Some solvable stochastic control problems in noncompact symmetric spaces of rank one

Abstract

A family of stochastic control problems for diffusion processes in noncompact symmetric spaces of rank one are described and explicitly solved. The problem is to control Brownian motion in a noncompact symmetric space of rank one by a drift so that it remains close to a fixed point in the space called the origin. The solution is obtained by finding a smooth solution to the Hamilton-Jaobbi or dynamic programming equation for the control problem. All noncompact symmetric spaces of rank one and dimension ≥2 are included and for each of these symmetric spaces a countably infinite family of different, explicitly solvable control problems are given. The cost functional for these control problems are obtained from some spherical functions on these spaces. These spherical functions are monotone increasing functions of the distance of a point from the origin.