Analytical solution to an LQG homing problem in two dimensions

Abstract

An analytical solution is found to the problem of maximising the time spent in the first quadrant by the two-dimensional diffusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the first quadrant through the y-axis. This type of problem is known as LQG homing and is very difficult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisfied by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an infinite sum of Airy functions.

Keywords

• first-exit time
• brownian motion
• partial differential equations
• method of separation of variables