Prediction Processes and an Autonomous Germ-Markov Property

Abstract

Let $X(t)$ be a measurable stochastic process on a countably generated space $(E, \mathscr{E})$, and let $G(t) = \cap_{\delta>o} \mathscr{F}^\circ (t, t + \delta)$ be its germ field. By transferring the probabilities to a representation space, we define and analyze the class of such processes which are Markovian relative to $G(t)$ and autonomous, in the sense that they have a stationary transition mechanism. These processes are reduced to Ray processes on an abstract space with a certain weak topology. Five kinds of examples are indicated.