A New Algorithm and Worst Case Complexity for Feynman–Kac Path Integration

We study algorithms for approximation of Feynman–Kac path integrals in the worst case setting. The algorithms use a finite number of samples of the initial condition and potential functions. We present a new algorithm and an explicit bound on its cost to compute an ε-approximation to the Feynman–Kac path integral. We also establish bounds on the worst case complexity of Feynman–Kac path integration. The upper bound is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem. The lower bound is given in terms of the complexity of a certain weighted integration problem. For some classes of functions, these two bounds coincide modulo a multiplicative factor. In this case, the new algorithm is almost optimal. The new algorithm requires precomputation of some real coefficients that are combinations of multivariate integrals with special weights. This precomputation is difficult and limits the application of the new algorithm. We report the results of the precomputation for specific cases.