  We consider a class of stochastic control problems which has been widely used
in optimal foraging theory. The state processes have two distinct dynamics,
characterized by two pairs of drift and diffusion coefficients, depending on
whether it takes values bigger or smaller than a threshold value. Adopting a
perturbation type approach, we find an expression for potential measure of the
optimal state process. We then obtain an expression for the transition density
of the optimal state process by inverting the associated Laplace transform.
Properties including the stationary distribution of the optimal state process
are discussed. Finally, the expression of the value function is given for this
class stochastic control problems.
