Optimal stopping and maximal inequalities for linear diffusions (Q1383185)

The subject of this paper is optimal stopping [cf. \textit{Y. S. Chow, H. Robbins} and \textit{D. Siegmund}, ``Great expectations: The theory of optimal stopping'' (1971; Zbl 0233.60044)], and in particular the optimal stopping of a linear diffusion [cf. \textit{L. E. Dubins, L. A. Shepp} and \textit{A. N. Shiryaev}, Theory Probab. Appl. 38, No. 2, 226-261 (1993); translation from Teor. Veroyatn. Primen. 38, No. 2, 288-330 (1993; Zbl 0807.60040)] where the objective is to maximize the expected value of the maximum of the process up to the time of stopping, less a cost proportional to time of stopping. The optimal stopping boundary is shown to be the maximal solution of a nonlinear differential equation expressed in terms of the scale function and the speed measure. Main tools include proof that the value function \(V(\cdot)\) is continuous and increasing, satisfies \(V(x)>x\) for \(x>0\), and has derivative satisfying certain integral equations. Applications to maximal inequalities are given.