On optimal consumption, investment, and life insurance under consumption path-dependent preferences

Abstract

The continuous time optimal consumption and investment problem with path-dependent reference has been extensively investigated by incorporating various model generalizations in the past half-century. On the other hand, optimal life insurance under utility maximization has become a mainstream research topic among academics and practitioners. Different problem formulations and characterizations of consumption behavior pose exciting challenges and opportunities for stochastic control and analysis, coupled with new modelling and computing implementation. The thesis consists of three parts solving different important stochastic control problems, to interpret and guide the consumption, investment, and life insurance premium in the market, either theoretically or computationally. Part I focuses on an optimal consumption problem for a loss-averse agent with reference to the past consumption maximum. To account for loss aversion on rel­ative consumption, an S-shaped utility is adopted that measures the difference be­tween the nonnegative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the realization utility with respect to consumption, allowing us to focus on an auxiliary HJB equation on the strength of the concavification principle and dynamic programming arguments. By applying the dual transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in closed-form piecewisely, and some thresholds of the wealth variable are obtained. The optimal consumption and investment control of the original problem can be derived analytically in piecewise feedback form. Rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analyses and financial implications are also presented. Part II focuses on a life-cycle optimal portfolio-consumption problem when the consumption performance is measured by a shortfall aversion preference under an additional drawdown constraint on the consumption rate. Meanwhile, the agent also dynamically chooses her life insurance premium to maximize the expected bequest at death time. By using dynamic programming arguments and the dual transform, we solve the HJB variational inequality explicitly in a piecewise form across different regions and derive some thresholds of the wealth variable for the piecewise optimal feedback controls. Taking advantage of our analytical results, we are able to numeri­cally illustrate some quantitative impacts on optimal consumption and life insurance by model parameters and discuss their financial implications. Part III focuses on an optimal consumption and life insurance problem under habit formation preference when the return and volatility of the stock price dynam­ics are unknown. An offline reinforcement learning algorithm is proposed based on a policy improvement result and the evaluation of the policy by minimizing the mar­tingale loss. We illustrate by some simulated examples that the algorithm provides satisfactory performance after combing it with the estimation of volatility. In real data analysis, it is also shown that the proposed algorithm outperforms the conven­tional least square estimation method on the unknown return and volatility.