An optimal control problem without control costs

A two-dimensional diffusion process is controlled until it enters a given subset of R2. The aim is to find the control that minimizes the expected value of a cost function in which there are no control costs. The optimal control can be expressed in terms of the value function, which gives the smallest value that the expected cost can take. To obtain the value function, one can make use of dynamic programming to find the differential equation it satisfies. This differential equation is a non-linear second-order partial differential equation. We find explicit solutions to this non-linear equation, subject to the appropriate boundary conditions, in important particular cases. The method of similarity solutions is used.

Automated summary

This article investigates a two-dimensional diffusion process and finds a control method to minimize the expected cost. It also obtains explicit solutions to the value function in specific cases and boundary conditions, using the method of similarity solutions for a non-linear second-order partial differential equation.