Backward Stochastic Riccati Equation with Jumps Associated with Stochastic Linear Quadratic Optimal Control with Jumps and Random Coefficients

Fu, Zhang; Dong, Yuchao; Meng, Qingxin

doi:10.1137/18m1209684

Cited by 25 publications

(26 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, LQ stochastic optimal control for jump-diffusion models with random coefficients was studied in [38,39]. However, the corresponding SDE does not have the additive signal terms (equivalently, f = σ i = φ j = 0 for i = 1, .…”

Section: Literature Review and Comparisonmentioning

confidence: 99%

“…, l). Moreover, [38,39] did not consider the characterization of the explicit state-feedback optimal solution using the completion of squares approach and the validation of the proposed solution via the verification theorem. We mention that the problem formulation and the approaches established in our paper are different from those of [38,39].…”

Section: Literature Review and Comparisonmentioning

confidence: 99%

“…Moreover, [38,39] did not consider the characterization of the explicit state-feedback optimal solution using the completion of squares approach and the validation of the proposed solution via the verification theorem. We mention that the problem formulation and the approaches established in our paper are different from those of [38,39]. Furthermore, the completion of squares method and the validation of the optimal solution for (P) via the verification theorem have not been studied in the existing literature, which we address in our paper.…”

Section: Literature Review and Comparisonmentioning

confidence: 99%

“…We note that our paper can be viewed as a generalization of [32,38]. The main comparisons of (P) with [32,38] are stated as follows:…”

Section: Literature Review and Comparisonmentioning

confidence: 99%

See 3 more Smart Citations

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

Moon

Chung

2021

Mathematics

View full text Add to dashboard Cite

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.

show abstract

Section: Literature Review and Comparisonmentioning

confidence: 99%

Section: Literature Review and Comparisonmentioning

confidence: 99%

Section: Literature Review and Comparisonmentioning

confidence: 99%

“…We note that our paper can be viewed as a generalization of [32,38]. The main comparisons of (P) with [32,38] are stated as follows:…”

Section: Literature Review and Comparisonmentioning

confidence: 99%

See 2 more Smart Citations

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

Moon

Chung

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…We mention that in [30], the solvability of the IRDE in (6) was discussed when R 11 , R 21 , Q 1 , and M 1 were (uniformly) positive definite for all s ∈ [0, T]. However, this condition cannot be satisfied in the I-LQ-MF-SZSDG-JD.…”

Section: Coupled Integro-riccati Differential Equations: Solvabilitymentioning

confidence: 99%

Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

Moon

Kim

2020

Mathematics

View full text Add to dashboard Cite

We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

show abstract

Linear quadratic optimal control problems of infinite‐dimensional mean‐field type with jumps

Wang,

Tang,

Meng

2024

Asian Journal of Control

View full text Add to dashboard Cite

In this paper, we formulate and investigate a framework for the theory of the linear quadratic optimal control problem (LQ problem) for infinite‐dimensional mean‐field stochastic evolution systems with jumps. We ensure the well‐posedness of the investigated problems by establishing the existence, uniqueness, and a priori estimates for mild solutions to general infinite‐dimensional mean‐field forward stochastic evolution equations (MF‐SEE) and mean‐field backward stochastic evolution equations (MF‐BSEE) with jumps. Leveraging the Yosida approximation theory, we establish a dual theory between MF‐SEE and MF‐BSEE with jumps, overcoming the challenge posed by the inapplicability of Itô's formula in the context of mild solutions. Our main results regarding the existence and uniqueness of the optimal control, along with corresponding dual characterizations and state feedback representations, are obtained through convex analysis techniques, our established dual theory, and decoupling methods.

show abstract

Backward Stochastic Riccati Equation with Jumps Associated with Stochastic Linear Quadratic Optimal Control with Jumps and Random Coefficients

Cited by 25 publications

References 25 publications

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

Linear quadratic optimal control problems of infinite‐dimensional mean‐field type with jumps

Contact Info

Product

Resources

About