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Linear-Quadratic Stochastic Stackelberg Differential Games for Jump-Diffusion Systems
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2021-03-09 , DOI: 10.1137/20m1352314 Jun Moon
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2021-03-09 , DOI: 10.1137/20m1352314 Jun Moon
SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 954-976, January 2021.
This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback-type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem of the leader using the variational method. However, to obtain the state-feedback representation of the open-loop solution for the leader, there is a technical challenge due to the jump process. To overcome this limitation, we consider two different cases, in which the state-feedback-type optimal control for the leader in terms of the ISRDE can be characterized by generalizing the Four-Step Scheme. Finally, in these two cases, we show that the state-feedback representation of the open-loop optimal solutions for the leader and the follower constitutes the Stackelberg equilibrium. Note that the (indefinite) LQ control problem of the leader is new and nontrivial due to the coupled forward-backward stochastic differential equation constraint induced by the rational behavior of the follower.
中文翻译:
跳跃扩散系统的线性二次随机Stackelberg微分对策
SIAM控制与优化杂志,第59卷,第2期,第954-976页,2021年1月。
本文考虑具有随机系数的跳跃扩散系统的线性二次(LQ)随机前导跟随从Stackelberg微分博弈。我们首先使用随机最大原理解决跟随器的LQ问题,并根据积分随机Riccati微分方程(ISRDE)获得开环最优解的状态反馈表示,其中状态反馈型控制通过完成平方法证明是最优的。接下来,我们采用变分方法建立了领导者的不确定LQ随机最优控制问题的随机最大值原理。但是,要获得针对领导者的开环解决方案的状态反馈表示,由于跳转过程,存在技术上的挑战。为了克服此限制,我们考虑了两种不同的情况,其中,可以通过概括四步方案来表征针对ISRDE的领导者的状态反馈型最优控制。最后,在这两种情况下,我们证明了针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。我们表明,针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。我们表明,针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。
更新日期:2021-04-23
This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback-type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem of the leader using the variational method. However, to obtain the state-feedback representation of the open-loop solution for the leader, there is a technical challenge due to the jump process. To overcome this limitation, we consider two different cases, in which the state-feedback-type optimal control for the leader in terms of the ISRDE can be characterized by generalizing the Four-Step Scheme. Finally, in these two cases, we show that the state-feedback representation of the open-loop optimal solutions for the leader and the follower constitutes the Stackelberg equilibrium. Note that the (indefinite) LQ control problem of the leader is new and nontrivial due to the coupled forward-backward stochastic differential equation constraint induced by the rational behavior of the follower.
中文翻译:
跳跃扩散系统的线性二次随机Stackelberg微分对策
SIAM控制与优化杂志,第59卷,第2期,第954-976页,2021年1月。
本文考虑具有随机系数的跳跃扩散系统的线性二次(LQ)随机前导跟随从Stackelberg微分博弈。我们首先使用随机最大原理解决跟随器的LQ问题,并根据积分随机Riccati微分方程(ISRDE)获得开环最优解的状态反馈表示,其中状态反馈型控制通过完成平方法证明是最优的。接下来,我们采用变分方法建立了领导者的不确定LQ随机最优控制问题的随机最大值原理。但是,要获得针对领导者的开环解决方案的状态反馈表示,由于跳转过程,存在技术上的挑战。为了克服此限制,我们考虑了两种不同的情况,其中,可以通过概括四步方案来表征针对ISRDE的领导者的状态反馈型最优控制。最后,在这两种情况下,我们证明了针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。我们表明,针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。我们表明,针对领导者和跟随者的开环最优解的状态反馈表示构成了Stackelberg平衡。请注意,由于跟随者的有理行为引起的耦合的前向-后向随机微分方程约束,领导者的(不确定)LQ控制问题是新的且不平凡的。