Control and stopping of a diffusion process on an interval

Consider a process X(·) = X(t), 0 ≤ t < ∞ which takes values in the interval $I = (0, 1)$, satisfies a stochastic differential equation dX(t) = β(t)dt + σ(t)dW(t), X(0) = x ε I and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters β and σ are selected by a controller at each instant t ε [0, ∞) from a set depending on the current position. Assume also that the controller selects a stopping time τ for the process and seeks to maximize Eu(X(τ)), where u: [0, 1] → is a continuous "reward" function. If λ := infx ε I: u(x) = max u and ρ := x ε I: u(x) = max u, then, to the left of λ, it is best to maximize the mean-variance ratio (β/σ²) or to stop, and to the right of ρ, it is best to minimize the ratio (β/σ²) or to stop. Between λ and ρ, it is optimal to follow any policy that will bring the process X(·) to a point of maximum for the function u(·) with probability 1, and then stop.