MINIMIZING OR MAXIMIZING THE EXPECTED TIME TO REACH ZERO.

We treat the following control problems: the process X//1(t) with values in the interval ( minus infinity , 0 right bracket (or left bracket , infinity )) is given by the stochastic differential equation dX//1(t) equals mu (t) dt plus sigma (t) dW//t, X//1(0) equals x//1 where the nonanticipative controls mu and sigma are to be chosen so that ( mu (t), sigma (t)) remains in a given set S and the object is to minimize (or maximize) the expected time to reach the origin. The minimization problem had been discussed earlier under various restrictions on the set S. Here an improved verification lemma is established which is used to solve the minimization and maximization problems for any S. An application to a portfolio problem is discussed.