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

Abstract

In the recent article [Jentzen, A., Müller-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.