A system of integro-differential equations arising in a price formation model in mean field games introduced by Gomes and Saúde [21] is studied. The system consists of a Hamilton–Jacobi equation and a Fokker–Planck equation coupled through a time-dependent price variable determined by a market clearing condition requiring that the aggregate demand coincides with a prescribed supply. The main result establishes the existence and uniqueness of classical solutions in a multidimensional setting. The main difficulty lies in the global integral constraint determining the price variable, which couples the Hamilton–Jacobi and Fokker–Planck equations. The analysis relies on suitable a priori estimates for the Hamilton–Jacobi and Fokker–Planck equations with a fixed price parameter. In particular, the regularity estimates depend only on the \(L^\infty \) norm of the price parameter, which allows the construction of a fixed point argument for the full system. This provides a direct PDE approach for treating the multidimensional case and Hamiltonians depending on space and time.
