  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.
