Optimal surrender strategies for equity-indexed annuity investors with partial information

In this paper we consider an equity-indexed annuity (EIA) investor who wants to determine when he should surrender the EIA in order to maximize his logarithmic utility of the wealth at surrender time. We model the dynamics of the index using a geometric Brownian motion with regime switching. To be more realistic, we consider a finite time horizon and assume that the Markov chain is unobservable. This leads to the optimal stopping problem with partial information. We give a representation of the value function and an integral equation satisfied by the boundary. In the Bayesian case which is a special case of our model, we obtain analytical results for the value function and the boundary.