Continuous strong Markov processes in dimension one : a stochastic calculus approach / Sigurd Assing, Wolfgang M. Schmidt

It is given a unified stochastic calculus approach to a systematic treatment of arbitrary one-dimensional continuous homogeneous strong Markov processes. The main novelty of fundamental importance is the notion and properties of a weakly additive functional. It is proved that changing time in a strong Markov process by the right-inverse of a weakly additive functional inherits the strong Markov property. The scale function and the speed measure of an arbitrary strong Markov process are introduced and characterized. An explicit formula is derived for the finite variation part in the canonical decomposition of a continuous strong Markov semimartingale. Under slight technical condition it is shown that every continuous strong Markov process can be transformed into a continuous strong Markov semimartingale by a one-to-one transformation of the state interval, and an important occupation time formula is derived. Using this techniques a general approach is developed to the construction of arbitrary continuous strong Markov processes from the standard Brownian motion. Finally, necessary and sufficient conditions are given for a continuous strong Markov process to be a solution of a certain stochastic differential equation with generalized or ordinary drift. These conditions are expressed in the natural terms of the related scale function, speed measure and regular, right singular and left singular subsets of the state interval. (Zentralblatt)