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

Graeme Baker, Ankita Chatterjee

TL;DR

This work develops a minimal strong solution framework for a jump-driven Skorokhod reflection problem in the positive orthant, establishing a dual-cone criterion that governs extendability across jump times and characterizing the unique minimal reflection when extension is possible. The main theoretical contribution is the precise condition $X_{\tau-}-\Delta Z_\tau \in C^*$ that determines whether a solution can be extended to $[0,\tau]$, together with a constructive fixed-point method to compute the minimal jump $\Delta L_\tau$. The authors also show existence and uniqueness of a maximal stopping time $\tau^*$ yielding a unique minimal solution on $[0,\tau^*)$, generalizing previous Brownian and sub-stochastic-reflection results to jump-driven dynamics. The applied example on two interconnected insurance firms demonstrates how the reflection-mediated reinsurance mechanism affects ruin probabilities and the timing of contract failure, offering practical guidance for risk-sharing in jump-risk environments. Overall, the paper advances the mathematical understanding of multi-dimensional Skorokhod problems with jumps and provides a tractable, constructive approach with concrete implications for reinsurance design.

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.

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

TL;DR

This work develops a minimal strong solution framework for a jump-driven Skorokhod reflection problem in the positive orthant, establishing a dual-cone criterion that governs extendability across jump times and characterizing the unique minimal reflection when extension is possible. The main theoretical contribution is the precise condition that determines whether a solution can be extended to , together with a constructive fixed-point method to compute the minimal jump . The authors also show existence and uniqueness of a maximal stopping time yielding a unique minimal solution on , generalizing previous Brownian and sub-stochastic-reflection results to jump-driven dynamics. The applied example on two interconnected insurance firms demonstrates how the reflection-mediated reinsurance mechanism affects ruin probabilities and the timing of contract failure, offering practical guidance for risk-sharing in jump-risk environments. Overall, the paper advances the mathematical understanding of multi-dimensional Skorokhod problems with jumps and provides a tractable, constructive approach with concrete implications for reinsurance design.

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.
Paper Structure (6 sections, 3 theorems, 18 equations, 4 figures)

This paper contains 6 sections, 3 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Result
  3. Increments of the Skorokhod Map
  4. Proof of Theorem \ref{['thm']}
  5. Fixed Point Approach
  6. Application to Reinsurance

Key Result

Theorem 1

Consider the cone $C=\{u\in\mathbb{R}^n:u\ge 0, u^\top (I-Q)\le 0\}$ and its dual cone $C^*=\{y\in\mathbb{R}^n:u^\top y \ge 0 \text{ for all }u\in C\}$. A solution $(X,L)$ of eq:ps--eq:Ldef on an interval $[0,\tau)$ can be extended to $[0,\tau]$ if and only if $X_{\tau-}-\Delta Z_\tau \in C^*$. More

Figures (4)

  • Figure 1: Example primal and dual problems with $n=2$, $q_{12}=q_{21}=2$, and ${(X_{\tau-}^1-\Delta Z_{\tau}^1,X_{\tau-}^2-\Delta Z_{\tau}^2)=(-1,6)}$. In this case, the primal problem is feasible.
  • Figure 2: Ruin probabilities on the time interval $t\in[0,500]$ when $\alpha=0.05$.
  • Figure 3: Ruin probabilities on the time interval $t\in[0,500]$ when $\alpha=0.5$.
  • Figure 4: Ruin probabilities on the time interval $t\in[0,10]$ when $\alpha=0.5$.

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • proof : Proof of Corollary \ref{['coro']}
  • Lemma 1
  • proof
  • proof