A Relaxed Control Problem With $L^\infty$ Cost and Jump Dynamics Motivated by Cyber Risks Insurance

TL;DR

This work develops a rigorous relaxed-control framework for a jump-driven cyber-risk insurance model built on a star-shaped network with SIR-like intra-edge spreading. It replaces trajectory-based optimization by occupation measures and derives a dual characterization of the $\mathbb{L}^q$-approximating value functions, with a controlled limit to a nonlocal Hamilton-Jacobi–integro-differential inequality. The main theoretical contributions include a dual description, equi-continuity results, and a viscosity-solution formulation linking the relaxed value function to the classical HJB equation, plus an explicit connection back to the original cyber-risk model. Practically, the framework enables tractable analysis of insurer reputation optimization under jump risks and interdependent cyber exposures, while highlighting the role of $\mathbb{L}^\infty$ cost formulations and occupation measures in nonconvex, jump-dominated settings.

Abstract

This paper has a double aim. One the one hand, we introduce a uni-nodal network model for cyber risks with firewalled edges and SIR intra-edge spreading. In connection to this, we formulate an insurance problem in which one seeks the running maximal reputation index against all control strategies of the companies represented by edges. On the other hand, we seek to characterize the value function with $L^\infty$ cost through linear programming techniques and more standard Hamilton-Jacobi integro-differential inequalities.

Figures (1)

• Figure 1: SIR Model With Protection Network.

Theorems & Definitions (18)

• Remark 1
• Proposition 2
• Remark 3
• Theorem 4
• Remark 5
• Remark 6
• Theorem 7
• Theorem 8
• Remark 9
• Proposition 10