Robust Retirement With Return Ambiguity : Optimal G-Stopping Time in Dual Space

Consider a robust retirement decision problem for a risk- and ambiguity-averse investor concerned about return ambiguity in risky asset prices. When the investor aims to maximize the worst-case scenario of his/her utility derived from consumption and bequest, we propose an optimal G-stopping approach to the robust optimization in a dual space with risk ambiguity. Under the G-expectation framework, we consider a reflected $G$-BSDE with an upper obstacle, which allows us to formulate a parabolic obstacle problem to the dual optimal stopping problem. From the dynamic result in the dual space, we establish the duality theorem to link between the primal problem with return ambiguity and the dual optimal stopping problem with risk ambiguity. We characterize the robust retirement time using the free boundary and derive the robust consumption and investment for a general class of utility functions