Limits Of One Dimensional Diffusions

TL;DR

This paper investigates the limits of one-dimensional inhomogeneous diffusions, showing under certain conditions they converge to almost-continuous diffusions, which are strong Markov processes, contrasting with multidimensional cases.

Contribution

It establishes conditions under which limits of one-dimensional diffusions are almost-continuous diffusions, extending understanding of convergence properties in stochastic processes.

Findings

Limits are almost-continuous diffusions under Lipschitz drift and continuity in probability.

Provides a simple criterion for the limit to be a continuous diffusion.

Contrasts one-dimensional and multidimensional convergence behaviors.

Abstract

In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.