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Keywords = backward stochastic Riccati equations

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18 pages, 369 KiB  
Article
Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint
by Yanrong Lu, Jize Li and Yonghui Zhou
Mathematics 2025, 13(8), 1327; https://doi.org/10.3390/math13081327 - 18 Apr 2025
Viewed by 267
Abstract
This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic [...] Read more.
This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article
(This article belongs to the Special Issue Stochastic Optimal Control, Game Theory, and Related Applications)
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18 pages, 345 KiB  
Article
The Linear Quadratic Optimal Control Problem for Stochastic Systems Controlled by Impulses
by Vasile Dragan and Ioan-Lucian Popa
Symmetry 2024, 16(9), 1170; https://doi.org/10.3390/sym16091170 - 6 Sep 2024
Cited by 2 | Viewed by 1809
Abstract
This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find [...] Read more.
This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)
24 pages, 404 KiB  
Article
The Equivalence Conditions of Optimal Feedback Control-Strategy Operators for Zero-Sum Linear Quadratic Stochastic Differential Game with Random Coefficients
by Chao Tang and Jinxing Liu
Symmetry 2023, 15(9), 1726; https://doi.org/10.3390/sym15091726 - 8 Sep 2023
Cited by 3 | Viewed by 1424
Abstract
From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As [...] Read more.
From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity. Full article
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25 pages, 396 KiB  
Article
Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps
by Zixuan Li and Jingtao Shi
Mathematics 2022, 10(21), 4062; https://doi.org/10.3390/math10214062 - 1 Nov 2022
Cited by 2 | Viewed by 1669
Abstract
The stochastic linear–quadratic optimal control problem with Poisson jumps is addressed in this paper. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed to be indefinite. The notion of closed-loop strategies is [...] Read more.
The stochastic linear–quadratic optimal control problem with Poisson jumps is addressed in this paper. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed to be indefinite. The notion of closed-loop strategies is introduced, and the sufficient and necessary conditions for the closed-loop solvability are given. The optimal closed-loop strategy is characterized by a Riccati integral–differential equation and a backward stochastic differential equation with Poisson jumps. A simple example is given to demonstrate the effectiveness of the main result. Full article
(This article belongs to the Special Issue Stochastic Control Systems: Theory and Applications)
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26 pages, 496 KiB  
Article
Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach
by Jun Moon and Jin-Ho Chung
Mathematics 2021, 9(22), 2918; https://doi.org/10.3390/math9222918 - 16 Nov 2021
Cited by 3 | Viewed by 2243
Abstract
In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the [...] Read more.
In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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