Discretionary Stopping of One-Dimensional Itô Diffusions with a Staircase Reward Function

We consider the problem of optimally stopping a general one-dimensional Ito diffusion X. In particular, we solve the problem that aims at maximising the performance criterion E-x exp(- integral(tau)(0) r(X-S) ds)f (X-tau) over all stopping times tau, where the reward function f can take only a finite number of values and has a 'staircase' form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem's value function is not C-1, which is due to the fact that the reward function f is not continuous