Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions
Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions
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DOI：
10.1051/cocv/2018013
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发表时间：
2017-01
期刊：
影响因子：
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通讯作者：
Qingmeng Wei;J. Yong;Zhiyong Yu
中科院分区：
文献类型：
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作者：
Qingmeng Wei;J. Yong;Zhiyong Yu
研究具有二次代价泛函的线性随机微分方程的最优控制问题。状态方程的系数和代价泛函中的权值是平方可积随机变量空间上的有界算子。我们研究的主要动机是平均场随机微分方程的线性二次（简称LQ）最优控制问题。该问题的开环可解性表征为带算子系数的线性耦合正反向随机微分方程（FBSDE，简称FBSDE）系统的可解性以及代价泛函的凸性条件。在适当的条件下，建立了该FBSDE的适定性，从而导致了开环最优控制的存在性。最后，作为我们主要成果的应用，我们解决了一个一般的平均场LQ控制问题和一个具体的开环情况下的均值-方差投资组合选择问题。
An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.