1}$ or on any nonsingularity condition on the jump multiplier $I_n+E$ in the state equation. Instead, we construct $P$ from the stochastic value flow, and derive the strong regularity of the Riccati solution by a small-interval localization method. In addition, sufficient conditions are obtained for uniform convexity, and examples are presented to illustrate indefinite terminal and control weighting matrices and a nonzero jump martingale component in the SRE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls Ding, Kai Wen, Jiaqiang Xiong, Jie Zhang, Xin Optimization and Control 49N10, 93E20 This paper studies finite-horizon stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps, where the weighting matrices may be random and indefinite. Under a uniform convexity condition on the cost functional, we prove that the associated stochastic Riccati equation (SRE) with jumps admits a unique strongly regular solution. As a consequence, the open-loop optimal control admits a closed-loop representation. The proof does not rely on a global representation of the form $P=\mathbf Y\mathbf X^{-1}$ or on any nonsingularity condition on the jump multiplier $I_n+E$ in the state equation. Instead, we construct $P$ from the stochastic value flow, and derive the strong regularity of the Riccati solution by a small-interval localization method. In addition, sufficient conditions are obtained for uniform convexity, and examples are presented to illustrate indefinite terminal and control weighting matrices and a nonzero jump martingale component in the SRE. |
| title | Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls |
| topic | Optimization and Control 49N10, 93E20 |
| url | https://arxiv.org/abs/2605.13204 |