On the optimal investment-consumption and life insurance selection problem with an external stochastic factor

TL;DR

This paper develops a stochastic control framework for optimal investment, consumption, and life insurance decisions under stochastic volatility, incorporating multiple insurance providers and jump-diffusion asset dynamics.

Contribution

It introduces a maximum principle for a complex control problem with stochastic volatility and applies it to a multi-insurance market setting.

Findings

Derived necessary and sufficient maximum principles for the control problem.

Solved the optimization problem explicitly under stochastic volatility.

Provided insights into optimal life insurance and investment strategies.

Abstract

In this paper, we study a stochastic optimal control problem with stochastic volatility. We prove the sufficient and necessary maximum principle for the proposed problem. Then we apply the results to solve an investment, consumption and life insurance problem with stochastic volatility, that is, we consider a wage earner investing in one risk-free asset and one risky asset described by a jump-diffusion process and has to decide concerning consumption and life insurance purchase. We assume that the life insurance for the wage earner is bought from a market composed of $M>1$ life insurance companies offering pairwise distinct life insurance contracts. The goal is to maximize the expected utilities derived from the consumption, the legacy in the case of a premature death and the investor's terminal wealth.