论文标题
连续和冲动可以通过NASH平衡和应用以有限的视野控制差异游戏
Continuous and Impulse Controls Differential Game in Finite Horizon with Nash-Equilibrium and Application
论文作者
论文摘要
本文考虑了一类新的确定性有限时间范围，两人零和差异游戏（DGS），其中允许最大化玩家可以进行连续和脉冲控制，而最小化的播放器仅允许使用Impulse Control。我们试图近似值函数，并为此类的DGS提供验证定理。我们首先，通过粘度解（VS）框架中的动态编程原理（DPP），将值函数描述为相关的汉密尔顿 - 雅各布斯 - 雅各布 - 贝尔曼-ISAACS（HJBI）双重启动方程的唯一VS。接下来，我们证明存在一个近似值函数，它是近似HJBI双键盘方程的唯一解决方案，当时间离散化步骤零时，局部将本地收敛于每个播放器的值函数。此外，我们提供了一个验证定理，该定理表征了考虑到DG控制问题的NASH平衡。最后，通过应用我们的结果，我们得出了一个新的连续时间组合优化模型，并提供相关的计算算法。
This paper considers a new class of deterministic finite-time horizon, two-player, zero-sum differential games (DGs) in which the maximizing player is allowed to take continuous and impulse controls whereas the minimizing player is allowed to take impulse control only. We seek to approximate the value function, and to provide a verification theorem for this class of DGs. We first, by means of dynamic programming principle (DPP) in viscosity solution (VS) framework, characterize the value function as the unique VS to the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) double-obstacle equation. Next, we prove that an approximate value function exists, that it is the unique solution to an approximate HJBI double-obstacle equation, and converges locally uniformly towards the value function of each player when the time discretization step goes to zero. Moreover, we provide a verification theorem which characterizes a Nash-equilibrium for the DG control problem considered. Finally, by applying our results, we derive a new continuous-time portfolio optimization model, and we provide related computational algorithms.
