# On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

## TL;DR

This paper demonstrates that stochastic differential equations in two and three dimensions can exhibit arbitrarily slow convergence rates for strong approximation, extending previous results from higher dimensions.

## Contribution

It extends prior work by showing slow convergence phenomena occur even in low-dimensional SDEs with smooth coefficients.

## Key findings

- Slow convergence also occurs in 2D and 3D SDEs
- Approximation methods cannot surpass arbitrary slow convergence rates
- Results hold for infinitely differentiable, globally bounded coefficients

## Abstract

In the recent article [Jentzen, A., M\"uller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14(6), 1477--1500, 2016] it has been established that for every arbitrarily slow convergence speed and every natural number $d \in \{4,5,\ldots\}$ there exist $d$-dimensional stochastic differential equations (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. In this paper we strengthen the above result by proving that this slow convergence phenomena also arises in two ($d=2$) and three ($d=3$) space dimensions.