A Two-Person Zero-Sum Game Approach for a Retirement Decision with Borrowing Constraints

SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 883-930, September 2024.
Abstract. We study an optimal consumption, investment, and retirement decision of an economic agent with borrowing constraints under a general class of utility functions. We transform the problem into a dual two-person zero-sum game, which involves two players: a stopper who is a maximizer and chooses a stopping time and a controller who is a minimizer and chooses a nonincreasing process. We derive the Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) of a max-min type from the dual two-person zero-sum game. We provide a solution to the HJBQVI and verify that the solution to the HJBQVI is the value of the dual two-person zero-sum game. We establish the duality result which allows us to derive the optimal strategies and value function of the primal problem from those of the dual problem. We provide examples for a class of utility functions.