On One Method Approximation of Diffusion Problems

Abstract

The paper describes methods for approximating functionals from diffusions and from an optimally controlled diffusion process, as well as methods for approximating diffusion processes that are solutions of stochastic differential equations of Ito, both controlled and uncontrolled. Since many of the functionals that we will calculate and approximate are in fact weak solutions of partial differential equations (the weak solution can be represented as some functional of a suitable diffusion process), the methods for approximating weak solutions are closely related to the methods for approximating diffusion processes and their functionals. In addition, the appearance of partial differential equations, which, at least formally, satisfy the functionals we are interested in, suggests numerical methods for solving these problems.