Optimal Retirement Problem Under Partial Information

The optimal retirement decision is essentially an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment and retirement problem when the growth rate is unobservable and is estimated by filtering from historical stock prices. To ensure both consumption rate and wealth of the representative agent be nonnegative, the analysis is based on a duality approach. We offer conditions for the filtering scheme for the unobserved growth rate so that the dual problem can be characterized by another optimal stopping problem with stochastic volatility. We further link the dual problem to an American option with stochastic volatility, and prove for the close of duality gap. The theory is then applied to a hidden Markov model for regime-switching growth rate with the Woham filter. We fully characterize the existence and uniqueness of the variational inequality in the dual, and the free boundary. Asymptotic closed-form solution is derived for the optimal retirement by a small-scale perturbation. We discuss the potential use of the result for other types of partial information settings