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January 2009 Limits of one-dimensional diffusions
George Lowther
Ann. Probab. 37(1): 78-106 (January 2009). DOI: 10.1214/08-AOP397


In this paper, we look at the properties of limits of a sequence of real valued 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 the 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.


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George Lowther. "Limits of one-dimensional diffusions." Ann. Probab. 37 (1) 78 - 106, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1210.60087
MathSciNet: MR2489160
Digital Object Identifier: 10.1214/08-AOP397

Primary: 60G44 , 60J60
Secondary: 60F99

Keywords: diffusion , finite-dimensional distributions , martingale , strong Markov

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 1 • January 2009
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