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

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.

26 pages

References in corpus (3)

• Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes
• Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes
• An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients

Cited by in corpus (14)

• Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
• On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift
• On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
• Convergence rate of EM algorithm for SDEs under integrability condition
• A generalized Avikainen's estimate and its applications
• On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values
• Stochastic differential equations with irregular coefficients:~mind the gap!
• On hard quadrature problems for marginal distributions of SDEs with bounded smooth coefficients
• Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives
• Strongly Asymptotically Optimal Schemes for the Strong Approximation of Stochastic Differential Equations with respect to the Supremum Error
• On construction of boundary preserving numerical schemes