Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps

We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$Ef(X(1)) of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case $$r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)$$r≤(⌊s/2⌋-1)·2d/(d+2), we almost achieve an error of order $$\min (r, s)/d$$min(r,s)/d in terms of the computational cost, which is optimal in a worst-case sense.