Characterization of spatial Poisson along optional increasing paths---a problem of dimension's reduction

A problem of reduction of dimension of a plane Poisson process is faced here by considering this process along optional increasing paths. A suitable reparametrization of an optional increasing path makes the process along the path a one-parameter Poisson process. A convenient characterization of the spatial Poisson process is so obtained. The key idea developed in this paper relates to the corresponding result which holds for one-parameter martingales stopped by a bounded stopping time