Stratification method for processes with independent increments

Abstract

Let$X\left( s \right) = \gamma \left( s \right) + W\left( {\sigma \left( s \right)} \right) + \int\limits_{ - \infty }^\infty {\mathop \smallint \limits^s } \ae \Pi \left( {d\ae ,ds} \right)$ be a process with independent increments, Let W be a Wiener process, and let Π be a Poisson measure with independent values. The quasiinvariant transformations

$$G_c X\left( s \right) = \gamma \left( s \right) + W\left( {\sigma \left( s \right)} \right) + \int\limits_{ - \infty }^\infty {\int\limits_o^s {g\left( {c, \ae , t} \right)} } \Pi \left( {d\ae ,ds} \right),$$

under an appropriate kernelg, form a one-parameter semigroup. One considers the partitions of a probability functional space into one-dimensional orbits of the semigroup G. One computes the conditional probabilities. The results of the computations can be used for the investigation of the distributions of the functionals of the process X. A series of results of the paper can be applied to a much wider class of processes and semigroups.