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In this paper we study the optimization problem of an economic agent whochooses a job and the time of retirement as well as consumption and portfolioof assets. The agent is constrained in the ability to borrow against futureincome. We transform the problem into a dual two-person zero-sum game, whichinvolves a controller, who is a minimizer and chooses a non-increasing process,and a stopper, who is a maximizer and chooses a stopping time. We derive theHamilton-Jacobi- Bellman quasi-variational inequality(HJBQV) of a max-min typearising from the game. We provide a solution to the HJBQV and verification thatit is the value of the game. We establish a duality result which allows toderive the optimal strategies and value function of the primal problem fromthose of the dual problem.

A problem of optimal switching and singular control with discretionary stopping in portfolio selection