| 王天啸.随机系数下线性二次控制问题的最优性条件及其应用*[J].数学年刊A辑,2021,42(3):331~348 |
| 随机系数下线性二次控制问题的最优性条件及其应用* |
| Optimality Conditions in Linear Quadratic Problems with Random Coefficients and Applications |
| Received:October 31, 2018 Revised:March 01, 2021 |
| DOI：10.16205/j.cnki.cama.2021.0026 |
| 中文关键词: 随机线性二次问题, 倒向随机Riccati方程, 闭环最优策略, 必要性最优条件 |
| 英文关键词:Stochastic linear quadratic problems, Backward stochastic Riccati equations, Closed-Loop optimal strategy, Necessary optimality conditions |
| 基金项目:国家自然科学基金 (No.11971332, No.11931011) |
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| 中文摘要: |
| 本文旨在研究随机系数下随机微分方程的线性二次最优控制问题.本文从闭环最优控制/策略存在的必要性条件的角度开展研究. 若闭环最优控制/策略存在, 得到其显示反馈表示、带伪逆运算的倒向随机Riccati方程的适定性及不同系数间满足的一些本质性条件. 此处结论本质地推广和改进了文[Ait Rami M, Moore J, Zhou X. Indefinite stochastic linear quadratic control and generalized differential Riccati equation [J]. {\it SIAM J Control Optim,} 2001, 40:1296--1311;Sun J, Yong J. Linear quadratic stochastic differential games: open-loop and closed-loop saddle points [J]. {\it SIAM J Control Optim,} 2014, 52:4082--4121;L\"{u} Q, Wang T, Zhang X. Characterization of optimal feedback for stochastic linear quadratic control problems,Probab Uncertain Quant Risk, 2017, 2017, 2:11, DOI 10.1186/s41546-017-0022-7]的相应结论.此外, 本文得到了一个关于倒向随机Riccati方程和二阶伴随方程两类方程适应解之间的微妙关系. 注意到,这一结论在现有文献中首次出现. 最后, 本文讨论了在均值方差对冲问题中的应用. |
| 英文摘要: |
| This paper is concerned with linear quadratic optimal control problems of stochastic differential equations with random coefficients. Instead of providing sufficiency conditions of closed-loop optimal controls/strategies, the author investigates the control problems from the necessary viewpoint. If closed-loop optimal controls/strategies exist, he obtains their explicit feedback representations, the well-posedness of backward stochastic Riccati equations (BSREs for short) with pesudo-inverse, as well as some intrinsic results among given coefficients. The conclusions substantially extend and improve the counterparts in [Ait Rami, M., Moore, J., and Zhou, X., Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., 2001, vol. 40, pp.1296–1311; Sun, J. and Yong, J., Linear quadratic stochastic differential games: open-loop and closed-loop saddle points, SIAM J. Control Optim., 2014, vol. 52, pp. 4082–4121;L¨u, Q., Wang, T., and Zhang, X., Characterization of optimal feedback for stochastic linear quadratic control problems, Probab Uncertain Quant Risk, 2017, vol. 2, pp. 11, DOI 10.1186/s41546-017-0022-7.] with distinctive tricks. Moreover, the author obtains one subtle necessity connection between the solution of BSREs and that of second-order adjoint equations, the result of which is new in related literature to our best. Eventually he indicates one relevant application in mean-variance hedging problems. |
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