The problem of optimally controlling one-dimensional diffusion processes until they leave a given interval is considered. By linearizing the Riccati differential equation satisfied by the derivative of the value function in the so-called linear quadratic Gaussian homing problem, we are able to obtain an exact expression for the solution to the general problem. Particular problems are solved explicitly.

Optimal control problems for one-dimensional diffusion processes in the interval (d1,d2) are considered. The aim is either to maximize or to minimize the time spent by the controlled processes in (d1,d2). Exact solutions are obtained when the processes are symmetrical with respect to d∗∈(d1,d2). Approximate solutions are derived in the asymmetrical case. The one-barrier cases are also treated. Examples are presented.

The matrix Riccati equation that must be solved to obtain the solution to stochastic optimal control problems known as LQG homing is linearized for a class of processes. The results generalize a theorem proved by Whittle and the one-dimensional case already considered by the authors. A particular two-dimensional problem is solved explicitly.

Summary Stochastic optimal control problems in which two‐dimensional diffusion processes are controlled until they enter a given set are solved explicitly. When symmetry can be used, exact solutions are obtained. The same arguments are valid for one‐dimensional processes. In the general case, it is shown how to obtain good approximate solutions. Various examples are presented.