Probability bounds for reflecting diffusion processes

Abstract

A solution to the optimal problem for determining vector fields which maximize (resp. minimize) the transition probabilities for a class of reflecting diffusion processes is obtained in this paper. The approach is based on a representation for the transition probability density functions. The optimal transition probabilities under the constraint that the drift vector field is bounded are studied in terms of the HJB equation. We demonstrate by simulations that, even in one dimension, by considering the nodal set of the solutions to the HJB equation, the optimal diffusion processes exhibit an interesting feature of phase transitions.