Robust Control for Marked Point Processes under Transition-Rate Uncertainty

2026-07-18

AI Summary

This study addresses the risk arising from uncertainty in cumulative transition intensities under non-Markovian marked point processes, focusing on utility maximization and robust valuation of insurance reserves. Within a finite state space and accounting for path-dependent upper and lower bounds, the framework employs control strategies to hedge against worst-case transition rates, while nature selects the most adverse biometric scenario. Methodologically, the paper introduces a class of non-standard worst-case backward stochastic differential equations (BSDEs), establishes their existence and uniqueness, and integrates the martingale optimality principle with power utility to develop a prospective reserve theory for life and health insurance contracts featuring reserve-dependent benefits. Key contributions include proving existence and uniqueness for the worst-case BSDE solution, formulating a robust insurance reserving framework, and deriving explicit solutions to the robust consumption–insurance problem.

Abstract

We consider a novel robust utility maximisation problem under bounded cumulative transition rate uncertainty within the class of non-Markovian marked point processes on a finite state-space. Utility is maximised over the class of admissible controls, while Nature chooses a worst-case biometric scenario from the class of admissible, path-dependent cumulative transition rates restricted by path-dependent upper and lower bounds. We prove a martingale optimality principle and a novel existence and uniqueness result for a non-standard worst-case backwards stochastic differential equation, which allows us to establish existence and uniqueness of worst-case and best-case prospective reserves of life and health insurance contracts with reserve-dependent payments. Finally, we find an explicit solution of a novel robust consumption-insurance problem with power utility preferences.

Problem

• robust control
• marked point processes
• transition-rate uncertainty
• utility maximisation
• insurance reserves

Innovation

• robust control
• marked point processes
• backward stochastic differential equations
• transition-rate uncertainty
• prospective reserves