A Risk Extended Version of Merton's Optimal Consumption and Portfolio Selection

A risk management version of the classical investment-consumption problem known as Merton's problem in the finance literature is proposed. Risk is measured by variance, which introduces a nonlinear function of the expected value into the control problem. Standard stochastic theory cannot properly handle this type of nonlinear stochastic optimization problem. Therefore, we study this time-inconsistent problem within the mean field-type control framework. We derive the sufficient condition of optimality and solve the problem completely. Numerical results illustrating the effect of risk on optimal policies are also presented. Applications can be numerous, including all kinds of investment decisions or operations decisions.