An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

Na Li (), Yuan Wang () and Zhen Wu ()
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Na Li: Shandong University of Finance and Economics
Yuan Wang: Shandong University
Zhen Wu: Shandong University

Journal of Optimization Theory and Applications, 2018, vol. 179, issue 2, No 15, 722-744

Abstract: Abstract In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.

Keywords: Stochastic linear quadratic problem; Stochastic differential delayed equations; Forward–backward stochastic differential equations; Hamiltonian system; 93E20; 60H10; 49N10 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-018-1237-1

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