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 $( { - \infty ,0} ]$ (or $[ {0,\infty } )$) is given by the stochastic differential equation \[dX_1 (t) = \mu (t)dt + \sigma (t)dW_t ,\quad X_1 (0) = x_1 \] where the nonanticipative controls $\mu $ and $\sigma $ are to be chosen so that $(\mu (t),\sigma (t))$ remains in a given set $\mathcal{S}$ and the object is to minimize (or maximize) the expected time to reach the origin. The minimization problem had been dicussed earlier by Heath, Pestien,and Sudderth under various restrictions on the set $\mathcal{S}$. Here an improved verification lemma is established which is used to solve the minimization and maximization problems for any $\mathcal{S}$. An application to a portfolio problem is discussed.