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 rate $O(h^4)$, where $h$ is
the distance between the nodes of the tree. We also show that this convergence estimate is
optimal in that 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.

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