Optimal Stopping of Regular Diffusions under Random Discounting

Let X be a one-dimensional regular diffusion, A a positive continuous additive functional of X, and h a measurable real-valued function. A method is proposed to determine a stopping rule $T^*$ that maximizes {\bf E}$\{e^{-A_T} h(X_T) \,1_{\{T < \infty\}}\}$ over all stopping times~T of~X. Several examples are discussed.