The Tree-Grid Method with Control-Independent Stencil

Abstract

The Tree-Grid method is a novel explicit convergent scheme for solving stochastic control problems or Hamilton-Jacobi-Bellman equations with one space dimension. One of the characteristics of the scheme is that the stencil size is dependent on space, control and possibly also on time. Because of the dependence on the control variable, it is not trivial to solve the optimization problem inside the method. Recently, this optimization part was solved by brute-force testing of all permitted controls. In this paper, we present a simple modification of the Tree-Grid scheme leading to a control-independent stencil. Under such modification an optimal control can be found analytically or with the Fibonacci search algorithm.