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We consider the convexity and comparative static properties of a class of r-harmonic mappings for a given linear, time-homogeneous and regular diffusion process. We present a set of weak conditions under which the minimal r-excessive mappings for the considered diffusion are convex and under which an arbitrary nontrivial r-excessive mapping is convex on the regions where it is r-harmonic. Consequently, we are able to present a set of usually satisfied conditions under which increased volatility increases the value of r-harmonic mappings. We apply our results to a class of optimal stopping problems arising frequently in studies considering the pricing of perpetual American contingent claims and state a set of conditions under which the value function is convex on the continuation region and, consequently, under which increased volatility unambiguously increases the value function and expands the continuation region, thus postponing the rational exercise of the claim.

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We consider the determination of the optimal stationary singular stochastic control of a linear diffusion for a class of average cumulative cost minimization problems arising in various financial and economic applications of stochastic control theory. We present a set of conditions under which the optimal policy is of the standard local time reflection type and state the first order conditions from which the boundaries can be determined. Since the conditions do not require symmetry or convexity of the costs, our results cover also the cases where costs are asymmetric and non-convex. We also investigate the comparative static properties of the optimal policy and delineate circumstances under which higher volatility expands the continuation region where utilizing the control is suboptimal.