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

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