A Lévy-type characterization of one-dimensional diffusions
The author gives a Lévy-type characterization for a one-dimensional time-homogeneous diffusion process \(X\), associated with the second-order differential operator \(L\). One introduces the scale function \(p\), such that \(Lp=0\) and the variance function \(s\), such that \(Ls=1\). These functions are defined in terms of the diffusion function and the drift function of \(X\). Then, one proves that \(X\) is a diffusion associated with \(L\) if and only if the processes \((p(X_{t}))_{t\geq 0}\) and \((s(X_{t})-t)_{t\geq 0}\) are local martingales.
scientific article; zbMATH DE number 1145761
scientific article