On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
This resolves the open question of whether slow convergence phenomena occurs in low dimensions, showing that the curse of dimensionality is not the cause.
The paper proves that for any arbitrarily slow convergence rate, there exist 2- and 3-dimensional SDEs with smooth bounded coefficients such that no method using finitely many Brownian motion observations can converge faster than that rate in absolute mean, extending a previous result for dimensions 4 and higher.
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.