A Predictive View of Continuous Time Processes

Abstract

Let $X(t), 0 \leqq t$, be an $\mathscr{L} \times \mathscr{F}$-measurable process on $(\Omega, \mathscr{F}, P)$ with state space $(E, \mathscr{E})$, where $\mathscr{L}$ is the Lebesgue $\sigma$-field and $\mathscr{E}$ is countably generated. Let $\mathscr{F}(t_1, t_2), 0 \leqq t_1 < t_2 \leqq \infty$, be the $\sigma$-field generated by $\{\int^t_{t_1}f(X(s)) ds, t_1 < t < t_2, 0 \leqq f \in \mathscr{E}\}$. A new process $Z(t)$ is constructed whose values consist of conditional probabilities in the wide sense over $\mathscr{F}(t, \infty)$ given $\mathscr{F}(0, t+)$. It is shown that $Z(t)$ is a homogeneous strong-Markov process on a compact metric space, with right-continuous paths having left limits for $t > 0. Z(t)$ determines $X(t) \mathrm{wp} 1$ except for $t$ in a Lebesgue-null set. We call $Z(t)$ the prediction process of $X(t)$. Some general properties of the construction are developed, followed by two applications.