Before revision: We consider an optimal consumption - investment problem for financial markets of Black-Scholes 's type with the random coefficients . The existence and uniqueness theorem for the Hamilton-Jacobi-Bellman (HJB) equation is shown.
After revision: We consider an optimal consumption - investment problem for financial markets of Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation is shown.