Stochastic maximum principle for delayed doubly stochastic control systems and their applications

Jun, Xu

doi:10.1080/00207179.2018.1508850

Cited by 12 publications

(21 citation statements)

References 17 publications

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“…By (H1) and Theorem 3.1.1 in [20], it is easy to see that there is a unique adapted solution to Equation (6).…”

Section: Necessary Maximum Principlementioning

confidence: 91%

“…f and g are continuously differentiable with respect to (y, y δ , z, z δ , v 1 , v 2 ), and their partial derivatives are bounded. Now, if both v 1 (•) and v 2 (•) are admissible controls, and assumption (H1) holds, then doubly stochastic differential equation with delay (3) admits a unique solution (y(•), z( [20]). The two players have their own benefits, which are described by the cost functional…”

Section: Notations and Formulation Of Problemsmentioning

confidence: 99%

“…Because of its importance to SPDEs, the interest in BDSDEs has increased considerably (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]). At the same time, the stochastic control problem of backward doubly stochastic systems has been studied extensively (see [16][17][18][19][20][21]).…”

Section: Introductionmentioning

confidence: 99%

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A Type of Time-Symmetric Stochastic System and Related Games

et al. 2021

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This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

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“…By (H1) and Theorem 3.1.1 in [20], it is easy to see that there is a unique adapted solution to Equation (6).…”

Section: Necessary Maximum Principlementioning

confidence: 91%

Section: Notations and Formulation Of Problemsmentioning

confidence: 99%

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A Type of Time-Symmetric Stochastic System and Related Games

et al. 2021

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“…Although we focus on this kind of game, we are also able to develop some results for optimal control of BDSDEs, for example Han et al [9], Shi and Zhu [18], Xu and Han [25,26], Zhang and Shi [28], Zhu and Shi [35].…”

Section: Noticing the Fact Thatmentioning

confidence: 99%

“…Chen and Yu [7] studied the nonzero-sum stochastic differential game of stochastic differential delay equation (SDDE), Shi and Wang [16] discussed the nonzero-sum differential game of backward stochastic differential equation (BSDE) with timedelayed generator. Xu and Han [25] and Xu [26] introduced one kind of delayed doubly stochastic differential equations and discussed the maximum principle for this kind delayed doubly stochastic control systems.…”

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confidence: 99%

Nonzero-sum differential game of backward doubly stochastic systems with delay and applications

Zhu¹,

Shi²

2021

Mathematical Control &Amp; Related Fields

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This paper is concerned with a kind of nonzero-sum differential game of backward doubly stochastic system with delay, in which the state dynamics follows a delayed backward doubly stochastic differential equation (SDE). To deal with the above game problem, it is natural to involve the adjoint equation, which is a kind of anticipated forward doubly SDE. We give the existence and uniqueness of solutions to delayed backward doubly SDE and anticipated forward doubly SDE. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a nonzero-sum differential game of linear-quadratic backward doubly stochastic system with delay.

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Forward‐backward doubly stochastic differential equations with random jumps and related games

Zhu

Shi

Teng

2020

Asian Journal of Control

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A type of forward‐backward doubly stochastic differential equations driven by Brownian motions and the Poisson process (FBDSDEP) is studied. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions of FBDSDEP are established via a method of continuation. Then the continuity and differentiability of the solutions to FBDSDEP depending on parameters is discussed. Furthermore, these results were applied to backward doubly stochastic linear quadratic (LQ) nonzero sum differential games with random jumps to get the explicit form of the open‐loop Nash equilibrium point by the solution of the FBDSDEP.

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Stochastic maximum principle for delayed doubly stochastic control systems and their applications

Cited by 12 publications

References 17 publications

A Type of Time-Symmetric Stochastic System and Related Games

A Type of Time-Symmetric Stochastic System and Related Games

Nonzero-sum differential game of backward doubly stochastic systems with delay and applications

Forward‐backward doubly stochastic differential equations with random jumps and related games

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