  Controlled one-dimensional diffusion processes, with infinitesimal variance
(instead of the infinitesimal mean) depending on the control variable, are
considered in an interval located on the positive half-line. The process is
controlled until it reaches either end of the interval. The aim is to minimize
the expected value of a cost criterion with quadratic control costs on the way
and a final cost equal to zero (resp. a large constant) if the process exits
the interval through its left (resp. right) end point. Explicit expressions are
obtained both for the optimal value of the control variable and the value
function when the infinitesimal parameters of the processes are proportional to
a power of the state variable.
