|
|
相关噪声下随机最优控制问题的最大值原理
|
Abstract:
本文研究了一类延迟信息下的随机最优控制问题，其中控制过程是关于延迟信息的滤子流适应，且系统中多个噪声不独立具有相关性。本文首先利用凸变分法建立必要最大值原理，进一步假设哈密尔顿函数和终端效用函数具有凹性得到充分最大值原理，最后将得到的充分必要最大值原理应用于一类资产组合配置问题中的均值–方差模型。
This paper studies a class of stochastic optimal control problems with delayed information, where the control process is adapted to the delayed filtration, which describes the delayed information, and that the noises in the system are not independent but correlated. The necessary maximum principle is established using the convex variational method. Furthermore, the sufficient maximum principle is given on the assumption that the Hamiltonian function and the terminal utility function are concave. Finally, the obtained maximum principles are applied to the mean-variance model for a class of asset portfolio allocation problems.
| [1] | Yüksel, S. (2025) Another Look at Partially Observed Optimal Stochastic Control: Existence, Ergodicity, and Approximations without Belief-Reduction. Applied Mathematics & Optimization, 91, Article No. 16. https://doi.org/10.1007/s00245-024-10211-9 |
| [2] | Han, B. and Wong, H.Y. (2019) Optimal Investment and Consumption Problems under Correlation Ambiguity. IMA Journal of Management Mathematics, 31, 69-89. https://doi.org/10.1093/imaman/dpz002 |
| [3] | Bertsekas, D.P. and White, C.C. (1977) Dynamic Programming and Stochastic Control. IEEE Transactions on Systems, Man, and Cybernetics, 7, 758-759. https://doi.org/10.1109/tsmc.1977.4309612 |
| [4] | Gamkrelidze, R.V. (1977) Principles of Optimal Control Theory. Springer. |
| [5] | Yong, J.M. and Zhou, X.Y. (1999) Stochastic Control: Hamiltonian Systems and HJB Equations. Springer. |
| [6] | Baghery, F. and Øksendal, B. (2007) A Maximum Principle for Stochastic Control with Partial Information. Stochastic Analysis and Applications, 25, 705-717. https://doi.org/10.1080/07362990701283128 |
| [7] | Li, N. and Wu, Z. (2016) Stochastic Linear-Quadratic Optimal Control Problems with Delay and Lévy Processes. 2016 35th Chinese Control Conference (CCC), Chengdu, 27-29 July 2016, 1758-1763. https://doi.org/10.1109/chicc.2016.7553347 |
| [8] | Wang, H., Zhang, H. and Xie, L. (2021) Optimal Control and Stabilization for Itô Systems with Input Delay. Journal of Systems Science and Complexity, 34, 1895-1926. https://doi.org/10.1007/s11424-021-1226-6 |
| [9] | Meng, W. and Shi, J. (2021) A Global Maximum Principle for Stochastic Optimal Control Problems with Delay and Applications. Systems & Control Letters, 150, Article ID: 104909. https://doi.org/10.1016/j.sysconle.2021.104909 |
| [10] | Zhang, S., Xiong, J. and Shi, J. (2021) A Linear-Quadratic Optimal Control Problem of Stochastic Differential Equations with Delay and Partial Information. Systems & Control Letters, 157, Article ID: 105046. https://doi.org/10.1016/j.sysconle.2021.105046 |
| [11] | Guo, H., Xiong, J. and Zheng, J. (2024) Stochastic Maximum Principle for Generalized Mean-Field Delay Control Problem. Journal of Optimization Theory and Applications, 201, 352-377. https://doi.org/10.1007/s10957-024-02398-2 |
| [12] | Meng, W., Shi, J., Wang, T. and Zhang, J. (2025) A General Maximum Principle for Optimal Control of Stochastic Differential Delay Systems. SIAM Journal on Control and Optimization, 63, 175-205. https://doi.org/10.1137/23m1552024 |
| [13] | Xiao, H. (2013) Optimality Conditions for Optimal Control of Jump-Diffusion SDEs with Correlated Observations Noises. Mathematical Problems in Engineering, 2013, Article ID: 613159. https://doi.org/10.1155/2013/613159 |
| [14] | Li, Z., Fu, M., Zhang, H. and Zhang, Z. (2024) Partially Observed Optimal Control with Correlated Noises. 2024 3rd Conference on Fully Actuated System Theory and Applications (FASTA), Shenzhen, 10-12 May 2024, 1328-1333. https://doi.org/10.1109/fasta61401.2024.10595170 |
| [15] | Wang, G.C. and Wu, Z. (2009) General Maximum Principles for Partially Observed Risk-Sensitive Optimal Control Problems and Applications to Finance. Journal of Optimization Theory and Applications, 141, 677-700. https://doi.org/10.1007/s10957-008-9484-1 |
| [16] | 郭云瑞, 梁晓青. Heston模型下DC型养老金鲁棒最优投资问题[J]. 应用概率统计, 2023, 39(4): 531-546. |
