CRC/TRR 388: Mean field games, rough analysis and optimal trading (SP B04)

This project combines the PIs’ expertise in rough analysis and game theory to analyze novel finite player games and mean field games (MFGs) with possibly singular controls and common noise driven rough paths and rough semi-martingales. We will establish new existence of equilibrium results for games with both finitely and infinitely many players and novel propagation of chaos-type results from which to deduce the convergence of equilibria in finite player games to equilibria in the corresponding MFG. For (mean-field) games with rough common noise we will analyze the stability of equilibria w.r.t. the common noise process for which rough path techniques seem to be tailor-made, at least as long as deterministic noise processes are considered. While we will initially focus on deterministic settings, we will collaborate with Project B05 in developing duality based methods for allowing non-anticipative equilibria in random settings. We will also introduce new signature-based methods for the efficient approximation of extended MFG with B04 common noise. Our game theoretic analysis requires novel existence and uniqueness of solutions results for forward-backward rough stochastic differential equations arising as adjoint dynamics when applying rough Pontryagin maximum principles as well as novel existence results for quadratic MF BSDEs. Our results will be applied to new interactive models of portfolio optimization with rough volatility and market impact with and without common noise, respectively. To gain additional insights into the nature of our MFG equilibria we will develop new approximation techniques for extended MFGs with common noise using signature methods. In the longer run our analysis will be extended to (mean-field) games with singular controls. When singular controls are considered one usually applies weaker solution concepts such as relaxed solutions or controlled martingale problems. It is an open problem to extend these solution concepts and to establish existence of equilibrium results to a rough path setting.