Diffusion control games with rank-based rewards

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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.