Diffusion control games with rank-based rewards

该项目的总体目标是了解相互作用和基于等级的奖励对经济主体如何选择扩散控制的影响，并提供计算和近似纳什均衡的方法。更具体地说，我们计划分析随机微分对策，其中每个参与者可以选择其状态过程的扩散强度。假设每个玩家的扩散强度控制在一个固定的有界区间内取值，并且相对于代表玩家信息流的过滤可逐步测量。此外，我们假设每个参与者的奖励取决于其在给定有限时间范围内的状态过程的排名。在第一部分中，我们的目标是分析一个程式化的2人零和游戏，其中两个玩家中的每个人都努力最大化领先对手的可能性。此外，假设每个玩家的状态都是一个布朗鞅。我们的第一个目标是提供纯马尔可夫控制下纳什均衡存在的充要条件。此外，在两个玩家的状态中，共同噪音的影响将被剔除。最后，我们的目标是研究一些推广，例如通过允许混合策略和非对称控制集。第二部分是对两个以上参与者的扩散控制博弈的分析。对于一些特定的基于等级的奖励，我们的目标是明确地确定纳什均衡。然而，基于排名的奖励的不连续性使得n人的情况很难分析。因此，该项目将有助于开放平均场技术，用于近似基于等级奖励的扩散控制博弈中的纳什均衡。为此，我们将考虑平均场控制问题，其解近似于n人扩散控制博弈的大n均衡策略。

The general aim of the project is to understand the impact of interaction and rank-based rewards on how economic agents choose diffusion controls, and to provide methods for computing and approximating Nash equilibria. More specifically, we plan to analyze stochastic differential games, where each player can choose the diffusion intensity of her state process. The diffusion intensity control of each player is assumed to take values in a fixed bounded interval and to be progressively measurable with respect to a filtration representing the player's information flow. Moreover, we assume that the reward of each player depends on the rank of her state process at a given finite time horizon. In the first part we aim at analyzing a stylized zero-sum 2-player game, where each of the two players strives to maximize the probability of being ahead of the opponent. Besides, each of the player's state is assumed to be a Brownian martingale. Our first aim is to provide sufficient and necessary conditions of Nash equilibria in pure Markov controls to exist. Moreover, the effect of common noise in both players' states will be carved out. Finally, we aim at investigating some generalizations, e.g. by allowing for mixed strategies and non-symmetric control sets. The second part is devoted to an analysis of diffusion control games with more than two players. For some specific rank-based rewards we aim at determining a Nash equilibrium explicitly. The discontinuity of rank-based reward, however, make the n-player case difficult to analyze in general. Therefore, the project will contribute towards opening mean field techniques for approximating Nash equilibria in diffusion control games with rank-based rewards. To this end, we will consider mean field control problems, the solution of which approximate, for large n, equilibrium strategies of n-player diffusion control games.