The Resolvent of a Degenerate Diffusion on the Plane, with Application to Partially Observed Stochastic Control

We compute the resolvent of the degenerate, two-dimensional diffusion process introduced by Benes, Karatzas and Rishel in the study of a stochastic control problem with partial observations. The explicit nature of our computations allows us to show that this diffusion can be constructed uniquely (in the sense of the probability law) starting at any point on the plane, including the origin, and to solve explicitly the control problem of Benes, Karatzas and Rishel for very general cost functions. Our derivation combines probabilistic techniques, with use of the so-called "principle of smooth fit."