LQG homing problem with a maximin cost

We consider the problem of optimally controlling a given two-dimensional diffusion process Q t = ( x t , y t ) starting at (x,y) in the interior of a domain D = { ( x , y ) ∈ R 2 : a < y < b } , until the process yt either crosses y=a or y=b for the first time. The aim of the optimizer is to maximize a worst case criterion that accounts for both the quadratic control and boundary crossing costs. Under certain conditions, we give explicit forms for the maximin optimal control in question.