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.
