Spectral properties of trinomial trees

Abstract

In this paper, we prove that the probability kernel of a random walk on a trinomial tree converges to the density of a Brownian motion with drift at the rateO(h4), wherehis the distance between the nodes of the tree. We also show that this convergence estimate is optimal in which the density of the random walk cannot converge at a faster rate. The proof is based on an application of spectral theory to the transition density of the random walk. This yields an integral representation of the discrete probability kernel that allows us to determine the convergence rate.