Discrete Feynman-Kac approximation for parabolic Anderson model using random walks

In this paper, we introduce a natively positive approximation method based on the Feynman-Kac representation using random walks, to approximate the solution to the one-dimensional parabolic Anderson model of Skorokhod type, with either a flat or a Dirac delta initial condition. Assuming the driving noise is a fractional Brownian sheet with Hurst parameters H≥12 and H∗≥12 in time and space, respectively, we also provide an error analysis of the proposed method. The error in Lp(Ω) norm is of order O(h 1 2[(2H+H∗−1)∧1]−ϵ), where h>0 is the step size in time (resp. h in space), and ϵ>0 can be chosen arbitrarily small. This error order matches the Hölder continuity of the solution in time with a correction order ϵ, making it ‘almost’ optimal. Furthermore, these results provide a quantitative framework for convergence of the partition function of directed polymers in Gaussian environments to the parabolic Anderson model.