Control and stopping of a diffusion process on an interval

Consider a process X(.) = {X(t), 0 less than or equal to t < infinity} whic h takes values in the interval I = (0, 1), satisfies a stochastic different ial equation dX(t) = beta(t) dt + sigma(t) dW(t), X(0) = x is an element of I and, when it reaches an endpoint of the interval I, it is absorbed there. S uppose that the parameters beta and sigma are selected by a controller at e ach instant t is an element of [0, infinity) from a set depending on the cu rrent position. Assume also that the controller selects a stopping time tau for the process and seeks to maximize Eu(X(tau)), where u: [0, 1] --> R is a continuous "reward" function. If lambda := inf{x is an element of I: u(x ) = max u} and rho := sup{x is an element of I: u(x) = max u}, then, to the left of lambda, it is best to maximize the mean-variance ratio (beta/sigma (2)) or to stop, and to the right of rho, it is best to minimize the ratio (beta/sigma(2)) or to stop. Between lambda and rho, 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.