The New Maximal Measures for Stochastic Processes

Abstract

In recent work the author proposed a reformed notion of stochastic processes, which in particular removes notorious problems with uncountable time domains. In case of a Polish state space the new stochastic processes are in one-to-one correspondence with the traditional ones. This implies for a stochastic process that the traditional canonical measure on the path space receives a certain distinguished maximal measure extension which has an immense domain. In the present paper we prove, under a certain local compactness condition on the Polish state space and for the time domain $[0,\infty [$, that the maximal domain in question has, for all stochastic processes, three distinguished members: the set of all continuous paths, the set of all paths with one-sided limits, and its subset of those paths which at each time are either left or right continuous. In all these cases the maximal measure of the set is equal to its outer canonical measure. However, the situation will be seen to be different for the set of the càdlàg paths, for example in the Poisson process.