A Linear-Quadratic Stackelberg Differential Game with Mixed Deterministic and Stochastic Controls
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866911174469091328 |
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| author | Shi, Jingtao Wang, Guangchen |
| author_facet | Shi, Jingtao Wang, Guangchen |
| contents | This paper is concerned with a linear-quadratic (LQ) leader-follower differential game with mixed deterministic and stochastic controls. In the game, the follower is a random controller which means that the follower can choose adapted stochastic processes, while the leader is a deterministic controller which means that the leader can choose only deterministic time functions. Such problem is motivated by a pension fund insurance problem, with government, supervisory or employer being a deterministic leader and individual producer or retail investor being a random follower. An open-loop Stackelberg equilibrium solution is considered. First, an optimal control process of the follower is characterized by a stationary condition of forward-backward stochastic differential equation (FBSDE) and a convexity condition of SDE. Then it is represented as a linear functional of optimal state variable of the follower and the leader's control variable, via a classical Riccati equation. Then an optimal control function of the leader is first characterized by a convexity condition of FBSDE and a stationary condition of mean-field type FBSDE. And it is represented as a functional of expectation of optimal state variable of the leader, with the help of a system consisting of two cross-coupled Riccati equations and a two-point boundary value problem of ordinary differential equations (ODEs). The solvabilities of this new system of Riccati equations and two-point boundary value problem and investigated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_00653 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Linear-Quadratic Stackelberg Differential Game with Mixed Deterministic and Stochastic Controls Shi, Jingtao Wang, Guangchen Optimization and Control 93E20, 49K45, 49N10, 49N70, 60H10 This paper is concerned with a linear-quadratic (LQ) leader-follower differential game with mixed deterministic and stochastic controls. In the game, the follower is a random controller which means that the follower can choose adapted stochastic processes, while the leader is a deterministic controller which means that the leader can choose only deterministic time functions. Such problem is motivated by a pension fund insurance problem, with government, supervisory or employer being a deterministic leader and individual producer or retail investor being a random follower. An open-loop Stackelberg equilibrium solution is considered. First, an optimal control process of the follower is characterized by a stationary condition of forward-backward stochastic differential equation (FBSDE) and a convexity condition of SDE. Then it is represented as a linear functional of optimal state variable of the follower and the leader's control variable, via a classical Riccati equation. Then an optimal control function of the leader is first characterized by a convexity condition of FBSDE and a stationary condition of mean-field type FBSDE. And it is represented as a functional of expectation of optimal state variable of the leader, with the help of a system consisting of two cross-coupled Riccati equations and a two-point boundary value problem of ordinary differential equations (ODEs). The solvabilities of this new system of Riccati equations and two-point boundary value problem and investigated. |
| title | A Linear-Quadratic Stackelberg Differential Game with Mixed Deterministic and Stochastic Controls |
| topic | Optimization and Control 93E20, 49K45, 49N10, 49N70, 60H10 |
| url | https://arxiv.org/abs/2004.00653 |