The Annals of Probability

Limits of one-dimensional diffusions

George Lowther

Full-text: Open access


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.

Article information

Ann. Probab., Volume 37, Number 1 (2009), 78-106.

First available in Project Euclid: 17 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60G44: Martingales with continuous parameter
Secondary: 60F99: None of the above, but in this section

Diffusion martingale strong Markov finite-dimensional distributions


Lowther, George. Limits of one-dimensional diffusions. Ann. Probab. 37 (2009), no. 1, 78--106. doi:10.1214/08-AOP397.

Export citation


  • [1] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing.
  • [2] Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 193–205.
  • [3] Lowther, G. (2008). Properties of expectations of functions of martingale diffusions. Preprint. Available at arXiv:0801.0330v1.
  • [4] Meyer, P.-A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 353–372.
  • [5] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Springer, Berlin.
  • [6] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [7] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. 2. Wiley, New York.