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

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].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].SIAM J Control Optim,2014,52:4082-4121;Lü Q,Wang T,Zhang X.Characterization of optimal feedback for stochastic linear quadratic control problems,Probab Uncertain Quant Risk,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.

This paper is concerned with linear quadratic optimal control problems of s-tochastic 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 a-mong 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;Lü,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.