Dynamic demand response (DR) management is becoming an integral part of power system and market operational practice. Motivated by the smart grid DR management problem, we propose a multi-resolution stochastic differential game-theoretic framework to model the players' intra-group and inter-group interactions in a large population regime. We study the game in both risk-neutral and risk-sensitive settings, and provide closed-form solutions for symmetric mean-field responses in the case of homogenous group population, and characterize the symmetric mean-field Nash equilibrium using the Hamilton-Jacobi-Bellman (HJB) equation together with the Fokker-Planck-Kolmogorov (FPK) equation. Finally, we apply the framework to the smart grid DR management problem and illustrate with a numerical example.