Reward functionals, salvage values, and optimal stopping

We consider the optimal stopping of a linear diffusion in a problem subject to both a cumulative term measuring the expected cumulative present value of a continuous and potentially state-dependent profit flow and an instantaneous payoff measuring the salvage or terminal value received at the optimally chosen stopping date. We derive an explicit representation of the value function in terms of the minimal r-excessive mappings for the considered diffusion, and state a set of necessary conditions for optimal stopping by applying the classical theory of linear diffusions and ordinary non-linear programming techniques. We also state a set of conditions under which our necessary conditions are also sufficient and prove that the smooth pasting principle follows directly from our approach, while the contrary is not necessarily true.