The method of stratification for processes with independent increments

M. A. Lifshits

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VIII, Tome 130 (1983), pp. 109-121

Résumé

Let $X(s)=\gamma(s)+W(\sigma(s))+\int_{-\infty}^\infty\int_0^s\ae\Pi(d\ae, ds)$ be a process with independent increments, where $\Pi$ is a Poisson measure, $W$ – Wiener process. The quasiinvariant transformations $$ G_cX(s)=\gamma(s)+W(\sigma(s))+\int_{-\infty}^\infty\int_0^sg(c, \ae, t)\Pi(d\ae, ds) $$ with suitable kernel $g$ form a one-parametric semigroup. Partition of probabilistic functional space into one-dimensional orbits of semigroup $G$ is considered. Conditional distributions and distributions of some functionals are calculated.