| Abstract: |
| Motivated by mean-field games (MFG) with common noise on the one hand and pathwise stochastic control theory on the other, we formulate here a linear-quadratic (LQ) MFG with rough common noise, along with a satisfactory well-posedness theory for the linear-quadratic case. A novel Volterra-type (or mild) formulation allows us to keep technical (rough-stochastic) considerations to a minimum. We derive a characterization of the optimal state and optimal control through a rough forward-backward SDE (rough FBSDE), and obtain existence and uniqueness. A number of stability estimates are established, and in particular we show continuous dependence of the It\^o-Lions-Lyons map (equilibrium law as a function of the rough common noise). In a final section, we discuss independent Brownian randomization of the rough common noise. The resulting stochastic problem can be seen as an MFG extension of pathwise stochastic control problems. |
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