By Cheng-ke Zhang, Huai-nian Zhu, Hai-ying Zhou, Ning Bin
This publication systematically reviews the stochastic non-cooperative differential video game concept of generalized linear Markov bounce structures and its program within the box of finance and coverage. The e-book is an in-depth study ebook of the continual time and discrete time linear quadratic stochastic differential video game, to be able to determine a comparatively entire framework of dynamic non-cooperative differential online game thought. It makes use of the strategy of dynamic programming precept and Riccati equation, and derives it into all types of life stipulations and calculating approach to the equilibrium options of dynamic non-cooperative differential video game. in response to the sport concept approach, this booklet reviews the corresponding powerful regulate challenge, specifically the life situation and layout approach to the optimum strong keep watch over approach. The ebook discusses the theoretical effects and its functions within the danger keep an eye on, choice pricing, and the optimum funding challenge within the box of finance and assurance, enriching the achievements of differential online game examine. This e-book can be utilized as a reference e-book for non-cooperative differential online game research, for graduate scholars majored in fiscal administration, technology and engineering of associations of upper learning.
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Extra resources for Non-cooperative Stochastic Differential Game Theory of Generalized Markov Jump Linear Systems
L: i¼1 ð3:2:6Þ P Similarly, we can prove that vÃ ðtÞ ¼ li¼1 F2iÃ ðtÞvrt ¼i ðtÞxðtÞ is the optimal control strategy of player P2. 1. 1 Problem Formulation In this subsection, we discuss the stochastic Nash differential games on time interval ½0; 1Þ. Before giving the problem to be discussed, ﬁrst deﬁne the following space m m Lloc 2 ðR Þ : ¼ f/ðÁ; ÁÞ : ½0; 1Þ Â X ! R j/ðÁ; ÁÞ is F t -adapted, Lebesgue meaRT 2 surable, and E 0 k/ðt; xÞk dt\1; 8T [ 0g. Consider the following Markov jump linear systems deﬁned by 8 < dxðtÞ ¼ ½Aðrt ÞxðtÞ þ B1 ðrt ÞuðtÞ þ B2 ðrt ÞvðtÞdt þ ½Cðrt ÞxðtÞ þ D1 ðrt ÞuðtÞ þ D2 ðrt ÞvðtÞdwðtÞ; ð3:2:7Þ : xð0Þ ¼ x0 : where Aðrt Þ ¼ AðiÞ, B1 ðrt Þ ¼ B1 ðiÞ, B2 ðrt Þ ¼ B2 ðiÞ, Cðrt Þ ¼ CðiÞ, D1 ðrt Þ ¼ D1 ðiÞ and D2 ðrt Þ ¼ D2 ðiÞ, when rt ¼ i, i ¼ 1; .
14) if there exist suitably smooth functions V i : ½t0 ; T Â Rn ! R, satisfying the semilinear parabolic partial differential equations ÀVti À n È Â Ã 1X Xhf ðt; xÞVxih xf ¼ max gi t; x; /Ã1 ðt; xÞ; . ; /ÃiÀ1 ðt; xÞ; ui ðtÞ; /Ãi þ 1 ðt; xÞ; . ; /Ãn ðt; xÞ ui 2 h;f Â ÃÉ þ Vxi ðt; xÞf t; x; /Ã1 ðt; xÞ; . ; /ÃiÀ1 ðt; xÞ; ui ðtÞ; /Ãi þ 1 ðt; xÞ; . ; /Ãn ðt; xÞ È iÂ Â ÃÉ ¼ g t; x; /Ã1 ðt; xÞ; /Ã2 ðt; xÞ; . ; /Ãn ðt; xÞ þ Vxi ðt; xÞf t; x; /Ã1 ðt; xÞ; . 1 and whose optimal solution (if it exists) is a feedback strategy.
2, similar to the open-loop situation, in closed-loop Nash equilibrium solution, we know that: First, given the optimal strategies of players, they should maximize the sum of the instantaneous payment and integration of state variation and covariate function in current time at every time point. That is, not only the instantaneous payment but also the whole payment influenced by state variation should be considered when one player chooses the optimal strategy. Second, the variation of optimal state depends on the optimal strategies of all the players, current time and state, and the optimal state of the beginning consistent with the initial state of the game.
Non-cooperative Stochastic Differential Game Theory of Generalized Markov Jump Linear Systems by Cheng-ke Zhang, Huai-nian Zhu, Hai-ying Zhou, Ning Bin