An optimal consumption model with stochastic volatility

We consider an optimal consumption and investment model in continuous time, which is an extension of the original Merton's problem. In the proposed model, the asset prices are affected by correlated economic factors, modelled as diffusion processes. Writing the value function in a special form, it can be seen that another optimal control problem is involved and studying its associated HJB equation smoothness properties of the original value function can be derived as well as optimal policies.