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February 2021 Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls
Jingrui Sun, Jie Xiong, Jiongmin Yong
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Ann. Appl. Probab. 31(1): 460-499 (February 2021). DOI: 10.1214/20-AAP1595

Abstract

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly more general than the classical conditions that the weighting matrix-valued processes are positive (semi-) definite.

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Jingrui Sun. Jie Xiong. Jiongmin Yong. "Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls." Ann. Appl. Probab. 31 (1) 460 - 499, February 2021. https://doi.org/10.1214/20-AAP1595

Information

Received: 1 August 2019; Revised: 1 March 2020; Published: February 2021
First available in Project Euclid: 8 March 2021

Digital Object Identifier: 10.1214/20-AAP1595

Subjects:
Primary: 49N10 , 93E20
Secondary: 49K45 , 49N35

Keywords: closed-loop representation , open-loop optimal control , random coefficient , Stochastic linear-quadratic optimal control problem , stochastic Riccati equation , value flow

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 1 • February 2021
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