Optimal Consumption, Life Insurance and Investment Strategies with Heterogeneous Discounting and Habit Formation under the HLSV Model

Abstract

The Heston local-stochastic volatility (HLSV) model is used to investigate optimal consumption, life insurance, and investment strategies for a decision maker with habit formation and heterogeneous discounting. The decision maker can allocate wealth in a financial market consisting of risk-free and risky assets, where the price process of the risky asset follows the HLSV model. The habit formation depends on historical consumption and satisfies a specific ordinary differential equation. To hedge against mortality risk, the decision maker purchases life insurance, with the core objective of maximizing the total expected utility of a state-consumption, bequest, and insurance wealth. Since heterogeneous discounting causes time-inconsistency and Bellman's optimality principle fails, we adopt the game-theoretic framework and an extended Hamilton-Jacobi-Bellman equation to pursue time-consistent equilibrium strategies. However, under the HLSV model, an analytical solution of strategies cannot be directly obtained due to the complicated nonlinearity of the partial differential equation. Therefore, we employ an asymptotic expansion technique to obtain an asymptotic solution of the time-consistent equilibrium strategy for the exponential utility function. Numerical simulations are provided to verify the validity of the theoretical results.