Before revision:  In the stochastic setting of the Black-Scholes complete market it turns out that the consumption and investment strategies have to satisfy a consistency condition (PDE) if they  come from a classical utility maximisation problem.
After revision:  In the stochastic setting of the Black-Scholes complete market it turns out that the consumption and investment strategies have to satisfy a consistency condition (PDE) if they are to come from a classical utility maximisation problem.