A Pair of Explicitly Solvable Singular Stochastic Control Problems

Abstract.

We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem'' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C <sup>2</sup> solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W <sub>loc</sub> <sup>2,p</sup>(R) with $p \geq 1$ , we prove a very general ``verification theorem'' by employing the generalized Meyer—Ito change of variables formula with local times. In what follows, we consider two special cases which we explicitly solve. These are the formal equivalent of the one-dimensional infinite time horizon LQG problem and a simple example with radial symmetry in an arbitrary Euclidean space. The value function of either of these problems is C ² and is expressed in terms of special functions, and, in particular, the confluent hypergeometric function and the modified Bessel function of the first kind, respectively.