Quadrature Formulas for the Wiener Measure

AbstractWe present a new method for the approximation of Wiener integrals and provide an explicit error bound for a class F of smooth integrands. The purely deterministic algorithm is a sequence of quadrature formulas for the Wiener measure, where the knots are piecewise linear functions. It uses ideas of Smolyak, as well as the multiscale decomposition of the Wiener measure due to Lévy and Ciesielski. For the class F we obtain n(ε)⩽max(1, 2ε−4), where n(ε) is the number of integrand evaluations needed to guarantee an error at most ε for f∈F