February 27, 2004:  Jose Perez, CIMS

Convergence of numerical schemes in the total variation sense
  

  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 L^1 (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. This is
 joint work with Jonathan Goodman and Peter Glynn.