Convergence of numerical schemes in the total variation sense.

摘要

There are various ways to measure the accuracy of numerical methods for solving stochastic differential equations. Because we typically average results from many generated (approximate) sample paths, we should be more interested in statistical measures of accuracy rather than in how well a particular approximate path shadows an exact path. For this reason, we study the L1 (or "total variation") difference between the joint PDF of the computed values at all time steps and the joint PDF for the values of the exact process at the same times. This is stronger than what the field calls "weak error" but different from "strong error", which requires coupling of discrete and continuous paths. This uncoupling allows us to consider new methods that have statistical properties similar to Milstein's method but are simpler. The main technical tool is short time approximation of the Green's function for the forward equation.