Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system

Kehan Si; Zhenda Xu; Ka Fai Cedric Yiu; Xun Li

doi:10.3934/jimo.2021074

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2022, Volume 18, Issue 4: 2415-2433. Doi: 10.3934/jimo.2021074 open access

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Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system

1.
School of Mathematics, Shandong University, Jinan, Shandong Province, 250100, China
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

^* Corresponding author: Ka Fai Cedric Yiu
^* Corresponding author: Ka Fai Cedric Yiu

Received: August 31, 2020

Revised: December 31, 2020

Published: July 2022

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Abstract

This paper investigates the mean-field stochastic linear quadratic optimal control problem of Markov regime switching system (M-MF-SLQ, for short). The representation of the cost functional for the M-MF-SLQ is derived using the technique of operators. It is shown that the convexity of the cost functional is necessary for the finiteness of the M-MF-SLQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. We demonstrate with a few examples that our results can be employed for tackling some financial problems such as mean-variance portfolio selection problem.

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Mathematics Subject Classification: Primary: 49N10, 49N35; Secondary: 93E20.

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