Extended mean-field games with multi-dimensional singular controls and non-linear jump impact

We establish a probabilistic framework for analysing extended mean-field games with multi-dimensional singular controls and state-dependent jump dynamics and costs. Two key challenges arise when analysing such games: the state dynamics may not depend continuously on the control and the reward function may not be u.s.c.~Both problems can be overcome by restricting the set of admissible singular controls to controls that can be approximated by continuous ones. We prove that the corresponding set of admissible weak controls is given by the weak solutions to a Marcus-type SDE and provide an explicit characterisation of the reward function. The reward function will in general only be u.s.c.~To address the lack of continuity we introduce a novel class of MFGs with a broader set of admissible controls, called MFGs of parametrisations. Parametrisations are laws of state/control processes that continuously interpolate jumps. We prove that the reward functional is continuous on the set of parametrisations, establish the existence of equilibria in MFGs of parametrisations, and show that the set of Nash equilibria in MFGs of parametrisations and in the underlying MFG with singular controls coincide. This shows that MFGs of parametrisations provide a canonical framework for analysing MFGs with singular controls and non-linear jump impact.