Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation
Haiyang Wang Zhen Wu
Mathematical Control & Related Fields 2015, 5(3): 651-678 doi: 10.3934/mcrf.2015.5.651
In this paper, we study a class of time-inconsistent optimal control problems with random coefficients. By the method of multi-person differential games, a family of parameterized backward stochastic partial differential equations, called the stochastic equilibrium Hamilton-Jacobi-Bellman equation, is derived for the equilibrium value function of this problem. Under appropriate conditions, we obtain the wellposedness of such an equation and construct the time-consistent equilibrium strategy of closed-loop. Besides, we investigate the linear-quadratic problem as a special and important case.
keywords: Time inconsistency equilibrium strategy stochastic partial differential equations equilibrium HJB equation linear-quadratic problem. multi-person differential games
Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs
Rui Mu Zhen Wu
Mathematical Control & Related Fields 2017, 7(2): 289-304 doi: 10.3934/mcrf.2017010

This paper is concerned with recursive nonzero-sum stochastic differential game problem in Markovian framework when the drift of the state process is no longer bounded but only satisfies the linear growth condition. The costs of players are given by the initial values of related backward stochastic differential equations which, in our case, are multidimensional with continuous coefficients, whose generators are of linear growth on the volatility processes and stochastic monotonic on the value processes. We finally show the well-posedness of the costs and the existence of a Nash equilibrium point for the game under the generalized Isaacs assumption.

keywords: Recursive utility nonzero-sum stochastic differential games Nash equilibrium point backward stochastic differential equations Isaacs condition
Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance
Dejian Chang Zhen Wu
Journal of Industrial & Management Optimization 2015, 11(1): 27-40 doi: 10.3934/jimo.2015.11.27
This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the game systems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls. Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibrium point of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investment portfolio and optimal impulse consumption strategy are obtained explicitly.
keywords: Non-zero sum stochastic differential game stochastic recursive utility. maximum principle open-loop Nash equilibrium point forward-backward stochastic differential equations impulse controls
Linear quadratic mean-field-game of backward stochastic differential systems
Kai Du Jianhui Huang Zhen Wu
Mathematical Control & Related Fields 2018, 8(3&4): 653-678 doi: 10.3934/mcrf.2018028

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

keywords: Backward mean-field game (BMFG) \begin{document}$\epsilon$\end{document}-Nash equilibrium Hamiltonian-type consistency condition (HCC) Riccati-type consistency condition (RCC) backward stochastic differential equation (BSDE)

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