Time-consistent consumption, investment, and proportional reinsurance in market models with Markovian regime switching

This paper outlines an analysis of equilibrium strategies within a game-theoretic framework addressing discounting stochastic scenarios involving consumption, investment, and reinsurance problems. The controlled state process follows a multi-dimensional linear stochastic differential equation influenced by Brownian motion and Poison jump process under a Markovian regime-switching environment. The objective functional encompasses both running and terminal costs explicitly linked to general discount functions, introducing time inconsistency in the model. Open-loop Nash equilibrium controls are detailed, supported by necessary and sufficient equilibrium conditions and a verification outcome. Furthermore, a state feedback equilibrium strategy is attained through a specific partial differential–difference equation. The study delves into investment consumption and equilibrium, reinsurance/new business strategies, specifically examining power and logarithmic utility functions in select cases. To validate the theoretical findings, a numerical example is presented, demonstrating their efficacy.