Arbitrary order total variation Convergence of Markov semigroups using random grids

TL;DR

This paper introduces a framework for achieving total variation convergence at arbitrary rates for numerical schemes solving SDEs, leveraging standard weak approximation properties like Euler and constructing specific SDE approximations.

Contribution

It presents a novel abstract framework that allows controlling the total variation convergence rate of numerical schemes for SDEs, extending beyond traditional weak convergence results.

Findings

Total variation convergence can be achieved at any desired rate.

Standard schemes like Euler can be adapted for total variation convergence.

A new approximation method for SDEs enhances convergence control.

Abstract

We provide a abstract framework to prove total variation convergence result with arbitrary rate for numerical scheme for SDE. In particular we show that under standard weak approximation properties of scheme such as Euler we can obtain total variation convergence with any desired rate by building a specific approximation for the SDE.