Optimal Control of Scalar Diffusions with Finite Action Sets

We study a scalar finite-horizon controlled diffusion with a finite action set. An action may affect the drift, the instantaneous variance, or both, and performance is assessed by a uniformly convex terminal deviation cost. The model represents systems in which decisions are made continuously in time but only finitely many operating settings are available. Under uniform nondegeneracy and an explicit shape condition on the terminal cost, we show that the value function is classical at positive times, establish global positive upper and lower bounds for its second state derivative, and prove that the ratio of its first and second state derivatives is strictly increasing. This ordering implies that each action's optimality region is connected, the active actions occur in monotone order, and adjacent regions are separated by unique switching boundaries. It also yields an exact coefficient-based screen that removes actions that cannot be optimal on any open state region. For smooth continuous-action benchmarks, both the value loss from restricting the action set and the additional loss from rounding the continuous optimal feedback to the nearest available action decrease quadratically with the action-set resolution. Numerical experiments illustrate the resulting action screens, threshold policies, and implementation losses.