Volume 25, 2019
|Number of page(s)||38|
|Published online||17 July 2019|
Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions*
School of Mathematics and Statistics, Northeast Normal University,
130024, P.R. China
2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
3 School of Mathematics, Shandong University, Jinan 250100, P.R. China
** Corresponding author: email@example.com
Accepted: 2 February 2018
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.
Mathematics Subject Classification: 93E20 / 91A23 / 49N70 / 49N10
Key words: 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
This work is supported in part by NSF Grant DMS-1406776, the National Natural Science Foundation of China (11471192, 11401091,11571203), the Nature Science Foundation of Shandong Province (JQ201401), the Fundamental Research Funds of Shandong University (2017JC016), and the Fundamental Research Funds for the Central Universities (2412017FZ008).
© EDP Sciences, SMAI 2019
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