Minimal Solutions to the Skorokhod Reflection Problem Driven by Jump Processes and an Application to Reinsurance

TL;DR

This paper develops minimal solutions for a jump-driven Skorokhod reflection problem and applies it to model the failure time of reinsurance agreements between interconnected insurance firms.

Contribution

It extends existing models to include jump processes and provides a unique minimal solution framework for the reflected process in this setting.

Findings

Existence of a unique minimal strong solution up to a maximal stopping time.

Application to modeling ruin times in interconnected insurance firms.

Extension of previous work to jump-driven processes and non-sub-stochastic reflection matrices.

Abstract

We consider a reflected process in the positive orthant driven by an exogenous jump process. For a given input process, we show that there exists a unique minimal strong solution to the given particle system up until a certain maximal stopping time, which is stated explicitly in terms of the dual formulation of a linear programming problem associated with the state of the system. We apply this model to study the ruin time of interconnected insurance firms, where the stopping time can be interpreted as the failure time of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to the case of jump driving processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.