Kernel Convergence Estimates for Diffusions with Continuous Coefficients

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 Hölder 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 Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.