Mathematics – Probability
Scientific paper
2007-12-14
Ann. Probab. Volume 37, Number 1 (2009), 78-106
Mathematics
Probability
32 pages. Updated to most recent version submitted to Annals of Probability
Scientific paper
10.1214/08-AOP397
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.
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