On hard quadrature problems for marginal distributions of SDEs with bounded smooth coefficients
Establishes fundamental limitations of Monte Carlo quadrature for SDEs with smooth coefficients, showing that poor performance is inherent to the problem class, not just specific algorithms.
The paper proves that for SDEs with smooth bounded coefficients, no Monte Carlo method using finitely many evaluations of coefficients and their derivatives can achieve polynomial convergence rates in the worst case, and arbitrarily slow convergence is possible.
In recent work of Hairer, Hutzenthaler and Jentzen, see [9], a stochastic differential equation (SDE) with infinitely often differentiable and bounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In the present paper we show that this type of bad performance for quadrature of SDEs with infinitely often differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely often differentiable and bounded by one coefficients such that the first order derivatives of all diffusion coefficients are bounded by one as well and the first order derivatives of all drift coefficients are uniformly dominated by a single real-valued function and such that the corresponding sequence of mean absolute errors for approximation of the expected value of the first component of the solution at the final time can not converge to zero faster than the given speed.