Kernel Convergence Estimates for Diffusions with Continuous Coefficients

TL;DR

This paper derives sharp uniform bounds for the convergence rates of discretized kernels of one-dimensional diffusions with continuous coefficients, depending on their smoothness, using a new path conditioning technique.

Contribution

It introduces a novel path conditioning method and provides explicit convergence rate bounds for discretized diffusion kernels based on coefficient smoothness.

Findings

Fastest convergence rate is O(h^2) with bounded second derivatives.

Convergence still occurs with less smooth coefficients, but at slower rates.

The bounds apply to the kernel and its derivatives, including time and space derivatives.

Abstract

We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the convergence rate as a function of the degree of smoothness which we conjecture. The bounds also apply to the time derivative of the kernel and its first two space derivatives. Our proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. Convergence rates depend on the degree of smoothness and H\"older differentiability of the coefficients. We find that the fastest convergence rate is of order $O(h^2)$ and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of H\"older differentiability except that the convergence rate is slower. H\"older continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.