New general methods for numerical stochastic differential equations

Abstract

The major purpose of this paper is to indicate that suitable, nontrivial modifications of numerical methods for solving deterministic ordinary differential equations leads to a general class of algorithms to solve stochastic differential equations. Basically, we replace Taylor series expansions with Itô expansions, obtain a numerical algorithm and corresponding local error function, then sum up the local error to get a global error. Our methods have several advantages. The first is that the algorithms appear naturally using Itô expansions. The second is the immediate use of a local truncation error function which insures a consistency for the algorithms, i.e., guarantees that the algorithms make sense. In addition, the order of the global error can be obtained simply from the order of the local error. This makes it easy to determine the expansions necessary to achieve a given order of error. Finally, our tools are the basic tools of the mathematical theory and the justification of our results follow in a straightforward way from these tools.