Quadrature formulas for the Wiener measure
This paper is concerned with path integrals \(I_\infty (f) =\int _Cf(x) dw(x)\) with respect to the Wiener measure on \(C=C[0,1]\). The author presents a new method for the approximation of \(I_\infty (f)\) by means of a sequence \((A_s)_{s=1,2,\dots }\) of quadrature formulas with the error bound \[ \sup _{f\in {\mathcal F} }|I_\infty (f) -A_s(f) |<2^{-1/4} n_s ^{-1/4}, \] where \({\mathcal F}\) is a class of smooth functions. The quadrature formulas \(A_s \) use \(n_s \) knots and have the form \[ A_s(f)= \sum _{i=1}^{n_s}a_i F(x_i), \] where the \(x_i \) are piecewise linear functions. For the class \({\mathcal F}\) it is proved that \(n(\varepsilon) \leq \max (1,2\varepsilon ^{-4})\) where \(n(\varepsilon)\) is the number of integrand evaluations needed to guarantee an error at most \(\varepsilon \) for \( f\in {\mathcal F}\). The effectiveness of the proposed method is illustrated with test examples.
scientific article; zbMATH DE number 1499175
scientific article