Stochastic Graphon Games: II. The Linear-Quadratic Case
Stochastic Graphon Games: II. The Linear-Quadratic Case
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DOI：
10.1007/s00245-022-09839-2
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发表时间：
2021-05
影响因子：
1.8
通讯作者：
Alexander Aurell;R. Carmona;M. Laurière
中科院分区：
文献类型：
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作者：
Alexander Aurell;R. Carmona;M. Laurière
在本文中，我们分析了线性二次随机微分对策，其中连续体参与者通过石墨集合相互作用，每个状态都受到特殊布朗冲击。主要的技术问题是与样本和玩家标签相关的玩家状态轨迹的联合可测量性，这需要计算涉及石墨烯聚合的成本。为了解决这个问题，我们将游戏设置在一个乘积概率空间的Fubini扩展中。我们提供了石墨聚集体是确定性的条件，并且连续体中所有参与者的线性状态方程是唯一可解的。庞特里亚金极大值原理给出了石墨博弈的平衡条件，并以正-倒向随机微分方程的形式给出了存在唯一性。然后，我们研究图形博弈如何在随机权重的图上近似具有有限多玩家的博弈。我们用一个数值例子来说明其中的一些结果。
In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.