Consumption–Investment and Reinsurance Problem Under Markovian Regime Switching: Time-Consistent Solution

Abstract

This paper presents a characterization of equilibrium in a game theoretic description of discounting stochastic consumption, investment and reinsurance problem, in which the controlled state process evolves according to a multi-dimensional linear stochastic differential equation, when the noise is driven by a Brownian motion under the effect of a Markovian regime switching. The running and the terminal costs in the objective functional, are explicitly depended on some general discount functions, which create the time inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions as well as a verification result. A state feedback equilibrium strategy is achieved via certain partial differential-difference equation. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for some particular cases of power and logarithmic utility functions. A numerical example is provided to demonstrate the efficacy of theoretical results.

Keywords

Stochastic optimization / Investment–consumption problem / Merton portfolio problem / Non-exponential discounting / Time inconsistency / Equilibrium strategies / Stochastic maximum principle / 93E20 / 60H30 / 93E99 / 60H10

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature