Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On stochastic differential equations with arbitrary slow convergence
  rates for strong approximation   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.

26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph