Convergence of discrete-time deterministic games to path-dependent Isaacs partial differential equations with quadratically growing Hamiltonians
Convergence of discrete-time deterministic games to path-dependent Isaacs partial differential equations with quadratically growing Hamiltonians
复制标题
离散时间确定性博弈收敛于具有二次增长哈密顿量的路径相关 Isaacs 偏微分方程
DOI：
10.1007/s00245-022-09829-4
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发表时间：
2022
影响因子：
1.8
通讯作者：
H. Kaise
中科院分区：
文献类型：
--
作者：
GAINA Daniel;BADIA Guillermo;KOWALSKI Tomasz;H. Kaise
考虑了二次增长条件下确定性微分对策的路径依赖Isaacs偏微分方程(PDE)的离散时间逼近问题，其中包括具有分布时滞和离散时滞的线性/二次问题。由于系统的路径依赖性，Isaacs偏微分方程组被定义在过去状态轨迹的无限维空间上。利用Lukoyanov基于路径空间的共不变导数提出的无穷维空间粘性解的概念，证明了离散时间路径依赖动态对策收敛于Isaacs偏微分方程解的唯一性.注意到这些对策实际上可以定义在有限维空间上，我们讨论了路径依赖的Isaacs偏微分方程粘性解的有限维近似。给出了一个例子，我们推导出离散时间的Riccati型递推方程来计算路径相关的线性/二次问题的显式离散时间近似。
We consider discrete-time approximations for path-dependent Isaacs partial differential equations (PDEs) of deterministic differential games under quadratic growth conditions including linear/quadratic problems with distributed and discrete delays. Owing to the path-dependence of the system, the Isaacs PDEs are defined on infinite-dimensional spaces of past state trajectories. Using the notion of viscosity solutions on the infinite-dimensional spaces as proposed by Lukoyanov based on co-invariant derivatives of path spaces, we show that the discrete-time path-dependent dynamic games converge to a unique viscosity solution for the Isaacs PDEs. Noting that these games can be practically defined on finite-dimensional spaces, we discuss finite-dimensional approximations of viscosity solutions of path-dependent Isaacs PDEs. Given an example, we derive discrete-time Riccati-type recursive equations to calculate explicit discrete-time approximations for the path-dependent linear/quadratic problems.