On stochastic differential equations with arbitrary slow convergence rates for strong approximation

TL;DR

This paper demonstrates that for certain smooth and bounded stochastic differential equations, no approximation method based on finitely many observations can achieve polynomial or faster convergence rates in strong approximation.

Contribution

It proves the existence of SDEs where all finite observation-based methods cannot surpass arbitrary slow convergence rates, extending previous results on Euler scheme limitations.

Findings

No polynomial rate of convergence for some SDEs.

Existence of SDEs with arbitrarily slow convergence.

Limitations apply to all finite observation-based methods.

Abstract

In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that the Euler scheme converges to the solution in the strong sense but with no polynomial rate. Hairer et al.'s result naturally leads to the question whether this slow convergence phenomenon can be overcome by using a more sophisticated approximation method than the simple Euler scheme. In this article we answer this question to the negative. We prove that there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion converges in absolute mean to the solution with a polynomial rate. Even worse, we prove that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.