A New Algorithm for Computing Path Integrals and Weak Approximation of SDEs Inspired by Large Deviations and Malliavin Calculus

The paper gives a novel path integral formula inspired by large deviation theory and Malliavin calculus. The proposed finite-dimensional approximation of integrals on path space will be a new higher-order weak approximation of multidimensional stochastic differential equations where the dominant part of the local expansion is governed by Varadhan's geodesic distance and the correction terms are given as Malliavin weights. An optimal truncation of asymptotic expansion is used to reduce computational effort. Kusuoka's estimate is applied to justify the finite-dimensional approximation of path integrals. An efficient simulation method is provided with the algorithm. Numerical results are shown to verify the effectiveness