Optimal investment, retirement, and life insurance with consumption ratcheting and time-inhomogeneous utility

In this paper, we study a portfolio selection problem of an investor with a retirement option, who has possibility to buy life insurance and does not tolerate any decline in consumption. The agent's optimization problem can be viewed as a mixed singular control and optimal stopping problem with time-inhomogeneous utility functions. The closed-form optimal solution is not available in general. We use the dual control method to convert the original problem into two classes of optimal stopping problems in finite and infinite horizons. We show that the optimal consumption strategy and the best retirement time depend on the free-boundary functions which satisfy Fredholm and Volterra integral equations. We derive the closed-form formulas for these two free boundaries for some special cases and develop numerical methods to solve the integral equations of the free boundaries for general cases.