Minimal solutions to the Skorokhod reflection problem driven by càdlàg processes and an application to reinsurance

We consider a reflected process in the positive orthant driven by an exogenous càdlàg process. For a given input process, we show that solutions to the reflection problem can be continued at a jump time if and only if the proposed jump of the unregulated process is within the dual cone of a linear programming problem associated with the state of the system. As a consequence, for piecewise non-decreasing driving processes there exists a unique minimal strong solution to the given particle system up until the stopping time at which such a non-allowed jump first occurs. We apply this model to study the ruin of interconnected insurance firms, where the stopping time can be interpreted as the failure of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to a class of Lévy processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.