Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 17.

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.

DOI : 10.1051/cocv/2018013
Classification : 93E20, 91A23, 49N70, 49N10
Mots-clés : Linear stochastic differential equation with operator coefficients, open-loop solvability, forward-backward stochastic differential equations, mean-field linear quadratic control problem, mean-variance portfolio selection
Wei, Qingmeng 1 ; Yong, Jiongmin 1 ; Yu, Zhiyong 1

1 
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     title = {Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
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Wei, Qingmeng; Yong, Jiongmin; Yu, Zhiyong. Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 17. doi : 10.1051/cocv/2018013. http://www.numdam.org/articles/10.1051/cocv/2018013/

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