A sequential entry problem with forced exits

We consider a continuous time optimal stopping problem with multiple entries and forced exits. The value for such an optimization problem with a general payoff function is solved in closed form under the assumption that the state process is a geometric Brownian motion and the forced exits come in according to a Poisson process. The effect due to the forced exits is analyzed. It is shown that the presence of the forced exits is a true risk (meaning that it will reduce the value and enlarge the “continuation” region) if and only if the entry cost is large enough compared to the running cost.