Absolute continuity of measures in the class of Markov and semi-Markov processes of diffusion type (Q5921646)

Generally, for semi-Markov processes the Markov property with respect to any fixed time is not valid. For this reason, well-known results about absolute continuity based on the classical definition of the Itô stochastic integral are not applicable in their exact form. However, a semi-Markov process has a Markov ``track''. A track is a class of trajectories differing only by monotone continuous time changes. A semi-Markov process possesses the Markov property with respect to the intrinsic Markov times analogous to the first exit time or to an iteration of these times. In this way a distribution induced by the map ``trajectory \(\rightarrow\) track'' can be generated. The author defines and uses the stochastic integral along a track in terms of the well-known curvilinear integral with respect to some additive functional. The density of a semi-Markov process with respect to another process is derived. This density is represented in the form of a product of two densities. The first density is based on the asymptotic of the distribution of the first exit point for the process exiting from an ellipsoidal neighborhood of the initial point. In terms of the associated Markov process and the induced Wiener process, this formula coincides with the known formula for the density of a diffusion-type Markov process measure. The second density is based on the semi-Markov property, which implies that the conditional distribution of the time run along a given track is the distribution of a monotone process with independent increments.