Absolute continuity of measures in the class of semi-Markov processes of diffusion type

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 5, Tome 294 (2002), pp. 216-244

Résumé

The property of absolute continuity of measures in the class of semi-Markov processes of diffusion type is investigated. The measure of such a process can be represented in the form of a composition of two measures. The first one is a distribution of a random track, and the second one is a conditional distribution of a time run along the track. The desired density (if it exists) is represented in the form of product of two corresponding densities. The first density is based on the asymptotic of the distribution density of the first exit point for the process, exiting from an ellipsoidal neighborhood of its initial point. In terms of the associated Markov process and the induced Wiener process this formula coincides with the known formula for a 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 given track is a distribution of a monotone process with independent increments.