Diffusions as a limit of stretched Brownian motions
The authors show that any regular diffusion on the line can be written as a weak limit of a class of processes called stretched Brownian motions. These are diffusions whose scale functions are piecewise-linear and whose speed measures have densities which are step-functions. The canonical example of such a process is the skew Brownian motion described in the book of \textit{K. Itô} and \textit{H. P. McKean}, Diffusion processes and their sample paths. (1965; Zbl 0127.095), 2nd corrected printing 1974. The stretched Brownian motions are in turn shown to be weak limits of certain random walks.
scientific article; zbMATH DE number 3862185